Preliminaries Review Set existence? History Philosophy Statements reversing to WKL over RCA 0 The Infinity Lemma [can be applied in] the most diverse mathematical disciplines, since it often furnishes a useful method of carrying over certain results from the finite to the infinite . . . Some applications of the Infinity Lemma are analogous to applications of the Heine-Borel covering theorem. Because of this it seems interesting to remark that, from a certain standpoint, the Infinity Lemma can be thought of as the proper foundation of this covering theorem. König 1927/1936 9/25
Preliminaries Review Set existence? History Philosophy Statements reversing to WKL over RCA 0 The Infinity Lemma [can be applied in] the most diverse mathematical disciplines, since it often furnishes a useful method of carrying over certain results from the finite to the infinite . . . Some applications of the Infinity Lemma are analogous to applications of the Heine-Borel covering theorem. Because of this it seems interesting to remark that, from a certain standpoint, the Infinity Lemma can be thought of as the proper foundation of this covering theorem. König 1927/1936 Reversals to WKL 0 : § Heine-Borel Covering Lemma, Peano existence lemma, Brouwer fixed point theorem. 9/25
Preliminaries Review Set existence? History Philosophy Statements reversing to WKL over RCA 0 The Infinity Lemma [can be applied in] the most diverse mathematical disciplines, since it often furnishes a useful method of carrying over certain results from the finite to the infinite . . . Some applications of the Infinity Lemma are analogous to applications of the Heine-Borel covering theorem. Because of this it seems interesting to remark that, from a certain standpoint, the Infinity Lemma can be thought of as the proper foundation of this covering theorem. König 1927/1936 Reversals to WKL 0 : § Heine-Borel Covering Lemma, Peano existence lemma, Brouwer fixed point theorem. § Every countable consistent set of first-order sentences has a countable model. (Gödel) 9/25
Preliminaries Review Set existence? History Philosophy Statements reversing to WKL over RCA 0 The Infinity Lemma [can be applied in] the most diverse mathematical disciplines, since it often furnishes a useful method of carrying over certain results from the finite to the infinite . . . Some applications of the Infinity Lemma are analogous to applications of the Heine-Borel covering theorem. Because of this it seems interesting to remark that, from a certain standpoint, the Infinity Lemma can be thought of as the proper foundation of this covering theorem. König 1927/1936 Reversals to WKL 0 : § Heine-Borel Covering Lemma, Peano existence lemma, Brouwer fixed point theorem. § Every countable consistent set of first-order sentences has a countable model. (Gödel) § If ϕ p x q and ψ p x q are Σ 0 1 s.t. �D x p ϕ p x q ^ ψ p x qq , then there is X s.t. @ x p ϕ p x q Ñ x P X ^ ψ p x q Ñ x R X q . ( Σ 0 1 - Separation ) 9/25
Preliminaries Review Set existence? History Philosophy Existence simpliciter and conditional existence Orthodox view of “ontological commitment” (Quine 1948): A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. 10/25
Preliminaries Review Set existence? History Philosophy Existence simpliciter and conditional existence Orthodox view of “ontological commitment” (Quine 1948): A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. § E.g. I Σ 0 1 $ D x p Prime p x q ^ 17 ă x q or RCA 0 $ D X p x P X Ø Prime p x qq . 10/25
Preliminaries Review Set existence? History Philosophy Existence simpliciter and conditional existence Orthodox view of “ontological commitment” (Quine 1948): A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. § E.g. I Σ 0 1 $ D x p Prime p x q ^ 17 ă x q or RCA 0 $ D X p x P X Ø Prime p x qq . § Conditional existence assertions : § If there exists a tree greater than 100m, then there exists the trunk of such a tree. 10/25
Preliminaries Review Set existence? History Philosophy Existence simpliciter and conditional existence Orthodox view of “ontological commitment” (Quine 1948): A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. § E.g. I Σ 0 1 $ D x p Prime p x q ^ 17 ă x q or RCA 0 $ D X p x P X Ø Prime p x qq . § Conditional existence assertions : § If there exists a tree greater than 100m, then there exists the trunk of such a tree. § If there exists a greatest perfect number, then there exists the successor of such a number. 10/25
Preliminaries Review Set existence? History Philosophy Existence simpliciter and conditional existence Orthodox view of “ontological commitment” (Quine 1948): A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. § E.g. I Σ 0 1 $ D x p Prime p x q ^ 17 ă x q or RCA 0 $ D X p x P X Ø Prime p x qq . § Conditional existence assertions : § If there exists a tree greater than 100m, then there exists the trunk of such a tree. § If there exists a greatest perfect number, then there exists the successor of such a number. § If god exists, then there exists a cure for cancer. 10/25
Preliminaries Review Set existence? History Philosophy Existence simpliciter and conditional existence Orthodox view of “ontological commitment” (Quine 1948): A theory is committed to those and only those entities to which the bound variables of the theory must be capable of referring in order that the affirmations made in the theory be true. § E.g. I Σ 0 1 $ D x p Prime p x q ^ 17 ă x q or RCA 0 $ D X p x P X Ø Prime p x qq . § Conditional existence assertions : § If there exists a tree greater than 100m, then there exists the trunk of such a tree. § If there exists a greatest perfect number, then there exists the successor of such a number. § If god exists, then there exists a cure for cancer. § If S is consistent, then there exists M | ù S . 10/25
Preliminaries Review Set existence? History Philosophy Comprehension and separation § Two means of asserting the existence of sets: 1) By comprehension for a class of formulas Γ : ( Γ -AC ) For all ϕ p x q P Γ not containing X free, D X @ x p x P X Ø ϕ p x qq . 11/25
Preliminaries Review Set existence? History Philosophy Comprehension and separation § Two means of asserting the existence of sets: 1) By comprehension for a class of formulas Γ : ( Γ -AC ) For all ϕ p x q P Γ not containing X free, D X @ x p x P X Ø ϕ p x qq . 2) By separation for a class of formulas Γ : ( Γ - Sep ) For all ϕ p x q , ψ p x q P Γ not containing X free, �D x p ϕ p x q ^ ψ p x qq Ñ D X @ x p ϕ p x q Ñ x P X ^ ψ p x q Ñ x R X q . 11/25
Preliminaries Review Set existence? History Philosophy Comprehension and separation § Two means of asserting the existence of sets: 1) By comprehension for a class of formulas Γ : ( Γ -AC ) For all ϕ p x q P Γ not containing X free, D X @ x p x P X Ø ϕ p x qq . 2) By separation for a class of formulas Γ : ( Γ - Sep ) For all ϕ p x q , ψ p x q P Γ not containing X free, �D x p ϕ p x q ^ ψ p x qq Ñ D X @ x p ϕ p x q Ñ x P X ^ ψ p x q Ñ x R X q . § Recall the logical form of WKL : @ T p 0-1-Tree p T q & Infinite p T q Ñ D g p g is a path through T qq 11/25
Preliminaries Review Set existence? History Philosophy Comprehension and separation § Two means of asserting the existence of sets: 1) By comprehension for a class of formulas Γ : ( Γ -AC ) For all ϕ p x q P Γ not containing X free, D X @ x p x P X Ø ϕ p x qq . 2) By separation for a class of formulas Γ : ( Γ - Sep ) For all ϕ p x q , ψ p x q P Γ not containing X free, �D x p ϕ p x q ^ ψ p x qq Ñ D X @ x p ϕ p x q Ñ x P X ^ ψ p x q Ñ x R X q . § Recall the logical form of WKL : @ T p 0-1-Tree p T q & Infinite p T q Ñ D g p g is a path through T qq § WKL does not have the “surface grammar” of either 1) or 2). 11/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? 12/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? § Note that since ACA 0 $ WKL , such a Γ would have to be a sub-schema of arithmetical comprehension. 12/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? § Note that since ACA 0 $ WKL , such a Γ would have to be a sub-schema of arithmetical comprehension. § But if RCA 0 $ WKL Ø Γ -AC , then there is a single arithmetical formula ϕ p x, X q s.t. (1) RCA 0 $ WKL Ø @ X D Y @ n p n P Y Ø ϕ p n, X qq . 12/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? § Note that since ACA 0 $ WKL , such a Γ would have to be a sub-schema of arithmetical comprehension. § But if RCA 0 $ WKL Ø Γ -AC , then there is a single arithmetical formula ϕ p x, X q s.t. (1) RCA 0 $ WKL Ø @ X D Y @ n p n P Y Ø ϕ p n, X qq . § In this case by extensionality p 2 q RCA 0 $ WKL Ø @ X D ! Y @ n p n P Y Ø ϕ p n, X qq 12/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? § Note that since ACA 0 $ WKL , such a Γ would have to be a sub-schema of arithmetical comprehension. § But if RCA 0 $ WKL Ø Γ -AC , then there is a single arithmetical formula ϕ p x, X q s.t. (1) RCA 0 $ WKL Ø @ X D Y @ n p n P Y Ø ϕ p n, X qq . § In this case by extensionality p 2 q RCA 0 $ WKL Ø @ X D ! Y @ n p n P Y Ø ϕ p n, X qq § Simpson, Tanaka, Yamazaki (2002): for all arith. ψ p X, Y q (3) If WKL 0 $ @ X D ! Y ψ p X, Y q , then RCA 0 $ @ X D Y ψ p X, Y q . 12/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? § Note that since ACA 0 $ WKL , such a Γ would have to be a sub-schema of arithmetical comprehension. § But if RCA 0 $ WKL Ø Γ -AC , then there is a single arithmetical formula ϕ p x, X q s.t. (1) RCA 0 $ WKL Ø @ X D Y @ n p n P Y Ø ϕ p n, X qq . § In this case by extensionality p 2 q RCA 0 $ WKL Ø @ X D ! Y @ n p n P Y Ø ϕ p n, X qq § Simpson, Tanaka, Yamazaki (2002): for all arith. ψ p X, Y q (3) If WKL 0 $ @ X D ! Y ψ p X, Y q , then RCA 0 $ @ X D Y ψ p X, Y q . § (2) implies WKL 0 $ @ X D ! Y @ n p n P Y Ø ϕ p n, X qq and hence by (3) RCA 0 $ @ X D Y @ n p n P Y Ø ϕ p n, X qq . 12/25
Preliminaries Review Set existence? History Philosophy WKL and comprehension Is there a Γ such that RCA 0 $ WKL Ø Γ -AC ? § Note that since ACA 0 $ WKL , such a Γ would have to be a sub-schema of arithmetical comprehension. § But if RCA 0 $ WKL Ø Γ -AC , then there is a single arithmetical formula ϕ p x, X q s.t. (1) RCA 0 $ WKL Ø @ X D Y @ n p n P Y Ø ϕ p n, X qq . § In this case by extensionality p 2 q RCA 0 $ WKL Ø @ X D ! Y @ n p n P Y Ø ϕ p n, X qq § Simpson, Tanaka, Yamazaki (2002): for all arith. ψ p X, Y q (3) If WKL 0 $ @ X D ! Y ψ p X, Y q , then RCA 0 $ @ X D Y ψ p X, Y q . § (2) implies WKL 0 $ @ X D ! Y @ n p n P Y Ø ϕ p n, X qq and hence by (3) RCA 0 $ @ X D Y @ n p n P Y Ø ϕ p n, X qq . § But then RCA 0 $ WKL by (1) . Contradiction. 12/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . 13/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . § The Kleene tree T S is defined as t P T iff @ x, y ă lh p t qp Proof S p y, x q Ñ t p x q “ 1 ^ Proof S p y, 9 � x q Ñ t p x q “ 0 q 13/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . § The Kleene tree T S is defined as t P T iff @ x, y ă lh p t qp Proof S p y, x q Ñ t p x q “ 1 ^ Proof S p y, 9 � x q Ñ t p x q “ 0 q § If S is consistent, then T S is infinite. 13/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . § The Kleene tree T S is defined as t P T iff @ x, y ă lh p t qp Proof S p y, x q Ñ t p x q “ 1 ^ Proof S p y, 9 � x q Ñ t p x q “ 0 q § If S is consistent, then T S is infinite. § Kleene (1952a): If S is essentially undecidable, then T S has no recursive path. 13/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . § The Kleene tree T S is defined as t P T iff @ x, y ă lh p t qp Proof S p y, x q Ñ t p x q “ 1 ^ Proof S p y, 9 � x q Ñ t p x q “ 0 q § If S is consistent, then T S is infinite. § Kleene (1952a): If S is essentially undecidable, then T S has no recursive path. § But Modulo RCA 0 , “ T S exists” is a constructive claim. 13/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . § The Kleene tree T S is defined as t P T iff @ x, y ă lh p t qp Proof S p y, x q Ñ t p x q “ 1 ^ Proof S p y, 9 � x q Ñ t p x q “ 0 q § If S is consistent, then T S is infinite. § Kleene (1952a): If S is essentially undecidable, then T S has no recursive path. § But Modulo RCA 0 , “ T S exists” is a constructive claim. § So modulo, WKL and Σ 0 1 - Sep both have the form 13/25
Preliminaries Review Set existence? History Philosophy WKL and separation Over RCA 0 , WKL is equivalent to Σ 0 1 - Sep . § Canonical example: Let S be a recursively axiomatized theory. ϕ p x q “ D y Proof S p y, x q , ψ p x q “ D y Proof S p y, 9 � x q . § The Kleene tree T S is defined as t P T iff @ x, y ă lh p t qp Proof S p y, x q Ñ t p x q “ 1 ^ Proof S p y, 9 � x q Ñ t p x q “ 0 q § If S is consistent, then T S is infinite. § Kleene (1952a): If S is essentially undecidable, then T S has no recursive path. § But Modulo RCA 0 , “ T S exists” is a constructive claim. § So modulo, WKL and Σ 0 1 - Sep both have the form If something X exists (constructive), then something Y exists (possibly non-constructive). 13/25
Preliminaries Review Set existence? History Philosophy Is WKL a “set existence axiom”? (3) § Observations: 1) While WKL is not a set existence principle simpliciter , it is a conditional set existence principle. 2) RCA 0 proves the existence of all recursive trees. 3) So modulo RCA 0 , WKL does have “existential import”. 14/25
Preliminaries Review Set existence? History Philosophy Is WKL a “set existence axiom”? (3) § Observations: 1) While WKL is not a set existence principle simpliciter , it is a conditional set existence principle. 2) RCA 0 proves the existence of all recursive trees. 3) So modulo RCA 0 , WKL does have “existential import”. § Question: Is the import “innocent”? § Finitism: no, because there are no infinite trees (or paths). § Predicativism: yes, because ACA 0 $ WKL . § “Finitistic reductionism”: yes, because of conservativity. (?) § Constructivism: complicated, because of the minimal non-constructivity of WKL . 14/25
Preliminaries Review Set existence? History Philosophy Is WKL a “set existence axiom”? (3) § Observations: 1) While WKL is not a set existence principle simpliciter , it is a conditional set existence principle. 2) RCA 0 proves the existence of all recursive trees. 3) So modulo RCA 0 , WKL does have “existential import”. § Question: Is the import “innocent”? § Finitism: no, because there are no infinite trees (or paths). § Predicativism: yes, because ACA 0 $ WKL . § “Finitistic reductionism”: yes, because of conservativity. (?) § Constructivism: complicated, because of the minimal non-constructivity of WKL . § Plan: Use the equivalence of WKL and the Completeness Theorem over RCA 0 to illustrate what’s at issue with respect to Hilbert’s dictum “consistency implies existence”. 14/25
Preliminaries Review Set existence? History Philosophy Frege vs Hilbert (1899) on model existence Frege’s dictum: “Existence entails consistency.” [What] I call axioms [are] propositions that are true but are not proved because our knowledge of them flows from a source very different from the logical source, a source which might be called spatial intuition. From the truth of the axioms it follows that they do not contradict one another. 15/25
Preliminaries Review Set existence? History Philosophy Frege vs Hilbert (1899) on model existence Frege’s dictum: “Existence entails consistency.” [What] I call axioms [are] propositions that are true but are not proved because our knowledge of them flows from a source very different from the logical source, a source which might be called spatial intuition. From the truth of the axioms it follows that they do not contradict one another. Hilbert’s dictum: “Consistency entails existence.” I found it very interesting to read this very sentence in your letter, for as long as I have been thinking, writing and lecturing on these things, I have been saying the exact reverse: if the arbitrarily given axioms do not contradict each other with all their consequences, then they are true and the things defined by the axioms exist . This is for me the criterion of truth and existence. 15/25
Preliminaries Review Set existence? History Philosophy Gödel 1929 L.E.J. Brouwer, in particular, has emphatically stressed that from the consistency of an axiom system we cannot conclude without further ado that a model can be constructed. 16/25
Preliminaries Review Set existence? History Philosophy Gödel 1929 L.E.J. Brouwer, in particular, has emphatically stressed that from the consistency of an axiom system we cannot conclude without further ado that a model can be constructed. But one might perhaps think that the existence of the notions introduced through an axiom system is to be defined outright by the consistency of the axioms and that, therefore, a proof [of completeness] has to be rejected out of hand . . . 16/25
Preliminaries Review Set existence? History Philosophy Gödel 1929 L.E.J. Brouwer, in particular, has emphatically stressed that from the consistency of an axiom system we cannot conclude without further ado that a model can be constructed. But one might perhaps think that the existence of the notions introduced through an axiom system is to be defined outright by the consistency of the axioms and that, therefore, a proof [of completeness] has to be rejected out of hand . . . This definition . . . however, manifestly presupposes the axiom that every mathematical problem is solvable . . . For, if the unsolvability of some problem . . . were proved , then . . . there would follow the existence of two non-isomorphic realizations of the axiom system . . . 16/25
Preliminaries Review Set existence? History Philosophy Gödel 1929 L.E.J. Brouwer, in particular, has emphatically stressed that from the consistency of an axiom system we cannot conclude without further ado that a model can be constructed. But one might perhaps think that the existence of the notions introduced through an axiom system is to be defined outright by the consistency of the axioms and that, therefore, a proof [of completeness] has to be rejected out of hand . . . This definition . . . however, manifestly presupposes the axiom that every mathematical problem is solvable . . . For, if the unsolvability of some problem . . . were proved , then . . . there would follow the existence of two non-isomorphic realizations of the axiom system . . . These reflections . . . are intended only to properly illuminated the difficulties that would be connected with such a definition of the notion of existence, without any definitive assertion being made about its possibility or impossibility. 1929, p. 63 16/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . 17/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . Gödel (1929) constructed a sequence of Herbrand models for the Skolem normal form of ϕ . 17/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . Gödel (1929) constructed a sequence of Herbrand models for the Skolem normal form of ϕ . § Hilbert & Bernays (1934) formalized Gödel’s proof in Z 2 and thus obtained arithmetical models M | ù ϕ such that M “ N and P M Ď N a i . i 17/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . Gödel (1929) constructed a sequence of Herbrand models for the Skolem normal form of ϕ . § Hilbert & Bernays (1934) formalized Gödel’s proof in Z 2 and thus obtained arithmetical models M | ù ϕ such that M “ N and P M Ď N a i . i § Kleene (1952) observed that since the construction is recursive in the Σ 0 1 -definition of derivability, the P M are ∆ 0 2 -definable. i 17/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . Gödel (1929) constructed a sequence of Herbrand models for the Skolem normal form of ϕ . § Hilbert & Bernays (1934) formalized Gödel’s proof in Z 2 and thus obtained arithmetical models M | ù ϕ such that M “ N and P M Ď N a i . i § Kleene (1952) observed that since the construction is recursive in the Σ 0 1 -definition of derivability, the P M are ∆ 0 2 -definable. i § Kriesel (1953) and Mostowski (1953) observed that this couldn’t be strengthened to ∆ 0 1 because there are finite theories with no recursive models. 17/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . Gödel (1929) constructed a sequence of Herbrand models for the Skolem normal form of ϕ . § Hilbert & Bernays (1934) formalized Gödel’s proof in Z 2 and thus obtained arithmetical models M | ù ϕ such that M “ N and P M Ď N a i . i § Kleene (1952) observed that since the construction is recursive in the Σ 0 1 -definition of derivability, the P M are ∆ 0 2 -definable. i § Kriesel (1953) and Mostowski (1953) observed that this couldn’t be strengthened to ∆ 0 1 because there are finite theories with no recursive models. § Subsequent work on Π 0 1 -classes and the basis theorems grew out of this – e.g. Shoenfield (1960) “The degrees of models”. 17/25
Preliminaries Review Set existence? History Philosophy The arithmetized completeness theorem (1934-1972) § Suppose { $ Fol � ϕ . Gödel (1929) constructed a sequence of Herbrand models for the Skolem normal form of ϕ . § Hilbert & Bernays (1934) formalized Gödel’s proof in Z 2 and thus obtained arithmetical models M | ù ϕ such that M “ N and P M Ď N a i . i § Kleene (1952) observed that since the construction is recursive in the Σ 0 1 -definition of derivability, the P M are ∆ 0 2 -definable. i § Kriesel (1953) and Mostowski (1953) observed that this couldn’t be strengthened to ∆ 0 1 because there are finite theories with no recursive models. § Subsequent work on Π 0 1 -classes and the basis theorems grew out of this – e.g. Shoenfield (1960) “The degrees of models”. § Jockush & Soare (1972) showed that every recursive theory i q 1 “ 0 1 . has a low model – i.e. deg p P M 17/25
Preliminaries Review Set existence? History Philosophy Completeness for intuitionistic logic ( HPC ) § Kleene (1952a) used the Kleene tree to show that Brouwer’s Fan Theorem fails if restricted to recursive choice sequences. 18/25
Preliminaries Review Set existence? History Philosophy Completeness for intuitionistic logic ( HPC ) § Kleene (1952a) used the Kleene tree to show that Brouwer’s Fan Theorem fails if restricted to recursive choice sequences. § Beth (1947, 1956) proposed a completeness proof for ( HPC ) based on Beth models – i.e. infinite “tableau-like” trees. 18/25
Preliminaries Review Set existence? History Philosophy Completeness for intuitionistic logic ( HPC ) § Kleene (1952a) used the Kleene tree to show that Brouwer’s Fan Theorem fails if restricted to recursive choice sequences. § Beth (1947, 1956) proposed a completeness proof for ( HPC ) based on Beth models – i.e. infinite “tableau-like” trees. § Gödel & Kreisel (1958, 1961, 1962) raised doubts about proof. 18/25
Preliminaries Review Set existence? History Philosophy Completeness for intuitionistic logic ( HPC ) § Kleene (1952a) used the Kleene tree to show that Brouwer’s Fan Theorem fails if restricted to recursive choice sequences. § Beth (1947, 1956) proposed a completeness proof for ( HPC ) based on Beth models – i.e. infinite “tableau-like” trees. § Gödel & Kreisel (1958, 1961, 1962) raised doubts about proof. § Kreisel (1970) showed “ HPC is complete” implies the negation of intuitionistic Church’s Thesis ( CT 0 ). 18/25
Preliminaries Review Set existence? History Philosophy Completeness for intuitionistic logic ( HPC ) § Kleene (1952a) used the Kleene tree to show that Brouwer’s Fan Theorem fails if restricted to recursive choice sequences. § Beth (1947, 1956) proposed a completeness proof for ( HPC ) based on Beth models – i.e. infinite “tableau-like” trees. § Gödel & Kreisel (1958, 1961, 1962) raised doubts about proof. § Kreisel (1970) showed “ HPC is complete” implies the negation of intuitionistic Church’s Thesis ( CT 0 ). § “This shows that the completeness of HPC is a rather dubious commodity.” van Dalen (1973), p. 87 18/25
Preliminaries Review Set existence? History Philosophy Completeness for intuitionistic logic ( HPC ) § Kleene (1952a) used the Kleene tree to show that Brouwer’s Fan Theorem fails if restricted to recursive choice sequences. § Beth (1947, 1956) proposed a completeness proof for ( HPC ) based on Beth models – i.e. infinite “tableau-like” trees. § Gödel & Kreisel (1958, 1961, 1962) raised doubts about proof. § Kreisel (1970) showed “ HPC is complete” implies the negation of intuitionistic Church’s Thesis ( CT 0 ). § “This shows that the completeness of HPC is a rather dubious commodity.” van Dalen (1973), p. 87 § Yamazaki (2001) showed that the strong completeness of HPC wrt Kripke models is equivalent over RCA 0 to ACA 0 . 18/25
Preliminaries Review Set existence? History Philosophy Friedman (1974) ACA 0 is obviously sufficient to explicitly define a nonrecursive set (e.g., the jump). WKL 0 is not sufficient, and so the following theorem provides us with an illustration of our theme II. 19/25
Preliminaries Review Set existence? History Philosophy Friedman (1974) ACA 0 is obviously sufficient to explicitly define a nonrecursive set (e.g., the jump). WKL 0 is not sufficient, and so the following theorem provides us with an illustration of our theme II. Theorem 1.7 Suppose A p X q is a Σ 1 1 -formula with X as the only free set variable and WKL 0 $ pD X qp A p X q ^ X is not recursive q then WKL 0 $ @ Y D X p A p X q ^ X is not recursive ^ @ n p Y n ‰ X qq . 19/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q 20/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q § To show that { | ù ϕ – i.e. ϕ is not a logical truth à la Tarski – requires that D M s.t. M | ù � ϕ . 20/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q § To show that { | ù ϕ – i.e. ϕ is not a logical truth à la Tarski – requires that D M s.t. M | ù � ϕ . § Such an M must have an infinite domain . 20/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q § To show that { | ù ϕ – i.e. ϕ is not a logical truth à la Tarski – requires that D M s.t. M | ù � ϕ . § Such an M must have an infinite domain . § Similarly, to invalidate @ xR p x, x q requires the existence of an irreflexive relation. 20/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q § To show that { | ù ϕ – i.e. ϕ is not a logical truth à la Tarski – requires that D M s.t. M | ù � ϕ . § Such an M must have an infinite domain . § Similarly, to invalidate @ xR p x, x q requires the existence of an irreflexive relation. § Etchemendy: the extensional adequacy of Tarski’s definition of logical truth has “extralogical” – i.e. set theoretic – commitments. 20/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q § To show that { | ù ϕ – i.e. ϕ is not a logical truth à la Tarski – requires that D M s.t. M | ù � ϕ . § Such an M must have an infinite domain . § Similarly, to invalidate @ xR p x, x q requires the existence of an irreflexive relation. § Etchemendy: the extensional adequacy of Tarski’s definition of logical truth has “extralogical” – i.e. set theoretic – commitments. Similarly for the Completeness Theorem. 20/25
Preliminaries Review Set existence? History Philosophy Etchemendy (1990) contra Tarksi (1935) on logical truth § Consider the following sentence: ϕ “ p@ x @ y @ z p R p x, y q ^ R p y, z q Ñ R p x, z qq ^ @ x � R p x, x qq Ñ �@ x D yR p x, y q § To show that { | ù ϕ – i.e. ϕ is not a logical truth à la Tarski – requires that D M s.t. M | ù � ϕ . § Such an M must have an infinite domain . § Similarly, to invalidate @ xR p x, x q requires the existence of an irreflexive relation. § Etchemendy: the extensional adequacy of Tarski’s definition of logical truth has “extralogical” – i.e. set theoretic – commitments. Similarly for the Completeness Theorem. § Question: How far do these commitments extend? 20/25
Preliminaries Review Set existence? History Philosophy From consistency to non-constructive existence § Fnitely axiomatizable theories with no recursive models: § EFA ` � Con p EFA q (Tennenbaum 1959, MacAloon 1982) § GB ´ Inf “ t ϕ 1 , . . . , ϕ n u (Rabin 1958) 21/25
Preliminaries Review Set existence? History Philosophy From consistency to non-constructive existence § Fnitely axiomatizable theories with no recursive models: § EFA ` � Con p EFA q (Tennenbaum 1959, MacAloon 1982) § GB ´ Inf “ t ϕ 1 , . . . , ϕ n u (Rabin 1958) § In order to show § { | ù EFA Ñ Con p EFA q § { | ù p ϕ 1 ^ . . . ^ ϕ n ´ 1 q Ñ � ϕ n requires the existence of non-recursive countermodels. 21/25
Preliminaries Review Set existence? History Philosophy From consistency to non-constructive existence § Fnitely axiomatizable theories with no recursive models: § EFA ` � Con p EFA q (Tennenbaum 1959, MacAloon 1982) § GB ´ Inf “ t ϕ 1 , . . . , ϕ n u (Rabin 1958) § In order to show § { | ù EFA Ñ Con p EFA q § { | ù p ϕ 1 ^ . . . ^ ϕ n ´ 1 q Ñ � ϕ n requires the existence of non-recursive countermodels. § So the extra-logical commitments implicit in Tarski’s definitions (and Completeness) extend to non-recursive sets. 21/25
Preliminaries Review Set existence? History Philosophy From consistency to non-constructive existence § Fnitely axiomatizable theories with no recursive models: § EFA ` � Con p EFA q (Tennenbaum 1959, MacAloon 1982) § GB ´ Inf “ t ϕ 1 , . . . , ϕ n u (Rabin 1958) § In order to show § { | ù EFA Ñ Con p EFA q § { | ù p ϕ 1 ^ . . . ^ ϕ n ´ 1 q Ñ � ϕ n requires the existence of non-recursive countermodels. § So the extra-logical commitments implicit in Tarski’s definitions (and Completeness) extend to non-recursive sets. § Revised Hilbert’s dictum: “Consistency implies existence non-constructively.” 21/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. 22/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. § RCA 0 $ Comp Ø WKL 22/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. § RCA 0 $ Comp Ø WKL § On the “minimal non-constructivity” of WKL 0 à la Friedman: ù WKL 0 then, there exists M 1 Ď ω M such that § If M | M 1 | ù WKL 0 and S M 1 Ĺ S M . 22/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. § RCA 0 $ Comp Ø WKL § On the “minimal non-constructivity” of WKL 0 à la Friedman: ù WKL 0 then, there exists M 1 Ď ω M such that § If M | M 1 | ù WKL 0 and S M 1 Ĺ S M . § Rec “ Ş t S M : M | ù WKL 0 u . 22/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. § RCA 0 $ Comp Ø WKL § On the “minimal non-constructivity” of WKL 0 à la Friedman: ù WKL 0 then, there exists M 1 Ď ω M such that § If M | M 1 | ù WKL 0 and S M 1 Ĺ S M . § Rec “ Ş t S M : M | ù WKL 0 u . § There exists an ω -model M | ù WKL 0 such that all X P S M are low – i.e. X 1 “ 0 1 . 22/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. § RCA 0 $ Comp Ø WKL § On the “minimal non-constructivity” of WKL 0 à la Friedman: ù WKL 0 then, there exists M 1 Ď ω M such that § If M | M 1 | ù WKL 0 and S M 1 Ĺ S M . § Rec “ Ş t S M : M | ù WKL 0 u . § There exists an ω -model M | ù WKL 0 such that all X P S M are low – i.e. X 1 “ 0 1 . § If x A i : i P N y are non-recursive, then exists ω -model M | ù WKL 0 s.t. A i R S M , @ i P N . 22/25
Preliminaries Review Set existence? History Philosophy Minimal non-constructivity Completeness formalized in L 2 : p Comp q @ S p Con p S q Ñ D M @ n p Prov S p n q Ñ M p n q “ 1 qq where M p x q satisfies Tarski-like clauses. § RCA 0 $ Comp Ø WKL § On the “minimal non-constructivity” of WKL 0 à la Friedman: ù WKL 0 then, there exists M 1 Ď ω M such that § If M | M 1 | ù WKL 0 and S M 1 Ĺ S M . § Rec “ Ş t S M : M | ù WKL 0 u . § There exists an ω -model M | ù WKL 0 such that all X P S M are low – i.e. X 1 “ 0 1 . § If x A i : i P N y are non-recursive, then exists ω -model M | ù WKL 0 s.t. A i R S M , @ i P N . § So while Completeness entails non-constructive set existence, it does not require existence of specific non-recursive sets. 22/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) § 1.i) is a belief about a specific number (i.e. 33550336). 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) § 1.i) is a belief about a specific number (i.e. 33550336). § 1.ii) is a belief about an existential proposition (i.e. a “bare” existence claim). 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) § 1.i) is a belief about a specific number (i.e. 33550336). § 1.ii) is a belief about an existential proposition (i.e. a “bare” existence claim). 2) John believes that there are spies. § Two readings: 2.i) D x Bel p j, x Spy p 9 x q y q ( de re ) 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) § 1.i) is a belief about a specific number (i.e. 33550336). § 1.ii) is a belief about an existential proposition (i.e. a “bare” existence claim). 2) John believes that there are spies. § Two readings: 2.i) D x Bel p j, x Spy p 9 x q y q ( de re ) 2.ii) Bel p j, x D xSpy p x q y q ( de dicto ) 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) § 1.i) is a belief about a specific number (i.e. 33550336). § 1.ii) is a belief about an existential proposition (i.e. a “bare” existence claim). 2) John believes that there are spies. § Two readings: 2.i) D x Bel p j, x Spy p 9 x q y q ( de re ) 2.ii) Bel p j, x D xSpy p x q y q ( de dicto ) § 2.i) is a belief about a specific person requiring knowledge of identifying features – e.g. height, gender, nationality. 23/25
Preliminaries Review Set existence? History Philosophy Belief, de dicto and de re 1) John believes that there exists a perfect number ą 100000 . § Two readings: 1.i) D x Bel p j, x Perfect p 9 x q ^ 9 x ą 1000000 y q ( de re ) 1.ii) Bel p j, x D xPerfect p x q ^ x ą 1000000 y q ( de dicto ) § 1.i) is a belief about a specific number (i.e. 33550336). § 1.ii) is a belief about an existential proposition (i.e. a “bare” existence claim). 2) John believes that there are spies. § Two readings: 2.i) D x Bel p j, x Spy p 9 x q y q ( de re ) 2.ii) Bel p j, x D xSpy p x q y q ( de dicto ) § 2.i) is a belief about a specific person requiring knowledge of identifying features – e.g. height, gender, nationality. § 2.ii) is a belief about a “bare” existential proposition. 23/25
Preliminaries Review Set existence? History Philosophy Ontological commitment, de dicto and de re 3) John is a finitist/predicativist/constructivist, . . . 24/25
Preliminaries Review Set existence? History Philosophy Ontological commitment, de dicto and de re 3) John is a finitist/predicativist/constructivist, . . . i) He is committed to the existence of type Φ p X q sets. ii) But denies/is agnostic about the existence non- Φ p X q sets. 24/25
Preliminaries Review Set existence? History Philosophy Ontological commitment, de dicto and de re 3) John is a finitist/predicativist/constructivist, . . . i) He is committed to the existence of type Φ p X q sets. ii) But denies/is agnostic about the existence non- Φ p X q sets. § E.g. Φ p X q “ finite, recursive, arithmetical, hyperarithmetical, . . . , countable, Borel, analytic, coanalytic, projective, . . . 24/25
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