Indefiniteness, definiteness and semi-intuitionistic theories of sets
Michael Rathjen
Department of Pure Mathematics University of Leeds
Brouwer Symposium
Amsterdam
December 9th, 2016
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Second version: The cardinality of R is ℵ1 ℵ0 (or shorter: 2 = ℵ1).
The continuum hypothesis, CH
First version: Every infinite A ⊆ R is either of size N or of the same size as R.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The continuum hypothesis, CH
First version: Every infinite A ⊆ R is either of size N or of the same size as R.
Second version: The cardinality of R is ℵ1 ℵ0 (or shorter: 2 = ℵ1).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Die moglichen¨ Machtigkeiten¨ 1908. “Von anderen unendlichen Machtigkeiten,¨ als die abzahlbare,¨ die abzahlbar¨ unfertige, und die continuierliche, kann gar keine Rede sein.” • 1914: “Wir sahen oben dass das Cantorsche Haupttheorem fur¨ den Intuitionisten keines Beweises bedarf.” • Begrundung¨ der Mengenlehre unabhangig¨ vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919) • Gielen, de Swart, Veldman: The continuum hypothesis in intuitionism (1981)
Brouwer
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • 1914: “Wir sahen oben dass das Cantorsche Haupttheorem fur¨ den Intuitionisten keines Beweises bedarf.” • Begrundung¨ der Mengenlehre unabhangig¨ vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919) • Gielen, de Swart, Veldman: The continuum hypothesis in intuitionism (1981)
Brouwer
• Die moglichen¨ Machtigkeiten¨ 1908. “Von anderen unendlichen Machtigkeiten,¨ als die abzahlbare,¨ die abzahlbar¨ unfertige, und die continuierliche, kann gar keine Rede sein.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Begrundung¨ der Mengenlehre unabhangig¨ vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919) • Gielen, de Swart, Veldman: The continuum hypothesis in intuitionism (1981)
Brouwer
• Die moglichen¨ Machtigkeiten¨ 1908. “Von anderen unendlichen Machtigkeiten,¨ als die abzahlbare,¨ die abzahlbar¨ unfertige, und die continuierliche, kann gar keine Rede sein.” • 1914: “Wir sahen oben dass das Cantorsche Haupttheorem fur¨ den Intuitionisten keines Beweises bedarf.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Gielen, de Swart, Veldman: The continuum hypothesis in intuitionism (1981)
Brouwer
• Die moglichen¨ Machtigkeiten¨ 1908. “Von anderen unendlichen Machtigkeiten,¨ als die abzahlbare,¨ die abzahlbar¨ unfertige, und die continuierliche, kann gar keine Rede sein.” • 1914: “Wir sahen oben dass das Cantorsche Haupttheorem fur¨ den Intuitionisten keines Beweises bedarf.” • Begrundung¨ der Mengenlehre unabhangig¨ vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Brouwer
• Die moglichen¨ Machtigkeiten¨ 1908. “Von anderen unendlichen Machtigkeiten,¨ als die abzahlbare,¨ die abzahlbar¨ unfertige, und die continuierliche, kann gar keine Rede sein.” • 1914: “Wir sahen oben dass das Cantorsche Haupttheorem fur¨ den Intuitionisten keines Beweises bedarf.” • Begrundung¨ der Mengenlehre unabhangig¨ vom logischen Satz vom ausgeschlossenen Dritten. Zweiter Teil: Theorie der Punktmengen. (1919) • Gielen, de Swart, Veldman: The continuum hypothesis in intuitionism (1981)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Hilbert (1900): List of 23 problems.CH is number one. • Hilbert (1925): Uber¨ das Unendliche. In it, Hilbert sketched a ‘proof’ of CH. Instead of R he considers the set NN of all functions from N to N. • ,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen.” “If we want to order the set of these functions in the way required by the problem of the continuum, we must consider how an individual function is generated.”
Hilbert
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Hilbert (1925): Uber¨ das Unendliche. In it, Hilbert sketched a ‘proof’ of CH. Instead of R he considers the set NN of all functions from N to N. • ,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen.” “If we want to order the set of these functions in the way required by the problem of the continuum, we must consider how an individual function is generated.”
Hilbert
• Hilbert (1900): List of 23 problems.CH is number one.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • ,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen.” “If we want to order the set of these functions in the way required by the problem of the continuum, we must consider how an individual function is generated.”
Hilbert
• Hilbert (1900): List of 23 problems.CH is number one. • Hilbert (1925): Uber¨ das Unendliche. In it, Hilbert sketched a ‘proof’ of CH. Instead of R he considers the set NN of all functions from N to N.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS “If we want to order the set of these functions in the way required by the problem of the continuum, we must consider how an individual function is generated.”
Hilbert
• Hilbert (1900): List of 23 problems.CH is number one. • Hilbert (1925): Uber¨ das Unendliche. In it, Hilbert sketched a ‘proof’ of CH. Instead of R he considers the set NN of all functions from N to N. • ,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Hilbert
• Hilbert (1900): List of 23 problems.CH is number one. • Hilbert (1925): Uber¨ das Unendliche. In it, Hilbert sketched a ‘proof’ of CH. Instead of R he considers the set NN of all functions from N to N. • ,,Wenn wir die Menge dieser Funktionen im Sinne des Kontinuumproblems ordnen wollen, so bedarf es dazu der Bezugnahme auf die Erzeugung der einzelnen Funktionen.” “If we want to order the set of these functions in the way required by the problem of the continuum, we must consider how an individual function is generated.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Step 1. Produce a set-theoretic assertion Φ expressing a natural and “ intuitively true” set-theoretic principle.
Step 2. Prove that Φ determines CH. That is, prove Φ ⇒ CH or prove that Φ ⇒ ¬CH
137 years and still going strong(?): Cantor’s continuum problem
The dream solution template a` la Hamkins for determining truth in V . E.g. CH.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Step 2. Prove that Φ determines CH. That is, prove Φ ⇒ CH or prove that Φ ⇒ ¬CH
137 years and still going strong(?): Cantor’s continuum problem
The dream solution template a` la Hamkins for determining truth in V . E.g. CH.
Step 1. Produce a set-theoretic assertion Φ expressing a natural and “ intuitively true” set-theoretic principle.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 137 years and still going strong(?): Cantor’s continuum problem
The dream solution template a` la Hamkins for determining truth in V . E.g. CH.
Step 1. Produce a set-theoretic assertion Φ expressing a natural and “ intuitively true” set-theoretic principle.
Step 2. Prove that Φ determines CH. That is, prove Φ ⇒ CH or prove that Φ ⇒ ¬CH
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • A universe view: ( Woodin) CH has a truth value in the universe V . • A multiverse view: ( Hamkins) There are many universes that set theorists study. They all exist. CH has different truth values in different universes. • ( Feferman) CH is neither a definite mathematical problem nor a definite logical problem.
Three responses
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • A multiverse view: ( Hamkins) There are many universes that set theorists study. They all exist. CH has different truth values in different universes. • ( Feferman) CH is neither a definite mathematical problem nor a definite logical problem.
Three responses
• A universe view: ( Woodin) CH has a truth value in the universe V .
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • ( Feferman) CH is neither a definite mathematical problem nor a definite logical problem.
Three responses
• A universe view: ( Woodin) CH has a truth value in the universe V . • A multiverse view: ( Hamkins) There are many universes that set theorists study. They all exist. CH has different truth values in different universes.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Three responses
• A universe view: ( Woodin) CH has a truth value in the universe V . • A multiverse view: ( Hamkins) There are many universes that set theorists study. They all exist. CH has different truth values in different universes. • ( Feferman) CH is neither a definite mathematical problem nor a definite logical problem.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Godel¨ 1964 says that: “axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation “set of”.” (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals and maintains that “[t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above” (260-261). Godel¨ later refers to such axioms as having an “intrinsic necessary” status.
Godel’s¨ intrinsic program: Reflection
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals and maintains that “[t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above” (260-261). Godel¨ later refers to such axioms as having an “intrinsic necessary” status.
Godel’s¨ intrinsic program: Reflection
Godel¨ 1964 says that: “axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation “set of”.” (260)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS and maintains that “[t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above” (260-261). Godel¨ later refers to such axioms as having an “intrinsic necessary” status.
Godel’s¨ intrinsic program: Reflection
Godel¨ 1964 says that: “axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation “set of”.” (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Godel¨ later refers to such axioms as having an “intrinsic necessary” status.
Godel’s¨ intrinsic program: Reflection
Godel¨ 1964 says that: “axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation “set of”.” (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals and maintains that “[t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above” (260-261).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Godel’s¨ intrinsic program: Reflection
Godel¨ 1964 says that: “axioms of set theory [ZFC] by no means form a system closed in itself, but, quite on the contrary, the very concept of set on which they are based suggests their extension by new axioms which assert the existence of still further iterations of the operation “set of”.” (260) He mentions as examples the axioms asserting the existence of inaccessible and Mahlo cardinals and maintains that “[t]hese axioms show clearly, not only that the axiomatic system of set theory as used today is incomplete, but also that it can be supplemented without arbitrariness by new axioms which only unfold the content of the concept of set as explained above” (260-261). Godel¨ later refers to such axioms as having an “intrinsic necessary” status.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Theorem ( Cohen; Levy and Solovay 1967): CH is consistent with and independent of all “small” and “large”) LCAs that have been considered to date, provided they are consistent with ZF. Proof. By Cohen’s method of forcing.
What Hope for Godel’s¨ Program to settle CH?
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Proof. By Cohen’s method of forcing.
What Hope for Godel’s¨ Program to settle CH?
Theorem ( Cohen; Levy and Solovay 1967): CH is consistent with and independent of all “small” and “large”) LCAs that have been considered to date, provided they are consistent with ZF.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS What Hope for Godel’s¨ Program to settle CH?
Theorem ( Cohen; Levy and Solovay 1967): CH is consistent with and independent of all “small” and “large”) LCAs that have been considered to date, provided they are consistent with ZF. Proof. By Cohen’s method of forcing.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Godel’s¨ Extrinsic Program (1947)
“There might exist axioms so abundant in their verifiable consequences, shedding so much light upon a whole discipline...that quite irrespective of their intrinsic necessity they would have to be assumed in the same sense as any well-established physical theory.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • “Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work.” • “I think the evidence now favors CH.” • “The picture that is emerging now [...] is as follows. The solution to the inner model problem for one supercompact cardinal yields the ultimate enlargement of L. This enlargement of L is compatible with all stronger large cardinal axioms and strong forms of covering hold relative to this inner model.”
• Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • “I think the evidence now favors CH.” • “The picture that is emerging now [...] is as follows. The solution to the inner model problem for one supercompact cardinal yields the ultimate enlargement of L. This enlargement of L is compatible with all stronger large cardinal axioms and strong forms of covering hold relative to this inner model.”
• Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition. • “Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • “The picture that is emerging now [...] is as follows. The solution to the inner model problem for one supercompact cardinal yields the ultimate enlargement of L. This enlargement of L is compatible with all stronger large cardinal axioms and strong forms of covering hold relative to this inner model.”
• Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition. • “Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work.” • “I think the evidence now favors CH.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Woodin: The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal (2010) The second edition. • “Ultimately of far more significance for this book is that recent results concerning the inner model program undermine the philosophical framework for this entire work.” • “I think the evidence now favors CH.” • “The picture that is emerging now [...] is as follows. The solution to the inner model problem for one supercompact cardinal yields the ultimate enlargement of L. This enlargement of L is compatible with all stronger large cardinal axioms and strong forms of covering hold relative to this inner model.”
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Exploring the frontiers of incompleteness. Peter Koellner’s Templeton project.
Solomon Feferman: Is the continuum hypothesis a definite mathematical problem?
Feferman on CH
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Solomon Feferman: Is the continuum hypothesis a definite mathematical problem?
Feferman on CH
Exploring the frontiers of incompleteness. Peter Koellner’s Templeton project.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Feferman on CH
Exploring the frontiers of incompleteness. Peter Koellner’s Templeton project.
Solomon Feferman: Is the continuum hypothesis a definite mathematical problem?
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 1 Notion of a definite totality/ domain of things.
2 Notion of a definite proposition/ definite predicate.
Two Informal Notions of Definiteness
Criteria for these can be given in logical terms
Feferman’s analysis
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 1 Notion of a definite totality/ domain of things.
2 Notion of a definite proposition/ definite predicate. Criteria for these can be given in logical terms
Feferman’s analysis
Two Informal Notions of Definiteness
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 2 Notion of a definite proposition/ definite predicate. Criteria for these can be given in logical terms
Feferman’s analysis
Two Informal Notions of Definiteness
1 Notion of a definite totality/ domain of things.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Criteria for these can be given in logical terms
Feferman’s analysis
Two Informal Notions of Definiteness
1 Notion of a definite totality/ domain of things.
2 Notion of a definite proposition/ definite predicate.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Feferman’s analysis
Two Informal Notions of Definiteness
1 Notion of a definite totality/ domain of things.
2 Notion of a definite proposition/ definite predicate. Criteria for these can be given in logical terms
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • A predicate P is definite over a domain D iff the Principle of Bivalence holds for it, i.e.
∀~x ∈ D [P(~x ) ∨ ¬P(~x )].
• A totality D is definite iff quantification over D is a definite logical operation, i.e., whenever R(~x, y) is definite over D, so are ∀y ∈ DR(y, ~x ) and ∃y ∈ DR(y, ~x ).
The Criteria
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • A totality D is definite iff quantification over D is a definite logical operation, i.e., whenever R(~x, y) is definite over D, so are ∀y ∈ DR(y, ~x ) and ∃y ∈ DR(y, ~x ).
The Criteria
• A predicate P is definite over a domain D iff the Principle of Bivalence holds for it, i.e.
∀~x ∈ D [P(~x ) ∨ ¬P(~x )].
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The Criteria
• A predicate P is definite over a domain D iff the Principle of Bivalence holds for it, i.e.
∀~x ∈ D [P(~x ) ∨ ¬P(~x )].
• A totality D is definite iff quantification over D is a definite logical operation, i.e., whenever R(~x, y) is definite over D, so are ∀y ∈ DR(y, ~x ) and ∃y ∈ DR(y, ~x ).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Proposed logical framework for what’s definite and what’s not: What’s definite is the domain of classical logic, what’s not is that of intuitionistic logic.
• ∈ and = are definite. Sets are viewed as definite domains. Hence Classical logic for bounded (∆0) formulas. Intuitionistic logic for unbounded quantification.
Feferman’s analysis
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • ∈ and = are definite. Sets are viewed as definite domains. Hence Classical logic for bounded (∆0) formulas. Intuitionistic logic for unbounded quantification.
Feferman’s analysis
• Proposed logical framework for what’s definite and what’s not: What’s definite is the domain of classical logic, what’s not is that of intuitionistic logic.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Feferman’s analysis
• Proposed logical framework for what’s definite and what’s not: What’s definite is the domain of classical logic, what’s not is that of intuitionistic logic.
• ∈ and = are definite. Sets are viewed as definite domains. Hence Classical logic for bounded (∆0) formulas. Intuitionistic logic for unbounded quantification.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • According to the (ultra) finitists, the natural numbers form an “unfinished” or indefinite totality, and quantification over the natural numbers is indefinite, while bounded quantification is definite.
• According to the predicativists, the natural numbers form a definite totality, but not the supposed collection of arbitrary sets of natural numbers.
Some Examples
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • According to the predicativists, the natural numbers form a definite totality, but not the supposed collection of arbitrary sets of natural numbers.
Some Examples
• According to the (ultra) finitists, the natural numbers form an “unfinished” or indefinite totality, and quantification over the natural numbers is indefinite, while bounded quantification is definite.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Some Examples
• According to the (ultra) finitists, the natural numbers form an “unfinished” or indefinite totality, and quantification over the natural numbers is indefinite, while bounded quantification is definite.
• According to the predicativists, the natural numbers form a definite totality, but not the supposed collection of arbitrary sets of natural numbers.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • According to some classical Descriptive Set Theorists, the set R of real numbers is a definite totality but not the supposed totality of arbitrary subsets of R.
• Set theory identifies definite totalities with sets. Then V is not a definite totality by Russell’s Paradox. (“Predicative” Set Theory)
More Examples
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Set theory identifies definite totalities with sets. Then V is not a definite totality by Russell’s Paradox. (“Predicative” Set Theory)
More Examples
• According to some classical Descriptive Set Theorists, the set R of real numbers is a definite totality but not the supposed totality of arbitrary subsets of R.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS More Examples
• According to some classical Descriptive Set Theorists, the set R of real numbers is a definite totality but not the supposed totality of arbitrary subsets of R.
• Set theory identifies definite totalities with sets. Then V is not a definite totality by Russell’s Paradox. (“Predicative” Set Theory)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) § 68. In Frege: Philosophy of Mathematics, Dummett’s diagnosis of the failure of Frege’s logicist project focusses on the adoption of classical quantification. Dummett argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept.
Indefinitely extensible concepts
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS In Frege: Philosophy of Mathematics, Dummett’s diagnosis of the failure of Frege’s logicist project focusses on the adoption of classical quantification. Dummett argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept.
Indefinitely extensible concepts
Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) § 68.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Dummett argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept.
Indefinitely extensible concepts
Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) § 68. In Frege: Philosophy of Mathematics, Dummett’s diagnosis of the failure of Frege’s logicist project focusses on the adoption of classical quantification.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Indefinitely extensible concepts
Ich setze voraus, dass man wisse, was der Umfang eines Begriffes sei. I assume that it is known what the extension of a concept is. Frege: Die Grundlagen der Arithmetik (Breslau 1884) § 68. In Frege: Philosophy of Mathematics, Dummett’s diagnosis of the failure of Frege’s logicist project focusses on the adoption of classical quantification. Dummett argues that classical quantification is illegitimate when the domain is given as the objects which fall under an indefinitely extensible concept.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Cantor saw far more deeply into the matter than Frege did: he was aware, long before, that one cannot simply assume every concept to have an extension with a determinate cardinality. (p. 26) The fact revealed by the set-theoretic paradoxes was the existence of indefinitely extensible concepts - a fact of which Frege did not dream and even Cantor had only an obscure perception.
Dummett: “What is Mathematics About?”, 1994
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The fact revealed by the set-theoretic paradoxes was the existence of indefinitely extensible concepts - a fact of which Frege did not dream and even Cantor had only an obscure perception.
Dummett: “What is Mathematics About?”, 1994
Cantor saw far more deeply into the matter than Frege did: he was aware, long before, that one cannot simply assume every concept to have an extension with a determinate cardinality. (p. 26)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Dummett: “What is Mathematics About?”, 1994
Cantor saw far more deeply into the matter than Frege did: he was aware, long before, that one cannot simply assume every concept to have an extension with a determinate cardinality. (p. 26) The fact revealed by the set-theoretic paradoxes was the existence of indefinitely extensible concepts - a fact of which Frege did not dream and even Cantor had only an obscure perception.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS There can be no objection to quantifying over all objects falling under some indefinitely extensible concept, say over everything we should, given an intelligible description of it, recognize as an ordinal number, provided that we do not think of the statements formed by means of such quantification as having determinate truth-conditions; we can understand them only as making claims of the kind already sketched. They will not then satisfy the laws of classical logic, but only the weaker laws of intuitionistic logic.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle. • One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist. • This is the approach Tait takes in his work on reflection principles. Feferman work on semi-intuitionistic systems of set theory can also be recast in those terms.
Actualism versus Potentialism
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist. • This is the approach Tait takes in his work on reflection principles. Feferman work on semi-intuitionistic systems of set theory can also be recast in those terms.
Actualism versus Potentialism
• This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • This is the approach Tait takes in his work on reflection principles. Feferman work on semi-intuitionistic systems of set theory can also be recast in those terms.
Actualism versus Potentialism
• This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle. • One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Actualism versus Potentialism
• This, to be sure, is a rough distinction. But it has a long history, going back to Aristotle. • One way of formally regimenting this informal distinction is by employing intuitionistic logic for domains for which one is a potentialist and reserving classic logic for domains for which one is an actualist. • This is the approach Tait takes in his work on reflection principles. Feferman work on semi-intuitionistic systems of set theory can also be recast in those terms.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Restrict quantifiers in the formulas that are supposed to represent definite properties, e.g. in Comprehension or Separation axioms. • Quantification over indefinite domains may still be regarded as meaningful, in order to state generic properties and closure properties of the ‘realm’ of sets, but is governed by intuitionistic logic.
Toward Axiomatic Formulations
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Quantification over indefinite domains may still be regarded as meaningful, in order to state generic properties and closure properties of the ‘realm’ of sets, but is governed by intuitionistic logic.
Toward Axiomatic Formulations
• Restrict quantifiers in the formulas that are supposed to represent definite properties, e.g. in Comprehension or Separation axioms.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Toward Axiomatic Formulations
• Restrict quantifiers in the formulas that are supposed to represent definite properties, e.g. in Comprehension or Separation axioms. • Quantification over indefinite domains may still be regarded as meaningful, in order to state generic properties and closure properties of the ‘realm’ of sets, but is governed by intuitionistic logic.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Extensionality Pairing and Union Infinity Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
• Feferman: On the strength of some semi-constructive theories (2012) • Basic axioms
The theory SCS
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Extensionality Pairing and Union Infinity Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
• Basic axioms
The theory SCS
• Feferman: On the strength of some semi-constructive theories (2012)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Extensionality Pairing and Union Infinity Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
The theory SCS
• Feferman: On the strength of some semi-constructive theories (2012) • Basic axioms
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Pairing and Union Infinity Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
The theory SCS
• Feferman: On the strength of some semi-constructive theories (2012) • Basic axioms Extensionality
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Infinity Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
The theory SCS
• Feferman: On the strength of some semi-constructive theories (2012) • Basic axioms Extensionality Pairing and Union
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
The theory SCS
• Feferman: On the strength of some semi-constructive theories (2012) • Basic axioms Extensionality Pairing and Union Infinity
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The theory SCS
• Feferman: On the strength of some semi-constructive theories (2012) • Basic axioms Extensionality Pairing and Union Infinity Set Induction
∀x [∀y ∈ x ϕ(y) → ϕ(x)] → ∀x ϕ(x) for any formula ϕ(x).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • LEM∆0 is the schema ϕ ∨ ¬ϕ
for ϕ ∆0. • BOS is the schema (for all formulas ϕ(x)): If ∀x ∈ a [ϕ(x) ∨ ¬ϕ(x)] then
∀x ∈ a ϕ(x) ∨ ∃x ∈ a ¬ϕ(x).
• ACfull is the schema (for all formulas ϕ(x, y)):
∀x ∈ a ∃y ϕ(x, y) → ∃f [dom(f ) = a ∧ ∀x ∈ a ϕ(x, f (x))]
• MP is the schema ¬¬∃x θ(x) → ∃x θ(x) for θ(x) ∆0.
SCS continued
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • BOS is the schema (for all formulas ϕ(x)): If ∀x ∈ a [ϕ(x) ∨ ¬ϕ(x)] then
∀x ∈ a ϕ(x) ∨ ∃x ∈ a ¬ϕ(x).
• ACfull is the schema (for all formulas ϕ(x, y)):
∀x ∈ a ∃y ϕ(x, y) → ∃f [dom(f ) = a ∧ ∀x ∈ a ϕ(x, f (x))]
• MP is the schema ¬¬∃x θ(x) → ∃x θ(x) for θ(x) ∆0.
SCS continued
• LEM∆0 is the schema ϕ ∨ ¬ϕ
for ϕ ∆0.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • ACfull is the schema (for all formulas ϕ(x, y)):
∀x ∈ a ∃y ϕ(x, y) → ∃f [dom(f ) = a ∧ ∀x ∈ a ϕ(x, f (x))]
• MP is the schema ¬¬∃x θ(x) → ∃x θ(x) for θ(x) ∆0.
SCS continued
• LEM∆0 is the schema ϕ ∨ ¬ϕ
for ϕ ∆0. • BOS is the schema (for all formulas ϕ(x)): If ∀x ∈ a [ϕ(x) ∨ ¬ϕ(x)] then
∀x ∈ a ϕ(x) ∨ ∃x ∈ a ¬ϕ(x).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • MP is the schema ¬¬∃x θ(x) → ∃x θ(x) for θ(x) ∆0.
SCS continued
• LEM∆0 is the schema ϕ ∨ ¬ϕ
for ϕ ∆0. • BOS is the schema (for all formulas ϕ(x)): If ∀x ∈ a [ϕ(x) ∨ ¬ϕ(x)] then
∀x ∈ a ϕ(x) ∨ ∃x ∈ a ¬ϕ(x).
• ACfull is the schema (for all formulas ϕ(x, y)):
∀x ∈ a ∃y ϕ(x, y) → ∃f [dom(f ) = a ∧ ∀x ∈ a ϕ(x, f (x))]
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS SCS continued
• LEM∆0 is the schema ϕ ∨ ¬ϕ
for ϕ ∆0. • BOS is the schema (for all formulas ϕ(x)): If ∀x ∈ a [ϕ(x) ∨ ¬ϕ(x)] then
∀x ∈ a ϕ(x) ∨ ∃x ∈ a ¬ϕ(x).
• ACfull is the schema (for all formulas ϕ(x, y)):
∀x ∈ a ∃y ϕ(x, y) → ∃f [dom(f ) = a ∧ ∀x ∈ a ϕ(x, f (x))]
• MP is the schema ¬¬∃x θ(x) → ∃x θ(x) for θ(x) ∆0.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974).
• Myhill’s Constructive Set Theory (1974).
• Feferman subjected these theories to functional interpretation (goes back a long way).
Why these axioms?
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Myhill’s Constructive Set Theory (1974).
• Feferman subjected these theories to functional interpretation (goes back a long way).
Why these axioms?
• Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • Feferman subjected these theories to functional interpretation (goes back a long way).
Why these axioms?
• Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974).
• Myhill’s Constructive Set Theory (1974).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Why these axioms?
• Older roots: Pozsgay (1971, 1972); Tharp (1971); Wolf (1974).
• Myhill’s Constructive Set Theory (1974).
• Feferman subjected these theories to functional interpretation (goes back a long way).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS + Let SCS be the theory SCS + ‘R is a set’. The formal version of the conjecture is that
SCS+ 6` CH ∨ ¬CH
• The theory SCS has too many axioms. • SCS+ is quite strong. It proves every theorem of (classical) second order arithmetic. In strength it resides strictly between second order arithmetic and Zermelo set theory.
Feferman’s Conjecture: CH is not a definite mathematical problem
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The formal version of the conjecture is that
SCS+ 6` CH ∨ ¬CH
• The theory SCS has too many axioms. • SCS+ is quite strong. It proves every theorem of (classical) second order arithmetic. In strength it resides strictly between second order arithmetic and Zermelo set theory.
Feferman’s Conjecture: CH is not a definite mathematical problem
+ Let SCS be the theory SCS + ‘R is a set’.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • The theory SCS has too many axioms. • SCS+ is quite strong. It proves every theorem of (classical) second order arithmetic. In strength it resides strictly between second order arithmetic and Zermelo set theory.
Feferman’s Conjecture: CH is not a definite mathematical problem
+ Let SCS be the theory SCS + ‘R is a set’. The formal version of the conjecture is that
SCS+ 6` CH ∨ ¬CH
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • SCS+ is quite strong. It proves every theorem of (classical) second order arithmetic. In strength it resides strictly between second order arithmetic and Zermelo set theory.
Feferman’s Conjecture: CH is not a definite mathematical problem
+ Let SCS be the theory SCS + ‘R is a set’. The formal version of the conjecture is that
SCS+ 6` CH ∨ ¬CH
• The theory SCS has too many axioms.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Feferman’s Conjecture: CH is not a definite mathematical problem
+ Let SCS be the theory SCS + ‘R is a set’. The formal version of the conjecture is that
SCS+ 6` CH ∨ ¬CH
• The theory SCS has too many axioms. • SCS+ is quite strong. It proves every theorem of (classical) second order arithmetic. In strength it resides strictly between second order arithmetic and Zermelo set theory.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS R: Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman. JSL 2016.
Techniques: Relativized constructible hierarchy L[A], forcing, realizability.
Remark: The machinery can also be applied to furnish other indefinite statements.
The conjecture is true
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Remark: The machinery can also be applied to furnish other indefinite statements.
The conjecture is true
R: Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman. JSL 2016.
Techniques: Relativized constructible hierarchy L[A], forcing, realizability.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The conjecture is true
R: Indefiniteness in semi-intuitionistic set theories: On a conjecture of Feferman. JSL 2016.
Techniques: Relativized constructible hierarchy L[A], forcing, realizability.
Remark: The machinery can also be applied to furnish other indefinite statements.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Theorem: (Koellner, Woodin) Let DWOR be the statement that that “there is a well-ordering of R in L(R)”. Assume Con(ZFC + “There are ω-many Woodin cardinals”). Then + SCS 0 DWOR ∨ ¬DWOR.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Theorem: (Koellner, Woodin) Let NCR be the statement “There is a non-constructible real”. Then SCS 0 NCR ∨ ¬NCR.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS One of the basic considerations leading to the SCS systems in general is that a statement ϕ is recognized to be definite relative to such just in case LEM holds for it, in other words just in case SCS proves ϕ ∨ ¬ϕ, and otherwise it is indefinite. Similarly, a formula ϕ(x) is recognized to be definite relative to SCS just in case it proves ∀x [ϕ(x) ∨ ¬ϕ(x)]. In the case of SCS, I conjecture that every such formula is equivalent to a ∆1 formula and hence is absolute for end-extensions. Furthermore, in the case of SCS + Pow(ω), the corresponding conjecture is that every such formula is ∆1 in the power set of ω.
Feferman’s second conjecture
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Feferman’s second conjecture
One of the basic considerations leading to the SCS systems in general is that a statement ϕ is recognized to be definite relative to such just in case LEM holds for it, in other words just in case SCS proves ϕ ∨ ¬ϕ, and otherwise it is indefinite. Similarly, a formula ϕ(x) is recognized to be definite relative to SCS just in case it proves ∀x [ϕ(x) ∨ ¬ϕ(x)]. In the case of SCS, I conjecture that every such formula is equivalent to a ∆1 formula and hence is absolute for end-extensions. Furthermore, in the case of SCS + Pow(ω), the corresponding conjecture is that every such formula is ∆1 in the power set of ω.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS ∆1
Definition: For a predicate given by a formula A(x) with at most x free we say that A(x) is ∆1 with respect to a theory T ⊇ SCS if there exist a Σ1-formula B(x) and a Π1-formula C(x) such that
T ` ∀x [A(x) ↔ B(x) ↔ C(x)].
A(x) is said to be definite or to satisfy LEM with respect to T if
T ` ∀x [A(x) ∨ ¬A(x)].
In constructive mathematics, the latter property is also know as decidability (with respect to T ).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The axiom of global choice, ACglobal, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R:
(i) ∀x∀y∀z[R(x, y) ∧ R(x, z) → y = z] (1) (ii) ∀x[x 6= ∅ → ∃y ∈ x R(x, y)]. (2)
Moreover, the axiom schemes of ∆0-Separation and ∆0-Collection are now formulated for ∆0-formulae of the extended language.
Global Choice
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS (i) ∀x∀y∀z[R(x, y) ∧ R(x, z) → y = z] (1) (ii) ∀x[x 6= ∅ → ∃y ∈ x R(x, y)]. (2)
Moreover, the axiom schemes of ∆0-Separation and ∆0-Collection are now formulated for ∆0-formulae of the extended language.
Global Choice
The axiom of global choice, ACglobal, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R:
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Moreover, the axiom schemes of ∆0-Separation and ∆0-Collection are now formulated for ∆0-formulae of the extended language.
Global Choice
The axiom of global choice, ACglobal, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R:
(i) ∀x∀y∀z[R(x, y) ∧ R(x, z) → y = z] (1) (ii) ∀x[x 6= ∅ → ∃y ∈ x R(x, y)]. (2)
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Global Choice
The axiom of global choice, ACglobal, is expressed in an extension of the language by a new binary relation symbol R, via the following axioms pertaining to R:
(i) ∀x∀y∀z[R(x, y) ∧ R(x, z) → y = z] (1) (ii) ∀x[x 6= ∅ → ∃y ∈ x R(x, y)]. (2)
Moreover, the axiom schemes of ∆0-Separation and ∆0-Collection are now formulated for ∆0-formulae of the extended language.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Theorem:
Suppose (∗) SCS + ACglobal ` ∀~x [∃yA(~x, y) ↔ ∀zB(~x, z)]
where A and B are ∆0. Then
SCS + ACglobal ` ∀~x [∃yA(~x, y) ∨ ¬∃yA(~x, y)].
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Proposition. There is an extension T of HA such that there is a formula A that is ∆1 relative to T but T does not prove A ∨ ¬A.
Counterexample
One might think there is a more general result to the effect that a theory with decidable atomic formulae and Collection has the property that ∆1 predicates provably satisfy excluded third.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Counterexample
One might think there is a more general result to the effect that a theory with decidable atomic formulae and Collection has the property that ∆1 predicates provably satisfy excluded third. Proposition. There is an extension T of HA such that there is a formula A that is ∆1 relative to T but T does not prove A ∨ ¬A.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Theorem: Suppose SCS ` ∀x [A(x) ∨ ¬A(x)].
Then there exist a Σ1-formula B(x) and a Π1-formula C(x) in
the language of SCS + ACglobal, L∈(R), such that
SCS + ACglobal ` ∀x [A(x) ↔ B(x) ↔ C(x)].
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS The previous result can also be slightly strengthened. Theorem:
Suppose SCS + ACglobal ` ∀x [A(x) ∨ ¬A(x)]. Then there exist a Σ1-formula B(x) and a Π1-formula C(x) in the language of
SCS + ACglobal, L∈(R), such that
SCS + ACglobal ` ∀x [A(x) ↔ B(x) ↔ C(x)].
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice. • One idea is to use class forcing (as in the case of ZFC) to
find an interpretation of SCS + ACglobal in SCS that preserves the formulae of L∈. In the case of ZFC the class of forcing conditions consists of local choice functions h, i.e., functions such that for all x in the domain of h, h(x) ∈ x if x 6= ∅ and h(x) = ∅ if x = ∅. • While forcing can be defined in SCS, the problem we encountered is that SCS seems to be too weak to be able to show that every theorem of SCS is forced, the particular culprit being Collection.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • One idea is to use class forcing (as in the case of ZFC) to
find an interpretation of SCS + ACglobal in SCS that preserves the formulae of L∈. In the case of ZFC the class of forcing conditions consists of local choice functions h, i.e., functions such that for all x in the domain of h, h(x) ∈ x if x 6= ∅ and h(x) = ∅ if x = ∅. • While forcing can be defined in SCS, the problem we encountered is that SCS seems to be too weak to be able to show that every theorem of SCS is forced, the particular culprit being Collection.
• One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • While forcing can be defined in SCS, the problem we encountered is that SCS seems to be too weak to be able to show that every theorem of SCS is forced, the particular culprit being Collection.
• One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice. • One idea is to use class forcing (as in the case of ZFC) to
find an interpretation of SCS + ACglobal in SCS that preserves the formulae of L∈. In the case of ZFC the class of forcing conditions consists of local choice functions h, i.e., functions such that for all x in the domain of h, h(x) ∈ x if x 6= ∅ and h(x) = ∅ if x = ∅.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS • One would like to get rid of global choice in the interpreting theory, i.e., weaken global choice to just choice. • One idea is to use class forcing (as in the case of ZFC) to
find an interpretation of SCS + ACglobal in SCS that preserves the formulae of L∈. In the case of ZFC the class of forcing conditions consists of local choice functions h, i.e., functions such that for all x in the domain of h, h(x) ∈ x if x 6= ∅ and h(x) = ∅ if x = ∅. • While forcing can be defined in SCS, the problem we encountered is that SCS seems to be too weak to be able to show that every theorem of SCS is forced, the particular culprit being Collection.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS + SCS = SCS + R is a set
Remark: Versions of the previous Theorem for stronger theories can also + be proved, for instance for the theory SCS = SCS + R is a set.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Theorem
Suppose A is a Π2 statement of L∈(R) that holds in L(b, ≺), where b is a transitive set and ≺ is well-ordering on b.
If SCS + ACglobal ` A ∨ ¬A, then
Lκ(b, ≺) |= A
holds for all admissible sets Lκ(b, ≺) with κ > ω.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 1 Comparability of well-orderings on subsets of ω. 2 ∆0-determinacy.
One can now, e.g., take any of the statements that are equivalent to ATR0 over ACA0 from Simpson’s book and conclude that they are indeterminate relative to SCS + ACglobal as they are of Π2 form, hold in L but fail to hold in L ck . This was also observed by Koellner and ω1 Woodin. Here are two examples.
One can also take the statement “Every real is constructible” since there are admissible sets of the form Lκ(x) where x is a constructible real but x fails to be constructible in Lκ(x).
Corollary:
Recall that a statement D is indefinite relative to SCS + ACglobal if SCS + ACglobal 0 D ∨ ¬D.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 1 Comparability of well-orderings on subsets of ω. 2 ∆0-determinacy. One can also take the statement “Every real is constructible” since there are admissible sets of the form Lκ(x) where x is a constructible real but x fails to be constructible in Lκ(x).
Corollary:
Recall that a statement D is indefinite relative to SCS + ACglobal if SCS + ACglobal 0 D ∨ ¬D. One can now, e.g., take any of the statements that are equivalent to ATR0 over ACA0 from Simpson’s book and conclude that they are indeterminate relative to SCS + ACglobal as they are of Π2 form, hold in L but fail to hold in L ck . This was also observed by Koellner and ω1 Woodin. Here are two examples.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS 2 ∆0-determinacy. One can also take the statement “Every real is constructible” since there are admissible sets of the form Lκ(x) where x is a constructible real but x fails to be constructible in Lκ(x).
Corollary:
Recall that a statement D is indefinite relative to SCS + ACglobal if SCS + ACglobal 0 D ∨ ¬D. One can now, e.g., take any of the statements that are equivalent to ATR0 over ACA0 from Simpson’s book and conclude that they are indeterminate relative to SCS + ACglobal as they are of Π2 form, hold in L but fail to hold in L ck . This was also observed by Koellner and ω1 Woodin. Here are two examples. 1 Comparability of well-orderings on subsets of ω.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS One can also take the statement “Every real is constructible” since there are admissible sets of the form Lκ(x) where x is a constructible real but x fails to be constructible in Lκ(x).
Corollary:
Recall that a statement D is indefinite relative to SCS + ACglobal if SCS + ACglobal 0 D ∨ ¬D. One can now, e.g., take any of the statements that are equivalent to ATR0 over ACA0 from Simpson’s book and conclude that they are indeterminate relative to SCS + ACglobal as they are of Π2 form, hold in L but fail to hold in L ck . This was also observed by Koellner and ω1 Woodin. Here are two examples. 1 Comparability of well-orderings on subsets of ω. 2 ∆0-determinacy.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Corollary:
Recall that a statement D is indefinite relative to SCS + ACglobal if SCS + ACglobal 0 D ∨ ¬D. One can now, e.g., take any of the statements that are equivalent to ATR0 over ACA0 from Simpson’s book and conclude that they are indeterminate relative to SCS + ACglobal as they are of Π2 form, hold in L but fail to hold in L ck . This was also observed by Koellner and ω1 Woodin. Here are two examples. 1 Comparability of well-orderings on subsets of ω. 2 ∆0-determinacy. One can also take the statement “Every real is constructible” since there are admissible sets of the form Lκ(x) where x is a constructible real but x fails to be constructible in Lκ(x).
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Remark. Likewise, the machinery can be applied to
SCS + ACglobal + Pow
to give indefinite statements.
Remark. The machinery can also be applied to
SCS + ACglobal + ‘R is a set’
to give indefinite statements relative to that theory.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS Remark. The machinery can also be applied to
SCS + ACglobal + ‘R is a set’
to give indefinite statements relative to that theory.
Remark. Likewise, the machinery can be applied to
SCS + ACglobal + Pow
to give indefinite statements.
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS dank u wel
INDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETSINDEFINITENESS, DEFINITENESS AND SEMI-INTUITIONISTICTHEORIESOFSETS