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