Peter Koellner

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Two Futures: Pattern and Chaos

Peter Koellner

Harvard University

November 18, 2018 Montseny, Catalonia

Peter Koellner Two Futures: Pattern and Chaos . Absolute Provability . L . Absolute Deﬁnability . HOD . Large Cardinals Let us review the historical development, from Gödel to the present.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel.

Peter Koellner Two Futures: Pattern and Chaos . L . Absolute Deﬁnability . HOD . Large Cardinals Let us review the historical development, from Gödel to the present.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel. . Absolute Provability

Peter Koellner Two Futures: Pattern and Chaos . Absolute Deﬁnability . HOD . Large Cardinals Let us review the historical development, from Gödel to the present.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel. . Absolute Provability . L

Peter Koellner Two Futures: Pattern and Chaos . HOD . Large Cardinals Let us review the historical development, from Gödel to the present.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel. . Absolute Provability . L . Absolute Deﬁnability

Peter Koellner Two Futures: Pattern and Chaos . Large Cardinals Let us review the historical development, from Gödel to the present.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel. . Absolute Provability . L . Absolute Deﬁnability . HOD

Peter Koellner Two Futures: Pattern and Chaos Let us review the historical development, from Gödel to the present.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel. . Absolute Provability . L . Absolute Deﬁnability . HOD . Large Cardinals

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Themes from Gödel

The two futures I shall be discussing involve several themes from Gödel. . Absolute Provability . L . Absolute Deﬁnability . HOD . Large Cardinals Let us review the historical development, from Gödel to the present.

Peter Koellner Two Futures: Pattern and Chaos 1931 . Relative versus absolute undecidability. 1938 . V = L provides a “natural completion” of ZFC. . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Gödel’s Views

Peter Koellner Two Futures: Pattern and Chaos . Relative versus absolute undecidability. 1938 . V = L provides a “natural completion” of ZFC. . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Gödel’s Views 1931

Peter Koellner Two Futures: Pattern and Chaos 1938 . V = L provides a “natural completion” of ZFC. . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Gödel’s Views 1931 . Relative versus absolute undecidability.

Peter Koellner Two Futures: Pattern and Chaos . V = L provides a “natural completion” of ZFC. . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Gödel’s Views 1931 . Relative versus absolute undecidability. 1938

Peter Koellner Two Futures: Pattern and Chaos . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Peter Koellner Two Futures: Pattern and Chaos . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Peter Koellner Two Futures: Pattern and Chaos . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Gödel’s Views 1931 . Relative versus absolute undecidability. 1938 . V = L provides a “natural completion” of ZFC. . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”.

Peter Koellner Two Futures: Pattern and Chaos . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Peter Koellner Two Futures: Pattern and Chaos . Upshot: ZFC + V = L + LCA is complete.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures Gödel’s Views 1931 . Relative versus absolute undecidability. 1938 . V = L provides a “natural completion” of ZFC. . Read: V = L provides a “complete picture” modulo height. . V = L is “absolutely consistent”. . Implicit: Absolute provability = ZFC + LCA. . So: The complete picture is not ruled out by considerations of height. . Upshot: ZFC + V = L + LCA is complete.

Peter Koellner Two Futures: Pattern and Chaos . V = L is “absolutely undecidable” and “set theory bifurcates”. . Upshot: ZFC + V = L + LCA is complete but not correct. 1946 . Proposal 1: Absolute provability = ZFC + LCA. . Proposal 2: Absolute deﬁnability = HOD. . Entertained possibility: ZFC + LCA is complete and correct. 1947 . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

1939

Peter Koellner Two Futures: Pattern and Chaos . Upshot: ZFC + V = L + LCA is complete but not correct. 1946 . Proposal 1: Absolute provability = ZFC + LCA. . Proposal 2: Absolute deﬁnability = HOD. . Entertained possibility: ZFC + LCA is complete and correct. 1947 . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

1939 . V = L is “absolutely undecidable” and “set theory bifurcates”.

Peter Koellner Two Futures: Pattern and Chaos . Proposal 1: Absolute provability = ZFC + LCA. . Proposal 2: Absolute deﬁnability = HOD. . Entertained possibility: ZFC + LCA is complete and correct. 1947 . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

1939 . V = L is “absolutely undecidable” and “set theory bifurcates”. . Upshot: ZFC + V = L + LCA is complete but not correct. 1946

Peter Koellner Two Futures: Pattern and Chaos . Proposal 2: Absolute deﬁnability = HOD. . Entertained possibility: ZFC + LCA is complete and correct. 1947 . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

1939 . V = L is “absolutely undecidable” and “set theory bifurcates”. . Upshot: ZFC + V = L + LCA is complete but not correct. 1946 . Proposal 1: Absolute provability = ZFC + LCA.

Peter Koellner Two Futures: Pattern and Chaos . Entertained possibility: ZFC + LCA is complete and correct. 1947 . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos 1947 . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

1939 . V = L is “absolutely undecidable” and “set theory bifurcates”. . Upshot: ZFC + V = L + LCA is complete but not correct. 1946 . Proposal 1: Absolute provability = ZFC + LCA. . Proposal 2: Absolute deﬁnability = HOD. . Entertained possibility: ZFC + LCA is complete and correct.

Peter Koellner Two Futures: Pattern and Chaos . Emphasis on extrinsic justiﬁcations. . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos . Argued there was evidence that V 6= L and ¬CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos . Scott: Measurable cardinals imply V 6= L . Gödel described this as a “beautiful result” and added “but the result does not surprise me.” 1963 . Cohen: ZFC cannot settle CH. 1967 . Levy-Solovay: Measurable cardinals cannot settle CH. . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961

Peter Koellner Two Futures: Pattern and Chaos . Gödel described this as a “beautiful result” and added “but the result does not surprise me.” 1963 . Cohen: ZFC cannot settle CH. 1967 . Levy-Solovay: Measurable cardinals cannot settle CH. . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961 . Scott: Measurable cardinals imply V 6= L

Peter Koellner Two Futures: Pattern and Chaos 1963 . Cohen: ZFC cannot settle CH. 1967 . Levy-Solovay: Measurable cardinals cannot settle CH. . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961 . Scott: Measurable cardinals imply V 6= L . Gödel described this as a “beautiful result” and added “but the result does not surprise me.”

Peter Koellner Two Futures: Pattern and Chaos . Cohen: ZFC cannot settle CH. 1967 . Levy-Solovay: Measurable cardinals cannot settle CH. . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961 . Scott: Measurable cardinals imply V 6= L . Gödel described this as a “beautiful result” and added “but the result does not surprise me.” 1963

Peter Koellner Two Futures: Pattern and Chaos 1967 . Levy-Solovay: Measurable cardinals cannot settle CH. . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961 . Scott: Measurable cardinals imply V 6= L . Gödel described this as a “beautiful result” and added “but the result does not surprise me.” 1963 . Cohen: ZFC cannot settle CH.

Peter Koellner Two Futures: Pattern and Chaos . Levy-Solovay: Measurable cardinals cannot settle CH. . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961 . Scott: Measurable cardinals imply V 6= L . Gödel described this as a “beautiful result” and added “but the result does not surprise me.” 1963 . Cohen: ZFC cannot settle CH. 1967

Peter Koellner Two Futures: Pattern and Chaos . Subsequent generalizations: LCA cannot settle CH.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

1961 . Scott: Measurable cardinals imply V 6= L . Gödel described this as a “beautiful result” and added “but the result does not surprise me.” 1963 . Cohen: ZFC cannot settle CH. 1967 . Levy-Solovay: Measurable cardinals cannot settle CH.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures The 1960s

Peter Koellner Two Futures: Pattern and Chaos . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Questions (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary

Peter Koellner Two Futures: Pattern and Chaos . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Questions (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete.

Peter Koellner Two Futures: Pattern and Chaos . ZFC + LCA cannot settle CH.

Questions (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L.

Peter Koellner Two Futures: Pattern and Chaos Questions (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Peter Koellner Two Futures: Pattern and Chaos (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Questions

Peter Koellner Two Futures: Pattern and Chaos (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Questions (1) How complete is ZFC + LCA?

Peter Koellner Two Futures: Pattern and Chaos (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Questions (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH?

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Summary . ZFC + LCA is not complete. . ZFC + LCA implies V 6= L. . ZFC + LCA cannot settle CH.

Questions (1) How complete is ZFC + LCA? (2) How can we go beyond ZFC + LCA and settle CH? (3) Is there a correct axiom A such that ZFC + A + LCA is complete?

Peter Koellner Two Futures: Pattern and Chaos . ZFC + LCA settles every statement of “complexity strictly below” that of CH. . There is a strong extrinsic case for ADL(R) (and its extensions). . The inner model L(R) is similar to L: . It is a paradise of understanding. . It is a partial paradise—it isn’t Cantor’s paradise.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (1): How complete is ZFC + LCA?

Peter Koellner Two Futures: Pattern and Chaos . There is a strong extrinsic case for ADL(R) (and its extensions). . The inner model L(R) is similar to L: . It is a paradise of understanding. . It is a partial paradise—it isn’t Cantor’s paradise.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (1): How complete is ZFC + LCA?

. ZFC + LCA settles every statement of “complexity strictly below” that of CH.

Peter Koellner Two Futures: Pattern and Chaos . The inner model L(R) is similar to L: . It is a paradise of understanding. . It is a partial paradise—it isn’t Cantor’s paradise.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (1): How complete is ZFC + LCA?

. ZFC + LCA settles every statement of “complexity strictly below” that of CH. . There is a strong extrinsic case for ADL(R) (and its extensions).

Peter Koellner Two Futures: Pattern and Chaos . It is a paradise of understanding. . It is a partial paradise—it isn’t Cantor’s paradise.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (1): How complete is ZFC + LCA?

. ZFC + LCA settles every statement of “complexity strictly below” that of CH. . There is a strong extrinsic case for ADL(R) (and its extensions). . The inner model L(R) is similar to L:

Peter Koellner Two Futures: Pattern and Chaos . It is a partial paradise—it isn’t Cantor’s paradise.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (1): How complete is ZFC + LCA?

. ZFC + LCA settles every statement of “complexity strictly below” that of CH. . There is a strong extrinsic case for ADL(R) (and its extensions). . The inner model L(R) is similar to L: . It is a paradise of understanding.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (1): How complete is ZFC + LCA?

Peter Koellner Two Futures: Pattern and Chaos . In the last 40 years there have been sophisticated arguments for forcing axioms like MM and (∗), each of which implies ¬CH. . In hindsight these arguments involve a curious oversight: the envelope perspective on forcing axioms. . Forcing axioms provide a paradise of understanding. . But it could very well be a partial paradise (the envelope) of the intended paradise.

Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

Peter Koellner Two Futures: Pattern and Chaos . In hindsight these arguments involve a curious oversight: the envelope perspective on forcing axioms. . Forcing axioms provide a paradise of understanding. . But it could very well be a partial paradise (the envelope) of the intended paradise.

Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

. In the last 40 years there have been sophisticated arguments for forcing axioms like MM and (∗), each of which implies ¬CH.

Peter Koellner Two Futures: Pattern and Chaos the envelope perspective on forcing axioms. . Forcing axioms provide a paradise of understanding. . But it could very well be a partial paradise (the envelope) of the intended paradise.

Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

. In the last 40 years there have been sophisticated arguments for forcing axioms like MM and (∗), each of which implies ¬CH. . In hindsight these arguments involve a curious oversight:

Peter Koellner Two Futures: Pattern and Chaos . Forcing axioms provide a paradise of understanding. . But it could very well be a partial paradise (the envelope) of the intended paradise.

Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

Peter Koellner Two Futures: Pattern and Chaos . But it could very well be a partial paradise (the envelope) of the intended paradise.

Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

. In the last 40 years there have been sophisticated arguments for forcing axioms like MM and (∗), each of which implies ¬CH. . In hindsight these arguments involve a curious oversight: the envelope perspective on forcing axioms. . Forcing axioms provide a paradise of understanding.

Peter Koellner Two Futures: Pattern and Chaos Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

Peter Koellner Two Futures: Pattern and Chaos Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

Perhaps it is time to standback and survey the possibilities.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Large Cardinals Beyond Choice Two Futures

Regarding (2): How can we go beyond ZFC + LCA and settle CH?

Perhaps it is time to standback and survey the possibilities. Perhaps ZFC + LCA can at least narrow the possibilities, and perhaps the partial paradises can serve as our guide.

Peter Koellner Two Futures: Pattern and Chaos . L is the most slender of inner models, while, in some sense, HOD is the broadest. . L is deﬁned locally, while HOD is deﬁned globally. . L cannot accomodate modest large cardinals, while HOD can accomodate all traditional large cardinals. . There are simple sets (such as real numbers like 0#) which are not set-generic over L, while every set is set-generic over HOD.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures The HOD Dichotomy

The inner models L and HOD are, in many respects, at opposite ends of the inner model spectrum.

Peter Koellner Two Futures: Pattern and Chaos . L is deﬁned locally, while HOD is deﬁned globally. . L cannot accomodate modest large cardinals, while HOD can accomodate all traditional large cardinals. . There are simple sets (such as real numbers like 0#) which are not set-generic over L, while every set is set-generic over HOD.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures The HOD Dichotomy

The inner models L and HOD are, in many respects, at opposite ends of the inner model spectrum.

. L is the most slender of inner models, while, in some sense, HOD is the broadest.

Peter Koellner Two Futures: Pattern and Chaos . L cannot accomodate modest large cardinals, while HOD can accomodate all traditional large cardinals. . There are simple sets (such as real numbers like 0#) which are not set-generic over L, while every set is set-generic over HOD.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures The HOD Dichotomy

The inner models L and HOD are, in many respects, at opposite ends of the inner model spectrum.

. L is the most slender of inner models, while, in some sense, HOD is the broadest. . L is deﬁned locally, while HOD is deﬁned globally.

Peter Koellner Two Futures: Pattern and Chaos . There are simple sets (such as real numbers like 0#) which are not set-generic over L, while every set is set-generic over HOD.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures The HOD Dichotomy

The inner models L and HOD are, in many respects, at opposite ends of the inner model spectrum.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures The HOD Dichotomy

The inner models L and HOD are, in many respects, at opposite ends of the inner model spectrum.

. L is the most slender of inner models, while, in some sense, HOD is the broadest. . L is deﬁned locally, while HOD is deﬁned globally. . L cannot accomodate modest large cardinals, while HOD can accomodate all traditional large cardinals. . There are simple sets (such as real numbers like 0#) which are not set-generic over L, while every set is set-generic over HOD.

Peter Koellner Two Futures: Pattern and Chaos Theorem (Jensen) Exactly one of the following hold. (1) For every singular cardinal γ, γ is singular in L and (γ+)L = γ+. (2) Every uncountable cardinal is an inaccessible cardinal in L

. In the ﬁrst alternative L is “close” to V and in the second alternative L is “far” from V .

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Jensen proved the following remarkable theorem—the L Dichotomy Theorem.

Peter Koellner Two Futures: Pattern and Chaos . In the ﬁrst alternative L is “close” to V and in the second alternative L is “far” from V .

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Jensen proved the following remarkable theorem—the L Dichotomy Theorem. Theorem (Jensen) Exactly one of the following hold. (1) For every singular cardinal γ, γ is singular in L and (γ+)L = γ+. (2) Every uncountable cardinal is an inaccessible cardinal in L

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

. In the ﬁrst alternative L is “close” to V and in the second alternative L is “far” from V .

Peter Koellner Two Futures: Pattern and Chaos Theorem (Woodin) Suppose that κ is an extendible cardinal. Then exactly one of the following hold. (1) For every singular cardinal γ > κ, γ is singular in HOD and (γ+)HOD = γ+. (2) Every regular cardinal γ ≥ κ is a measurable cardinal in HOD.

. In the ﬁrst alternative HOD is “close” to V and in the second alternative HOD is “far” from V .

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Surprisingly, Woodin proved a similar dichotomy theorem for HOD—the HOD Dichotomy Theorem, a weak version of which is the following:

Peter Koellner Two Futures: Pattern and Chaos . In the ﬁrst alternative HOD is “close” to V and in the second alternative HOD is “far” from V .

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Surprisingly, Woodin proved a similar dichotomy theorem for HOD—the HOD Dichotomy Theorem, a weak version of which is the following: Theorem (Woodin) Suppose that κ is an extendible cardinal. Then exactly one of the following hold. (1) For every singular cardinal γ > κ, γ is singular in HOD and (γ+)HOD = γ+. (2) Every regular cardinal γ ≥ κ is a measurable cardinal in HOD.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

. In the ﬁrst alternative HOD is “close” to V and in the second alternative HOD is “far” from V .

Peter Koellner Two Futures: Pattern and Chaos . In the case of the L Dichotomy, ZFC + LCA tells which side we are on. . In particular, if 0# exists, then we are on the “far” side. . In the case of the HOD Dichotomy, ZFC + LCA cannot force us into the “far” side. . So perhaps the “close” side holds. . Or perhaps there is a new stretch of the large cardinal hierarchy, which includes a higher analogue of 0# that forces us into the “far” side.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

There is an important foundational diﬀerence between these two dichotomy theorems.

Peter Koellner Two Futures: Pattern and Chaos . In particular, if 0# exists, then we are on the “far” side. . In the case of the HOD Dichotomy, ZFC + LCA cannot force us into the “far” side. . So perhaps the “close” side holds. . Or perhaps there is a new stretch of the large cardinal hierarchy, which includes a higher analogue of 0# that forces us into the “far” side.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

There is an important foundational diﬀerence between these two dichotomy theorems. . In the case of the L Dichotomy, ZFC + LCA tells which side we are on.

Peter Koellner Two Futures: Pattern and Chaos . In the case of the HOD Dichotomy, ZFC + LCA cannot force us into the “far” side. . So perhaps the “close” side holds. . Or perhaps there is a new stretch of the large cardinal hierarchy, which includes a higher analogue of 0# that forces us into the “far” side.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Peter Koellner Two Futures: Pattern and Chaos . So perhaps the “close” side holds. . Or perhaps there is a new stretch of the large cardinal hierarchy, which includes a higher analogue of 0# that forces us into the “far” side.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Peter Koellner Two Futures: Pattern and Chaos . Or perhaps there is a new stretch of the large cardinal hierarchy, which includes a higher analogue of 0# that forces us into the “far” side.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

There is an important foundational diﬀerence between these two dichotomy theorems. . In the case of the L Dichotomy, ZFC + LCA tells which side we are on. . In particular, if 0# exists, then we are on the “far” side. . In the case of the HOD Dichotomy, ZFC + LCA cannot force us into the “far” side. . So perhaps the “close” side holds.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

Peter Koellner Two Futures: Pattern and Chaos There is a new program that aims at establishing the second future—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice.

Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

There is a program that aims at establishing the ﬁrst future—the “close” side of the HOD Dichotomy. This is the program of inner model theory.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy L versus HOD Ultimate-L L Dichotomy Theorem Large Cardinals Beyond Choice HOD Dichotomy Theorem Two Futures

There is a program that aims at establishing the ﬁrst future—the “close” side of the HOD Dichotomy. This is the program of inner model theory.

There is a new program that aims at establishing the second future—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice.

Peter Koellner Two Futures: Pattern and Chaos Let us start with the oﬃcial version of the HOD Dichotomy.

Deﬁnition (Woodin) Let γ be an uncountable regular cardinal. Then γ is ω-strongly measurable in HOD if there exists κ < γ such that (1)( 2κ)HOD < γ and γ (2) There is no partition hSα : α < κi of Sω into stationary sets such that hSα : α < κi ∈ HOD.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures Ultimate-L

Peter Koellner Two Futures: Pattern and Chaos Deﬁnition (Woodin) Let γ be an uncountable regular cardinal. Then γ is ω-strongly measurable in HOD if there exists κ < γ such that (1)( 2κ)HOD < γ and γ (2) There is no partition hSα : α < κi of Sω into stationary sets such that hSα : α < κi ∈ HOD.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures Ultimate-L

Let us start with the oﬃcial version of the HOD Dichotomy.

Lemma (Woodin) Assume that γ is ω-strongly measurable in HOD. Then

HOD |= γ is a measurable cardinal.

Peter Koellner Two Futures: Pattern and Chaos . In the ﬁrst alternative HOD is “close” to V and in the second alternative HOD is “far” from V .

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem ((Woodin) The HOD Dichotomy Theorem) Suppose that κ is an extendible cardinal. Then exactly one of the following hold. (1) For every singular cardinal γ > κ, γ is singular in HOD and (γ+)HOD = γ+. (2) Every regular cardinal γ ≥ κ is ω-strongly measurable in HOD.

. In the ﬁrst alternative HOD is “close” to V and in the second alternative HOD is “far” from V .

Peter Koellner Two Futures: Pattern and Chaos Deﬁnition ((Woodin) The Weak HOD Conjecture) The Weak HOD Conjecture is the conjecture that

ZFC + “There is an extendible with a huge cardinal above it”

proves the HOD Hypothesis.

0 . Key: This is a Σ1-sentence. So it will not run up against the rock of undecidability.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Deﬁnition ((Woodin) The HOD Hypothesis) There exists a proper class of regular cardinals γ which are not ω-strongly measurable in HOD.

Peter Koellner Two Futures: Pattern and Chaos 0 . Key: This is a Σ1-sentence. So it will not run up against the rock of undecidability.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Deﬁnition ((Woodin) The HOD Hypothesis) There exists a proper class of regular cardinals γ which are not ω-strongly measurable in HOD.

Deﬁnition ((Woodin) The Weak HOD Conjecture) The Weak HOD Conjecture is the conjecture that

ZFC + “There is an extendible with a huge cardinal above it” proves the HOD Hypothesis.

Deﬁnition ((Woodin) The HOD Hypothesis) There exists a proper class of regular cardinals γ which are not ω-strongly measurable in HOD.

Deﬁnition ((Woodin) The Weak HOD Conjecture) The Weak HOD Conjecture is the conjecture that

ZFC + “There is an extendible with a huge cardinal above it” proves the HOD Hypothesis.

0 . Key: This is a Σ1-sentence. So it will not run up against the rock of undecidability.

Peter Koellner Two Futures: Pattern and Chaos . It is really quite a surprising conjecture to make. In fact, when I was a graduate student it was known by a diﬀerent name.It was known as “the silly conjecture.”

. But it is not so silly now. The reason has to do with recent developments in inner model theory.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. Notice: If the Weak HOD Conjecture is true then (assuming that there is an extendible cardinal with a huge cardinal above it) we must be in the “close” side of the HOD Dichotomy.

Peter Koellner Two Futures: Pattern and Chaos It was known as “the silly conjecture.”

. But it is not so silly now. The reason has to do with recent developments in inner model theory.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. Notice: If the Weak HOD Conjecture is true then (assuming that there is an extendible cardinal with a huge cardinal above it) we must be in the “close” side of the HOD Dichotomy.

. It is really quite a surprising conjecture to make. In fact, when I was a graduate student it was known by a diﬀerent name.

Peter Koellner Two Futures: Pattern and Chaos . But it is not so silly now. The reason has to do with recent developments in inner model theory.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. Notice: If the Weak HOD Conjecture is true then (assuming that there is an extendible cardinal with a huge cardinal above it) we must be in the “close” side of the HOD Dichotomy.

. It is really quite a surprising conjecture to make. In fact, when I was a graduate student it was known by a diﬀerent name.It was known as “the silly conjecture.”

. Notice: If the Weak HOD Conjecture is true then (assuming that there is an extendible cardinal with a huge cardinal above it) we must be in the “close” side of the HOD Dichotomy.

. It is really quite a surprising conjecture to make. In fact, when I was a graduate student it was known by a diﬀerent name.It was known as “the silly conjecture.”

. But it is not so silly now. The reason has to do with recent developments in inner model theory.

Peter Koellner Two Futures: Pattern and Chaos . This is what one would expect of an inner model targeting a supercompact.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures Weak Extender Models

Deﬁnition A transtive class N |= ZFC is a weak extender model for the supercompactness of κ if for every λ > κ there exists a λ-complete normal ﬁne measure U on Pκ(λ) such that

(1) N ∩ Pκ(λ) ∈ U and (2) U ∩ N ∈ N .

Deﬁnition A transtive class N |= ZFC is a weak extender model for the supercompactness of κ if for every λ > κ there exists a λ-complete normal ﬁne measure U on Pκ(λ) such that

(1) N ∩ Pκ(λ) ∈ U and (2) U ∩ N ∈ N .

. This is what one would expect of an inner model targeting a supercompact.

Peter Koellner Two Futures: Pattern and Chaos . In general, when one constructs an inner model targeting a given large cardinal, the existence of stronger large cardinals implies that the model is “far” from V . . One constructs an L-like paradise of understanding for the large cardinal, only to have it revealed as illusory—that is, “far” from V —by larger large cardinals. . But at the level of a supercompact something amazing happens ...

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Weak extender models for the supercompactness of κ have features that are quite remarkable:

Peter Koellner Two Futures: Pattern and Chaos . One constructs an L-like paradise of understanding for the large cardinal, only to have it revealed as illusory—that is, “far” from V —by larger large cardinals. . But at the level of a supercompact something amazing happens ...

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Peter Koellner Two Futures: Pattern and Chaos . But at the level of a supercompact something amazing happens ...

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Weak extender models for the supercompactness of κ have features that are quite remarkable: . In general, when one constructs an inner model targeting a given large cardinal, the existence of stronger large cardinals implies that the model is “far” from V . . One constructs an L-like paradise of understanding for the large cardinal, only to have it revealed as illusory—that is, “far” from V —by larger large cardinals.

Peter Koellner Two Futures: Pattern and Chaos . In other words, N is “close” to V regardless of which large cardinals live in V . . So if there is an L-like paradise of understanding at this level, then no large cardinal can reveal it to be illusory.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ. Then for every singular cardinal γ > κ, γ is singular in N and (γ+)N = γ+.

Peter Koellner Two Futures: Pattern and Chaos regardless of which large cardinals live in V . . So if there is an L-like paradise of understanding at this level, then no large cardinal can reveal it to be illusory.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ. Then for every singular cardinal γ > κ, γ is singular in N and (γ+)N = γ+.

. In other words, N is “close” to V

Peter Koellner Two Futures: Pattern and Chaos . So if there is an L-like paradise of understanding at this level, then no large cardinal can reveal it to be illusory.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ. Then for every singular cardinal γ > κ, γ is singular in N and (γ+)N = γ+.

. In other words, N is “close” to V regardless of which large cardinals live in V .

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ. Then for every singular cardinal γ > κ, γ is singular in N and (γ+)N = γ+.

. In other words, N is “close” to V regardless of which large cardinals live in V . . So if there is an L-like paradise of understanding at this level, then no large cardinal can reveal it to be illusory.

Peter Koellner Two Futures: Pattern and Chaos Theorem ((Woodin) Universality) Suppose that N is a weak extender model for the supercompactness of κ. Suppose that α > κ is an ordinal and

j : N ∩ Vα+1 N ∩ Vj (α)+1

is an elementary embedding such→ that κ ≤ crit(j ). Then j ∈ N.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

It turns out that such an N actually absorbs all of the large cardinal structure of V . The key is the following:

Theorem ((Woodin) Universality) Suppose that N is a weak extender model for the supercompactness of κ. Suppose that α > κ is an ordinal and

j : N ∩ Vα+1 N ∩ Vj (α)+1 is an elementary embedding such→ that κ ≤ crit(j ). Then j ∈ N.

Peter Koellner Two Futures: Pattern and Chaos Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is supercompact. Then N is a weak extender model for the supercompactness of δ.

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is an extendible cardinal. Then δ is an extendible cardinal in N .

And so on, up through the traditional large cardinal hierarchy ...

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

This has a series of corollaries:

Peter Koellner Two Futures: Pattern and Chaos Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is an extendible cardinal. Then δ is an extendible cardinal in N .

And so on, up through the traditional large cardinal hierarchy ...

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

This has a series of corollaries:

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is supercompact. Then N is a weak extender model for the supercompactness of δ.

Peter Koellner Two Futures: Pattern and Chaos And so on, up through the traditional large cardinal hierarchy ...

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

This has a series of corollaries:

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is supercompact. Then N is a weak extender model for the supercompactness of δ.

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is an extendible cardinal. Then δ is an extendible cardinal in N .

This has a series of corollaries:

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is supercompact. Then N is a weak extender model for the supercompactness of δ.

Theorem (Woodin) Suppose that N is a weak extender model for the supercompactness of κ and that δ > κ is an extendible cardinal. Then δ is an extendible cardinal in N .

And so on, up through the traditional large cardinal hierarchy ...

Peter Koellner Two Futures: Pattern and Chaos . So the Weak HOD Conjecture is really just the conjecture that inner model theory succeeds in this sense. Key Question: Is there an L-like candidate for a weak extender model for the supercompactness of κ?

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures The relevance of weak extender models to the HOD Dichotomy is the following:

Theorem (Woodin) Suppose that κ is an extendible cardinal. Then the following are equivalent. (1) The HOD Hypothesis holds. (2) There exists N ⊆ HOD which is a weak extender model for the supercompactness of κ.

Peter Koellner Two Futures: Pattern and Chaos Key Question: Is there an L-like candidate for a weak extender model for the supercompactness of κ?

. So the Weak HOD Conjecture is really just the conjecture that inner model theory succeeds in this sense.

. So the Weak HOD Conjecture is really just the conjecture that inner model theory succeeds in this sense. Key Question: Is there an L-like candidate for a weak extender model for the supercompactness of κ?

Peter Koellner Two Futures: Pattern and Chaos Fortunately, one can; curiously, it involves HOD, as computed in the paradises of understanding provided by determinacy. Deﬁnition ((Woodin)V = Ultimate-L)

(1) There is a proper class of Woodin cardinals.

(2) For each Σ2-sentence ϕ, if ϕ holds in V , then there exists a universally Baire set A ⊆ R such that

HODL(A,R) |= ϕ.

Peter Koellner Two Futures: Pattern and Chaos Deﬁnition ((Woodin)V = Ultimate-L)

(1) There is a proper class of Woodin cardinals.

(2) For each Σ2-sentence ϕ, if ϕ holds in V , then there exists a universally Baire set A ⊆ R such that

HODL(A,R) |= ϕ.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures The Weak Ultimate-L Conjecture Issue: How can one even state the axiom “V = Ultimate-L” prior to the construction? Fortunately, one can; curiously, it involves HOD, as computed in the paradises of understanding provided by determinacy.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures The Weak Ultimate-L Conjecture Issue: How can one even state the axiom “V = Ultimate-L” prior to the construction? Fortunately, one can; curiously, it involves HOD, as computed in the paradises of understanding provided by determinacy. Deﬁnition ((Woodin)V = Ultimate-L)

(1) There is a proper class of Woodin cardinals.

(2) For each Σ2-sentence ϕ, if ϕ holds in V , then there exists a universally Baire set A ⊆ R such that

HODL(A,R) |= ϕ.

Peter Koellner Two Futures: Pattern and Chaos Deﬁnition (Hamkins) A transitive class N is a ground of V if (1) N |= ZFC and (2) there exists a partial order P ∈ N and an N -generic G ⊆ P such that V = N [G].

Deﬁnition (Hamkins) The mantle of V is the intersection of the grounds of V .

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

The consequences are striking. But ﬁrst we need a few deﬁnitions from set-theoretic geology:

Peter Koellner Two Futures: Pattern and Chaos Deﬁnition (Hamkins) The mantle of V is the intersection of the grounds of V .

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

The consequences are striking. But ﬁrst we need a few deﬁnitions from set-theoretic geology:

Deﬁnition (Hamkins) A transitive class N is a ground of V if (1) N |= ZFC and (2) there exists a partial order P ∈ N and an N -generic G ⊆ P such that V = N [G].

The consequences are striking. But ﬁrst we need a few deﬁnitions from set-theoretic geology:

Deﬁnition (Hamkins) A transitive class N is a ground of V if (1) N |= ZFC and (2) there exists a partial order P ∈ N and an N -generic G ⊆ P such that V = N [G].

Deﬁnition (Hamkins) The mantle of V is the intersection of the grounds of V .

Peter Koellner Two Futures: Pattern and Chaos Then (1) CH. (2) V = HOD. (3) The Ω-Conjecture. (4) V has no non-trivial grounds (GA). (5) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Assume V = Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos (2) V = HOD. (3) The Ω-Conjecture. (4) V has no non-trivial grounds (GA). (5) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Assume V = Ultimate-L. Then (1) CH.

Peter Koellner Two Futures: Pattern and Chaos (3) The Ω-Conjecture. (4) V has no non-trivial grounds (GA). (5) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Assume V = Ultimate-L. Then (1) CH. (2) V = HOD.

Peter Koellner Two Futures: Pattern and Chaos (4) V has no non-trivial grounds (GA). (5) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Assume V = Ultimate-L. Then (1) CH. (2) V = HOD. (3) The Ω-Conjecture.

Peter Koellner Two Futures: Pattern and Chaos (5) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Theorem (Woodin) Assume V = Ultimate-L. Then (1) CH. (2) V = HOD. (3) The Ω-Conjecture. (4) V has no non-trivial grounds (GA).

Theorem (Woodin) Assume V = Ultimate-L. Then (1) CH. (2) V = HOD. (3) The Ω-Conjecture. (4) V has no non-trivial grounds (GA). (5) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Peter Koellner Two Futures: Pattern and Chaos 0 . Key: This is a Σ1-sentence. So it will not run up against the rock of undecidability.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

Deﬁnition ((Woodin) Weak Ultimate-L Conjecture) The Weak Ultimate-L Conjecture is the conjecture that

ZFC + “There is an extendible κ with a huge cardinal above it” proves: Then there exists a weak extender model N for the supercompactness of κ such that

(1) N is weakly Σ2-deﬁnable and N ⊆ HOD. (2) N |= “V = Ultimate-L”.

Deﬁnition ((Woodin) Weak Ultimate-L Conjecture) The Weak Ultimate-L Conjecture is the conjecture that

ZFC + “There is an extendible κ with a huge cardinal above it” proves: Then there exists a weak extender model N for the supercompactness of κ such that

(1) N is weakly Σ2-deﬁnable and N ⊆ HOD. (2) N |= “V = Ultimate-L”.

0 . Key: This is a Σ1-sentence. So it will not run up against the rock of undecidability.

Peter Koellner Two Futures: Pattern and Chaos . The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

0 . So, in this future (where the Σ1-sentence holds): . ZFC + LCA will imply that we are on the “close” side of the HOD Dichotomy. . We will have reached an L-like paradise of understanding that no large cardinal can transcend. . ZFC + V = Ultimate-L + LCA will be complete (but not necessarily correct).

This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

Peter Koellner Two Futures: Pattern and Chaos 0 . So, in this future (where the Σ1-sentence holds): . ZFC + LCA will imply that we are on the “close” side of the HOD Dichotomy. . We will have reached an L-like paradise of understanding that no large cardinal can transcend. . ZFC + V = Ultimate-L + LCA will be complete (but not necessarily correct).

This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

Peter Koellner Two Futures: Pattern and Chaos . ZFC + LCA will imply that we are on the “close” side of the HOD Dichotomy. . We will have reached an L-like paradise of understanding that no large cardinal can transcend. . ZFC + V = Ultimate-L + LCA will be complete (but not necessarily correct).

This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

0 . So, in this future (where the Σ1-sentence holds):

Peter Koellner Two Futures: Pattern and Chaos . We will have reached an L-like paradise of understanding that no large cardinal can transcend. . ZFC + V = Ultimate-L + LCA will be complete (but not necessarily correct).

This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

0 . So, in this future (where the Σ1-sentence holds): . ZFC + LCA will imply that we are on the “close” side of the HOD Dichotomy.

Peter Koellner Two Futures: Pattern and Chaos . ZFC + V = Ultimate-L + LCA will be complete (but not necessarily correct).

This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

Peter Koellner Two Futures: Pattern and Chaos (but not necessarily correct).

This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

Peter Koellner Two Futures: Pattern and Chaos This is the future where pattern prevails.

Themes from Gödel The HOD Dichotomy The HOD Dichotomy Ultimate-L Weak Extender Models Large Cardinals Beyond Choice The Weak Ultimate-L Conjecture Two Futures

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

. A great deal of work has been done toward proving the Weak Ultimate-L Conjecture.

. The Weak Ultimate-L Conjecture implies the Weak HOD Conjecture.

This is the future where pattern prevails.

Peter Koellner Two Futures: Pattern and Chaos The traditional large cardinals divide into the small and the large.

I will now describe the work that was jointly done with Bagaria and Woodin.

The traditional large cardinals divide into the small and the large.

Peter Koellner Two Futures: Pattern and Chaos . Inaccessible . Mahlo . Weakly Compact . Indescribable . Subtle . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals:

Peter Koellner Two Futures: Pattern and Chaos . Mahlo . Weakly Compact . Indescribable . Subtle . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals: . Inaccessible

Peter Koellner Two Futures: Pattern and Chaos . Weakly Compact . Indescribable . Subtle . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals: . Inaccessible . Mahlo

Peter Koellner Two Futures: Pattern and Chaos . Indescribable . Subtle . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals: . Inaccessible . Mahlo . Weakly Compact

Peter Koellner Two Futures: Pattern and Chaos . Subtle . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals: . Inaccessible . Mahlo . Weakly Compact . Indescribable

Peter Koellner Two Futures: Pattern and Chaos . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals: . Inaccessible . Mahlo . Weakly Compact . Indescribable . Subtle

Peter Koellner Two Futures: Pattern and Chaos Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Some small large cardinals: . Inaccessible . Mahlo . Weakly Compact . Indescribable . Subtle . Ineﬀable

Template #1: Reﬂection Principles:

α α V |= ϕ(A) ∃α Vα |= ϕ (A )

→

Peter Koellner Two Futures: Pattern and Chaos . In other words, measurable cardinals are large.

Theorem (Scott) Measurable cardinals imply V 6= L.

. In other words, measurable cardinals are large.

Peter Koellner Two Futures: Pattern and Chaos . Measurable . Strong . Supercompact . Huge . Rank to Rank

. I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals:

Peter Koellner Two Futures: Pattern and Chaos . Strong . Supercompact . Huge . Rank to Rank

. I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals: . Measurable

Peter Koellner Two Futures: Pattern and Chaos . Supercompact . Huge . Rank to Rank

. I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals: . Measurable . Strong

Peter Koellner Two Futures: Pattern and Chaos . Huge . Rank to Rank

. I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals: . Measurable . Strong . Supercompact

Peter Koellner Two Futures: Pattern and Chaos . Rank to Rank

. I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals: . Measurable . Strong . Supercompact . Huge

Peter Koellner Two Futures: Pattern and Chaos . I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals: . Measurable . Strong . Supercompact . Huge . Rank to Rank

Peter Koellner Two Futures: Pattern and Chaos Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M

where M is a transitive class. →

Some large large cardinals: . Measurable . Strong . Supercompact . Huge . Rank to Rank

. I0

Some large large cardinals: . Measurable . Strong . Supercompact . Huge . Rank to Rank

. I0

Template #2: Elementary Embeddings:There exists a non-trivial elementary embedding

j : V M where M is a transitive class. → Peter Koellner Two Futures: Pattern and Chaos In the limit, the ultimate large cardinal axiom would involve the ultimate degree of resemblance, where M = V . Reinhardt proposed this axiom in his dissertation.

Strength is obtained by demanding that M resemble V more and more.

In the limit, the ultimate large cardinal axiom would involve the ultimate degree of resemblance, where M = V . Reinhardt proposed this axiom in his dissertation.

Peter Koellner Two Futures: Pattern and Chaos . In other words, Reinhardt cardinals are very large.

Theorem (Kunen) Reinhardt cardinals imply ¬AC.

. In other words, Reinhardt cardinals are very large.

Peter Koellner Two Futures: Pattern and Chaos . Reinhardt . Super Reinhardt . Totally Reinhardt . Berkeley . Club Berkeley . Limit Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

Peter Koellner Two Futures: Pattern and Chaos . Super Reinhardt . Totally Reinhardt . Berkeley . Club Berkeley . Limit Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

. Reinhardt

Peter Koellner Two Futures: Pattern and Chaos . Totally Reinhardt . Berkeley . Club Berkeley . Limit Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

. Reinhardt . Super Reinhardt

Peter Koellner Two Futures: Pattern and Chaos . Berkeley . Club Berkeley . Limit Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

. Reinhardt . Super Reinhardt . Totally Reinhardt

Peter Koellner Two Futures: Pattern and Chaos . Club Berkeley . Limit Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

. Reinhardt . Super Reinhardt . Totally Reinhardt . Berkeley

Peter Koellner Two Futures: Pattern and Chaos . Limit Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

. Reinhardt . Super Reinhardt . Totally Reinhardt . Berkeley . Club Berkeley

The large cardinals I will be discussing today are very large. These are the large cardinals beyond choice.

. Reinhardt . Super Reinhardt . Totally Reinhardt . Berkeley . Club Berkeley . Limit Club Berkeley

Peter Koellner Two Futures: Pattern and Chaos . Φ1 reﬂects Φ2 if for all κ such that Φ1(κ) there exists κ¯ < κ such that Φ2(κ¯).

. Φ1 rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ ≤ κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

. Φ1 strongly rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ < κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

Deﬁnition

Suppose that Φ1 and Φ2 are large cardinal notions:

Peter Koellner Two Futures: Pattern and Chaos . Φ1 rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ ≤ κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

. Φ1 strongly rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ < κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

Deﬁnition

Suppose that Φ1 and Φ2 are large cardinal notions:

. Φ1 reﬂects Φ2 if for all κ such that Φ1(κ) there exists κ¯ < κ such that Φ2(κ¯).

Peter Koellner Two Futures: Pattern and Chaos . Φ1 strongly rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ < κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

Deﬁnition

Suppose that Φ1 and Φ2 are large cardinal notions:

. Φ1 reﬂects Φ2 if for all κ such that Φ1(κ) there exists κ¯ < κ such that Φ2(κ¯).

. Φ1 rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ ≤ κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

Deﬁnition

Suppose that Φ1 and Φ2 are large cardinal notions:

. Φ1 reﬂects Φ2 if for all κ such that Φ1(κ) there exists κ¯ < κ such that Φ2(κ¯).

. Φ1 rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ ≤ κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

. Φ1 strongly rank-reﬂects Φ2 if for all κ such that Φ1(κ) there are κ¯ < γ < κ such that (Vγ, Vγ+1) |= ZF2 + Φ2(κ¯).

Peter Koellner Two Futures: Pattern and Chaos Deﬁnition A cardinal κ is super Reinhardt if for all ordinals λ there exists a non-trivial elementary embedding j : V V such that crt(j ) = κ and j (κ) > λ. →

Deﬁnition A cardinal κ is Reinhardt if there exists a non-trivial elementary embedding j : V V such that crt(j ) = κ.

→

Deﬁnition A cardinal κ is Reinhardt if there exists a non-trivial elementary embedding j : V V such that crt(j ) = κ.

→ Deﬁnition A cardinal κ is super Reinhardt if for all ordinals λ there exists a non-trivial elementary embedding j : V V such that crt(j ) = κ and j (κ) > λ. →

Deﬁnition Let A be a proper class. A cardinal κ is A-super Reinhardt if for all ordinals λ there exists a non-trivial elementary embedding j : V V such that crt(j ) = κ, j (κ) > λ, and S j (A) = A, where j (A) = α∈On j (A ∩ Vα). A cardinal κ is totally Reinhardt→if for each A ∈ Vκ+1,

(Vκ, Vκ+1) |= ZF2 + “There is an A-super Reinhardt cardinal.”

Peter Koellner Two Futures: Pattern and Chaos . In other words, super Reinhardt cardinals strongly rank-reﬂect Reinhardt cardinals. . Trivially, totally Reinhardt cardinals rank-reﬂect super Reinhardt cardinals. . So, we have a proper hierarchy.

Theorem Suppose that κ is a super Reinhardt cardinal. Then there exists γ < κ such that

(Vγ, Vγ+1) |= ZF2 + “There is a Reinhardt cardinal.”

Peter Koellner Two Futures: Pattern and Chaos . Trivially, totally Reinhardt cardinals rank-reﬂect super Reinhardt cardinals. . So, we have a proper hierarchy.

Theorem Suppose that κ is a super Reinhardt cardinal. Then there exists γ < κ such that

(Vγ, Vγ+1) |= ZF2 + “There is a Reinhardt cardinal.”

. In other words, super Reinhardt cardinals strongly rank-reﬂect Reinhardt cardinals.

Peter Koellner Two Futures: Pattern and Chaos . So, we have a proper hierarchy.

Theorem Suppose that κ is a super Reinhardt cardinal. Then there exists γ < κ such that

(Vγ, Vγ+1) |= ZF2 + “There is a Reinhardt cardinal.”

. In other words, super Reinhardt cardinals strongly rank-reﬂect Reinhardt cardinals. . Trivially, totally Reinhardt cardinals rank-reﬂect super Reinhardt cardinals.

Theorem Suppose that κ is a super Reinhardt cardinal. Then there exists γ < κ such that

(Vγ, Vγ+1) |= ZF2 + “There is a Reinhardt cardinal.”

. In other words, super Reinhardt cardinals strongly rank-reﬂect Reinhardt cardinals. . Trivially, totally Reinhardt cardinals rank-reﬂect super Reinhardt cardinals. . So, we have a proper hierarchy.

Peter Koellner Two Futures: Pattern and Chaos . So Berkeley cardinals do not reﬂect extendible cardinals and hence they do not reﬂect super Reinhardt cardinals.

Themes from Gödel Traditional Large Cardinals The HOD Dichotomy The Reinhardt Segment Ultimate-L The Berkeley Segment Large Cardinals Beyond Choice Absolute Undecidability Two Futures Inner Model Theory The Berkeley Segment Deﬁnition A cardinal δ is a Berkeley cardinal if for every transitive set M such that δ ∈ M , and for every ordinal η < δ there exists j ∈ E (M ) with η < crt(j ) < δ.

Theorem

Suppose δ0 is the least Berkeley cardinal. Then there are no extendible cardinals ≤ δ0.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel Traditional Large Cardinals The HOD Dichotomy The Reinhardt Segment Ultimate-L The Berkeley Segment Large Cardinals Beyond Choice Absolute Undecidability Two Futures Inner Model Theory The Berkeley Segment Deﬁnition A cardinal δ is a Berkeley cardinal if for every transitive set M such that δ ∈ M , and for every ordinal η < δ there exists j ∈ E (M ) with η < crt(j ) < δ.

Theorem

Suppose δ0 is the least Berkeley cardinal. Then there are no extendible cardinals ≤ δ0.

. So Berkeley cardinals do not reﬂect extendible cardinals and hence they do not reﬂect super Reinhardt cardinals.

Peter Koellner Two Futures: Pattern and Chaos Theorem

Suppose that δ0 is the least Berkeley cardinal. Then there exists γ < δ0 such that

(Vγ, Vγ+1) |= ZF2 + “there exists an extendible cardinal with a Reinhardt cardinal below.”

. So Berkeley cardinals strongly rank-reﬂect extendible cardinals and Reinhardt cardinals.

Nevertheless Berkeley cardinals rank-reﬂect Reinhardt cardinals, along with any large cardinal in the traditional hierarchy. For example:

Peter Koellner Two Futures: Pattern and Chaos . So Berkeley cardinals strongly rank-reﬂect extendible cardinals and Reinhardt cardinals.

Nevertheless Berkeley cardinals rank-reﬂect Reinhardt cardinals, along with any large cardinal in the traditional hierarchy. For example: Theorem

Suppose that δ0 is the least Berkeley cardinal. Then there exists γ < δ0 such that

(Vγ, Vγ+1) |= ZF2 + “there exists an extendible cardinal with a Reinhardt cardinal below.”

Nevertheless Berkeley cardinals rank-reﬂect Reinhardt cardinals, along with any large cardinal in the traditional hierarchy. For example: Theorem

Suppose that δ0 is the least Berkeley cardinal. Then there exists γ < δ0 such that

(Vγ, Vγ+1) |= ZF2 + “there exists an extendible cardinal with a Reinhardt cardinal below.”

. So Berkeley cardinals strongly rank-reﬂect extendible cardinals and Reinhardt cardinals.

Peter Koellner Two Futures: Pattern and Chaos . Berkeley . Club Berkeley . Limit Club Berkeley

Instead of going into the details here is a summary of what is known:

The Berkeley hierarchy continues:

Peter Koellner Two Futures: Pattern and Chaos . Club Berkeley . Limit Club Berkeley

Instead of going into the details here is a summary of what is known:

The Berkeley hierarchy continues: . Berkeley

Peter Koellner Two Futures: Pattern and Chaos . Limit Club Berkeley

Instead of going into the details here is a summary of what is known:

The Berkeley hierarchy continues: . Berkeley . Club Berkeley

Peter Koellner Two Futures: Pattern and Chaos Instead of going into the details here is a summary of what is known:

The Berkeley hierarchy continues: . Berkeley . Club Berkeley . Limit Club Berkeley

Instead of going into the details here is a summary of what is known:

Peter Koellner Two Futures: Pattern and Chaos Theorem

Suppose that δ0 is the least Berkeley cardinal cof(δ0) = γ. Then γ-DC fails.

Basic Question: What is the coﬁnality?

It turns out that the coﬁnality of the least Berkeley cardinal is connected to the degree of the failure of choice.

Peter Koellner Two Futures: Pattern and Chaos Basic Question: What is the coﬁnality?

It turns out that the coﬁnality of the least Berkeley cardinal is connected to the degree of the failure of choice.

Theorem

Suppose that δ0 is the least Berkeley cardinal cof(δ0) = γ. Then γ-DC fails.

It turns out that the coﬁnality of the least Berkeley cardinal is connected to the degree of the failure of choice.

Theorem

Suppose that δ0 is the least Berkeley cardinal cof(δ0) = γ. Then γ-DC fails.

Basic Question: What is the coﬁnality?

Peter Koellner Two Futures: Pattern and Chaos Originally we showed this using a complicated Prikry forcing. But recently, Raﬀaella Cutolo (a student of mine and Woodin’s) found simpler proofs of sharper results.

Remarkably, it turns out that this basic question is independent!

Originally we showed this using a complicated Prikry forcing. But recently, Raﬀaella Cutolo (a student of mine and Woodin’s) found simpler proofs of sharper results.

Peter Koellner Two Futures: Pattern and Chaos Theorem (Cutolo) Assume ZF + BC. Then there is a forcing extension V [G] such that V [G] |= “cof(δ0) = ω”.

. This is just a sample, of a whole host of statements that would very likely be absolutely undecidable.

Themes from Gödel Traditional Large Cardinals The HOD Dichotomy The Reinhardt Segment Ultimate-L The Berkeley Segment Large Cardinals Beyond Choice Absolute Undecidability Two Futures Inner Model Theory Theorem (Cutolo) Assume ZF + DC + BC. Then there is a forcing extension V [G] such that

V [G] |= “cof(δ0) = ω1”.

Peter Koellner Two Futures: Pattern and Chaos . This is just a sample, of a whole host of statements that would very likely be absolutely undecidable.

Themes from Gödel Traditional Large Cardinals The HOD Dichotomy The Reinhardt Segment Ultimate-L The Berkeley Segment Large Cardinals Beyond Choice Absolute Undecidability Two Futures Inner Model Theory Theorem (Cutolo) Assume ZF + DC + BC. Then there is a forcing extension V [G] such that

V [G] |= “cof(δ0) = ω1”.

Theorem (Cutolo) Assume ZF + BC. Then there is a forcing extension V [G] such that V [G] |= “cof(δ0) = ω”.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel Traditional Large Cardinals The HOD Dichotomy The Reinhardt Segment Ultimate-L The Berkeley Segment Large Cardinals Beyond Choice Absolute Undecidability Two Futures Inner Model Theory Theorem (Cutolo) Assume ZF + DC + BC. Then there is a forcing extension V [G] such that

V [G] |= “cof(δ0) = ω1”.

Theorem (Cutolo) Assume ZF + BC. Then there is a forcing extension V [G] such that V [G] |= “cof(δ0) = ω”.

. This is just a sample, of a whole host of statements that would very likely be absolutely undecidable.

Peter Koellner Two Futures: Pattern and Chaos Theorem Suppose that Berkeley cardinals are consistent. Then the Weak HOD Conjecture fails, and so the Weak Ultimate-L Conjecture fails.

. In this scenario, inner model theory as we know it—even in its most basic respects—will fail.

This is the future where chaos prevails.

There is also a connection with inner model theory.

Peter Koellner Two Futures: Pattern and Chaos . In this scenario, inner model theory as we know it—even in its most basic respects—will fail.

This is the future where chaos prevails.

There is also a connection with inner model theory.

Theorem Suppose that Berkeley cardinals are consistent. Then the Weak HOD Conjecture fails, and so the Weak Ultimate-L Conjecture fails.

Peter Koellner Two Futures: Pattern and Chaos This is the future where chaos prevails.

There is also a connection with inner model theory.

Theorem Suppose that Berkeley cardinals are consistent. Then the Weak HOD Conjecture fails, and so the Weak Ultimate-L Conjecture fails.

. In this scenario, inner model theory as we know it—even in its most basic respects—will fail.

There is also a connection with inner model theory.

Theorem Suppose that Berkeley cardinals are consistent. Then the Weak HOD Conjecture fails, and so the Weak Ultimate-L Conjecture fails.

. In this scenario, inner model theory as we know it—even in its most basic respects—will fail.

This is the future where chaos prevails.

Peter Koellner Two Futures: Pattern and Chaos The program aimed at realizing this future is the program of inner model theory described above, in particular, the program to build Ultimate-L.

We shall assume ZFC + LCA in what follows.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures Two Futures

Future #1: Pattern

The ﬁrst future is the future in which the ﬁrst side of the HOD Dichotomy holds, where HOD is “close” to V .

Peter Koellner Two Futures: Pattern and Chaos We shall assume ZFC + LCA in what follows.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures Two Futures

Future #1: Pattern

The ﬁrst future is the future in which the ﬁrst side of the HOD Dichotomy holds, where HOD is “close” to V .

The program aimed at realizing this future is the program of inner model theory described above, in particular, the program to build Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures Two Futures

Future #1: Pattern

The ﬁrst future is the future in which the ﬁrst side of the HOD Dichotomy holds, where HOD is “close” to V .

The program aimed at realizing this future is the program of inner model theory described above, in particular, the program to build Ultimate-L.

We shall assume ZFC + LCA in what follows.

Peter Koellner Two Futures: Pattern and Chaos . The Weak HOD Conjecture holds. . HOD is “close” to V . . The choiceless hierarchy is obliterated. . HOD is a good candidate for absolute deﬁnability (since it cannot be “transcended”). . There is a paradise of understanding—namely, Ultimate-L—that cannot be revealed to be illusory by large cardinals. . ZFC + V = Ultimate-L + LCA is complete.

But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

Peter Koellner Two Futures: Pattern and Chaos . HOD is “close” to V . . The choiceless hierarchy is obliterated. . HOD is a good candidate for absolute deﬁnability (since it cannot be “transcended”). . There is a paradise of understanding—namely, Ultimate-L—that cannot be revealed to be illusory by large cardinals. . ZFC + V = Ultimate-L + LCA is complete.

But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

. The Weak HOD Conjecture holds.

Peter Koellner Two Futures: Pattern and Chaos . The choiceless hierarchy is obliterated. . HOD is a good candidate for absolute deﬁnability (since it cannot be “transcended”). . There is a paradise of understanding—namely, Ultimate-L—that cannot be revealed to be illusory by large cardinals. . ZFC + V = Ultimate-L + LCA is complete.

But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

. The Weak HOD Conjecture holds. . HOD is “close” to V .

Peter Koellner Two Futures: Pattern and Chaos . HOD is a good candidate for absolute deﬁnability (since it cannot be “transcended”). . There is a paradise of understanding—namely, Ultimate-L—that cannot be revealed to be illusory by large cardinals. . ZFC + V = Ultimate-L + LCA is complete.

But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

. The Weak HOD Conjecture holds. . HOD is “close” to V . . The choiceless hierarchy is obliterated.

Peter Koellner Two Futures: Pattern and Chaos . There is a paradise of understanding—namely, Ultimate-L—that cannot be revealed to be illusory by large cardinals. . ZFC + V = Ultimate-L + LCA is complete.

But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

. The Weak HOD Conjecture holds. . HOD is “close” to V . . The choiceless hierarchy is obliterated. . HOD is a good candidate for absolute deﬁnability (since it cannot be “transcended”).

Peter Koellner Two Futures: Pattern and Chaos . ZFC + V = Ultimate-L + LCA is complete.

But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

. The Weak HOD Conjecture holds. . HOD is “close” to V . . The choiceless hierarchy is obliterated. . HOD is a good candidate for absolute deﬁnability (since it cannot be “transcended”). . There is a paradise of understanding—namely, Ultimate-L—that cannot be revealed to be illusory by large cardinals.

Peter Koellner Two Futures: Pattern and Chaos But is it correct?

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

0 Stage 1: The Weak Ultimate-L Conjecture holds. (Σ1-fact)

But is it correct?

Peter Koellner Two Futures: Pattern and Chaos One strategy is to provide an extrinsic justiﬁcation of V = Ultimate-L in terms of its consequences.

The current scenario for how this might unfold involves consequences that involve set theoretic geology and the structure theory of ultrapowers.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: The case for V = Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos The current scenario for how this might unfold involves consequences that involve set theoretic geology and the structure theory of ultrapowers.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: The case for V = Ultimate-L.

One strategy is to provide an extrinsic justiﬁcation of V = Ultimate-L in terms of its consequences.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: The case for V = Ultimate-L.

One strategy is to provide an extrinsic justiﬁcation of V = Ultimate-L in terms of its consequences.

The current scenario for how this might unfold involves consequences that involve set theoretic geology and the structure theory of ultrapowers.

Peter Koellner Two Futures: Pattern and Chaos . The ultraﬁlters are assumed to be countably complete, and we identify the target model with the transitive collapse. . UA holds in all of the known canonical inner models. . UA is expected to hold in Ultimate-L as well.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Deﬁnition (Goldberg)

The Ultrapower Axiom (UA): If j0 : V P0 and j1 : V P1 are ultrapower embeddings, then there are internal ultrapower embeddings i0 : P0 N and i1 : P1 N→such that → i0 ◦ j0 = i1 ◦ j1. → →

Peter Koellner Two Futures: Pattern and Chaos . UA holds in all of the known canonical inner models. . UA is expected to hold in Ultimate-L as well.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Deﬁnition (Goldberg)

The Ultrapower Axiom (UA): If j0 : V P0 and j1 : V P1 are ultrapower embeddings, then there are internal ultrapower embeddings i0 : P0 N and i1 : P1 N→such that → i0 ◦ j0 = i1 ◦ j1. → → . The ultraﬁlters are assumed to be countably complete, and we identify the target model with the transitive collapse.

Peter Koellner Two Futures: Pattern and Chaos . UA is expected to hold in Ultimate-L as well.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Deﬁnition (Goldberg)

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Deﬁnition (Goldberg)

Peter Koellner Two Futures: Pattern and Chaos . The Mitchell order wellorders the class of normal ultrafilters. . Every countably complete ultrafilter is ordinal definable. . If there is a supercompact cardinal, then V is a set-generic extension of HOD. . If κ is strongly compact, then for any λ ≥ κ, 2λ = λ+. . Every strongly compact cardinal is either supercompact cardinal or a measurable limit of supercompact cardinals. This provides considerable extrinsic support for UA. So, through the expected implication, this extrinsic support transfers to V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Goldberg has shown that UA has some rather striking structural consequences:

Peter Koellner Two Futures: Pattern and Chaos . Every countably complete ultraﬁlter is ordinal deﬁnable. . If there is a supercompact cardinal, then V is a set-generic extension of HOD. . If κ is strongly compact, then for any λ ≥ κ, 2λ = λ+. . Every strongly compact cardinal is either supercompact cardinal or a measurable limit of supercompact cardinals. This provides considerable extrinsic support for UA. So, through the expected implication, this extrinsic support transfers to V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Goldberg has shown that UA has some rather striking structural consequences: . The Mitchell order wellorders the class of normal ultraﬁlters.

Peter Koellner Two Futures: Pattern and Chaos . If there is a supercompact cardinal, then V is a set-generic extension of HOD. . If κ is strongly compact, then for any λ ≥ κ, 2λ = λ+. . Every strongly compact cardinal is either supercompact cardinal or a measurable limit of supercompact cardinals. This provides considerable extrinsic support for UA. So, through the expected implication, this extrinsic support transfers to V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos . If κ is strongly compact, then for any λ ≥ κ, 2λ = λ+. . Every strongly compact cardinal is either supercompact cardinal or a measurable limit of supercompact cardinals. This provides considerable extrinsic support for UA. So, through the expected implication, this extrinsic support transfers to V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos . Every strongly compact cardinal is either supercompact cardinal or a measurable limit of supercompact cardinals. This provides considerable extrinsic support for UA. So, through the expected implication, this extrinsic support transfers to V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos This provides considerable extrinsic support for UA. So, through the expected implication, this extrinsic support transfers to V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Goldberg has shown that UA has some rather striking structural consequences: . The Mitchell order wellorders the class of normal ultrafilters. . Every countably complete ultrafilter is ordinal definable. . If there is a supercompact cardinal, then V is a set-generic extension of HOD. . If κ is strongly compact, then for any λ ≥ κ, 2λ = λ+. . Every strongly compact cardinal is either supercompact cardinal or a measurable limit of supercompact cardinals.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Now, recall the following consequences of V = Ultimate-L:

Theorem (Woodin) Assume V = Ultimate-L. Then (1) V = HOD. (2) V has no non-trivial grounds. (3) Suppose V [G] is a set-generic extension of V . Then V is the mantle of V [G].

Peter Koellner Two Futures: Pattern and Chaos Theorem (Usuba)

Assume that there is an extendible cardinal. Then M is a ground of V .

. So, LCA verify a predicted consequence of V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Let us focus on M, the mantle of V , and think of this as the ultimate inner model that V = Ultimate-L is targeting.

Peter Koellner Two Futures: Pattern and Chaos . So, LCA verify a predicted consequence of V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Let us focus on M, the mantle of V , and think of this as the ultimate inner model that V = Ultimate-L is targeting.

Theorem (Usuba)

Assume that there is an extendible cardinal. Then M is a ground of V .

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Let us focus on M, the mantle of V , and think of this as the ultimate inner model that V = Ultimate-L is targeting.

Theorem (Usuba)

Assume that there is an extendible cardinal. Then M is a ground of V .

. So, LCA verify a predicted consequence of V = Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos Theorem (Woodin)

Assume UA. Then M |= V = HOD.

. So, LCA + UA verify another predicted consequence of V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Let’s now us add UA to the mix.

Peter Koellner Two Futures: Pattern and Chaos . So, LCA + UA verify another predicted consequence of V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Let’s now us add UA to the mix.

Theorem (Woodin)

Assume UA. Then M |= V = HOD.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Let’s now us add UA to the mix.

Theorem (Woodin)

Assume UA. Then M |= V = HOD.

. So, LCA + UA verify another predicted consequence of V = Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos Conjecture (Mantle Conjecture) Assume UA and that there is an extendible cardinal. Then

M |= V = Ultimate-L.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

This suggests that LCA and UA are trying to tell us that M really is Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

This suggests that LCA and UA are trying to tell us that M really is Ultimate-L.

Conjecture (Mantle Conjecture) Assume UA and that there is an extendible cardinal. Then

M |= V = Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Here is another way to put it: We are assuming LCA and that 0 the Weak Ultimate-L Conjecture holds (a Σ1-fact). This implies that there is a canonical inner model of “V = Ultimate-L”. This axiom has two plausible consequences concerning this model—namely, GA and UA. The conjecture is (eﬀectively) that GA and UA recover V = Ultimate-L. Should this be the case we would have (the start of) a strong extrinsic case for V = Ultimate-L.

Peter Koellner Two Futures: Pattern and Chaos The program aimed at realizing this future is the program of large cardinals beyond choice.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Future #2: Chaos

The second future is the future in which the second side of the HOD Dichotomy holds, where HOD is “far” from V .

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Future #2: Chaos

The second future is the future in which the second side of the HOD Dichotomy holds, where HOD is “far” from V .

The program aimed at realizing this future is the program of large cardinals beyond choice.

Peter Koellner Two Futures: Pattern and Chaos . The Weak HOD Conjecture fails. . The Ultimate-L Conjecture fails. . There is no paradise of understanding that is not revealed to be illusory by large cardinals. . There are very likely absolutely undecidable sentences.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 1: Suppose Berkeley cardinals are consistent. (A 0 Π1-fact.)

Peter Koellner Two Futures: Pattern and Chaos . The Ultimate-L Conjecture fails. . There is no paradise of understanding that is not revealed to be illusory by large cardinals. . There are very likely absolutely undecidable sentences.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 1: Suppose Berkeley cardinals are consistent. (A 0 Π1-fact.) . The Weak HOD Conjecture fails.

Peter Koellner Two Futures: Pattern and Chaos . There is no paradise of understanding that is not revealed to be illusory by large cardinals. . There are very likely absolutely undecidable sentences.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 1: Suppose Berkeley cardinals are consistent. (A 0 Π1-fact.) . The Weak HOD Conjecture fails. . The Ultimate-L Conjecture fails.

Peter Koellner Two Futures: Pattern and Chaos . There are very likely absolutely undecidable sentences.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 1: Suppose Berkeley cardinals are consistent. (A 0 Π1-fact.) . The Weak HOD Conjecture fails. . The Ultimate-L Conjecture fails. . There is no paradise of understanding that is not revealed to be illusory by large cardinals.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Peter Koellner Two Futures: Pattern and Chaos Then, in addition, one has: . HOD is “far” from V . . HOD is not a good candidate for absolute deﬁnability.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: Suppose further that the “HOD analogues” of Berkeley cardinals exist and continue to work in ZFC + LCA.

Peter Koellner Two Futures: Pattern and Chaos . HOD is “far” from V . . HOD is not a good candidate for absolute deﬁnability.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: Suppose further that the “HOD analogues” of Berkeley cardinals exist and continue to work in ZFC + LCA. Then, in addition, one has:

Peter Koellner Two Futures: Pattern and Chaos . HOD is not a good candidate for absolute deﬁnability.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: Suppose further that the “HOD analogues” of Berkeley cardinals exist and continue to work in ZFC + LCA. Then, in addition, one has: . HOD is “far” from V .

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 2: Suppose further that the “HOD analogues” of Berkeley cardinals exist and continue to work in ZFC + LCA. Then, in addition, one has: . HOD is “far” from V . . HOD is not a good candidate for absolute deﬁnability.

Peter Koellner Two Futures: Pattern and Chaos . AC fails.

(True Chaos) Suppose ﬁnally that large cardinals beyond choice actually exist.

Then, of course, one has:

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 3:

Peter Koellner Two Futures: Pattern and Chaos Suppose ﬁnally that large cardinals beyond choice actually exist.

Then, of course, one has: . AC fails.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 3: (True Chaos)

Peter Koellner Two Futures: Pattern and Chaos Then, of course, one has: . AC fails.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 3: (True Chaos) Suppose ﬁnally that large cardinals beyond choice actually exist.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Stage 3: (True Chaos) Suppose ﬁnally that large cardinals beyond choice actually exist.

Then, of course, one has: . AC fails.

Peter Koellner Two Futures: Pattern and Chaos But either way, it’s going to be interesting.

Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

We don’t know which future holds. We don’t know whether pattern or chaos will prevail.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

We don’t know which future holds. We don’t know whether pattern or chaos will prevail.

But either way, it’s going to be interesting.

Peter Koellner Two Futures: Pattern and Chaos Themes from Gödel The HOD Dichotomy Ultimate-L Two Futures Large Cardinals Beyond Choice Two Futures

Happy Birthday, Joan!

Peter Koellner Two Futures: Pattern and Chaos