The Consistency of the Continuum Hypothesis Annals of Mathematical Studies, No. 3 Princeton University Press Princeton, N.J., 1940.


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1 TWO COMPUTATIONAL IDEAS Computations with Sets Union, intersection: computable Powerset: not computable Basic operations Absoluteness of Formulas A formula in the language of set theory is absolute if its truth value in a transitive class M, for some assignment v of objects from M to its free variables, depends only on v, but not on M. Both ideas introduced in: K. Gödel The Consistency of the Continuum Hypothesis Annals of Mathematical Studies, No. 3 Princeton University Press Princeton, N.J., 1940.
2 A Syntactic Approximation Every atomic formula is in 0. If ϕ and ψ are in 0, then so are ϕ, ϕ ψ, and ϕ ψ. If x and y are two different variables, and ϕ is in 0, then so is x yϕ.
3 Domain Independence Let σ be a signature without function symbols. Assume that σ includes P = {P 1,..., P n }. A query ϕ(x 1...., x n ) in σ is called P d.i. ( P domainindependent), if whenever: 1. S 1 and S 2 are structures for σ 2. S 1 is a substructure of S 2 3. Each P i P has in S 1 and S 2 the same interpretation then for all a 1 S 2,..., a n S 2 : S 2 = ϕ(a 1,..., a n ) a 1 S 1... a n S 1 S 1 = ϕ(a 1,..., a n )
4 The Class SS( P ) P i (t 1,..., t ni ) SS( P ) in case P i is in P. x = c and c = x are in SS( P ) ϕ ψ SS( P ) if ϕ SS( P ), ψ SS( P ), and the two formulas have the same free variables xϕ SS( P ) if ϕ SS( P ). If ϕ = ϕ 1 ϕ 2... ϕ k, then ϕ SS( P ) if the following conditions are met: 1. For each i, either ϕ i is atomic, or ϕ i is in SS( P ), or ϕ i is a negation of a formula of either type. 2. Every free variable x of ϕ is limited in ϕ. This means that there exists 1 i k such that x is free in ϕ i, and either ϕ i SS( P ), or there exists y which is already limited in ϕ, and ϕ i {x = y, y = x}.
5 Partial Domain Independence Let σ be a signature without function symbols. Assume that σ includes P = {P 1,..., P n }. A query ϕ(x 1...., x n, y 1...., y k ) in σ is called P d.i. with respect to {x 1,..., x n }, if whenever: 1. S 1 and S 2 are structures for σ 2. S 1 is a substructure of S 2 3. Each P i P has in S 1 and S 2 the same interpretation then for all a 1 S 2,..., a n S 2, b 1 S 1,..., b k S 1 : S 2 = ϕ(a 1,..., a n, b 1,..., b k ) a 1 S 1... a n S 1 S 1 = ϕ(a 1,..., a n, b 1,..., b k )
6 A Syntactic Approximation P 0. If ϕ P X and Z X, then ϕ P Z. 1. ϕ F v(ϕ) if ϕ is p(t 1,..., t n ) (where p P ). 2. x x P {x}, t = x P {x}, and x = t P {x} if x F v(t). 3. ϕ P if ϕ P. 4. ϕ ψ P X if ϕ P X and ψ P X. 5. ϕ ψ P X Y if ϕ P X, ψ P Y, and either Y F v(ϕ) = or X F v(ψ) =. 6. yϕ P X {y} if y X and ϕ P X. Notes: {ϕ ϕ P F v(ϕ)} strictly extends SS( P ) If ϕ P {x 1,..., x n }, and ψ, then: x 1... x n (ϕ ψ) P and x 1... x n (ϕ ψ) P.
7 Rich Languages for Set Theories Terms: Every variable is a term. The constant is a term. {x ϕ} is a term in case x is a variable, and ϕ is a formula such that ϕ ST {x}. F v({x ϕ}) = F v(ϕ) {x} Formulas: If t, s are terms than t = s and t s are formulas. If ϕ and ψ are formulas, and x and y are variables, then ϕ, (ϕ ψ), (ϕ ψ), and xϕ are formulas.
8 RST: Rudimentary Set Theory The safety relation: 1. ϕ RST if ϕ is atomic. 2. ϕ RST if ϕ RST. 3. ϕ RST {x} if x F v(t) and ϕ {x x, x = t, t = x, x t} 4. ϕ ψ RST X if ϕ RST X and ψ RST X. 5. ϕ ψ RST X Y if ϕ RST X, ψ RST Y and either Y F v(ϕ) = or X F v(ψ) =. 6. yϕ RST X {y} if y X and ϕ RST X. Axioms: Extensionality: y(y = {x x y}) Comprehension: x(x {x ϕ} ϕ)
9 Examples {t 1,..., t n } = Df {x x = t 1... x = t n } t, s = Df {{t}, {t, s}} {x t ϕ} = Df {x x t ϕ}, provided ϕ Rud {t x s} = Df {y x.x s y = t} s t = Df {x a b.a s b t x = a, b } s t = Df {x x s x t} s t = Df {x x s x t} S(x) = Df x {x} t = Df {x y.y t x y} t = Df {x x t y(y t (x y))} λx s.t = Df { x, t x s} (where x F v(s)) f(x) = Df F where F = {y z v(z f v z y v z = x, y )}
10 Basic Properties of RST Theorem 1. Let T be an extension of RST. 1. If t(x 1,..., x n ) is a term of RST, then t(a 1,..., a n ) has the same interpretation (which should be a set) in all transitive models of T which contain a 1,..., a n. 2. If ϕ(y 1,..., y n, x 1,..., x k ) RST {x 1,..., x k }, then { x 1,..., x k ϕ(a 1,..., a n, x 1,..., x k } has the same extension (which should be a set) in all transitive models of T which contain a 1,..., a n. Theorem 2 1. If F is an nary rudimentary function, then there exists a formula ϕ s. t.: (a) F v(ϕ) = {y, x 1,..., x n } (b) ϕ RST {y} (c) F (x 1,..., x n ) = {y ϕ}. 2. If ϕ is a formula such that: (a) F v(ϕ) = {y 1,..., y k, x 1,..., x n } (b) ϕ RST {y 1,..., y k } then there exists a rudimentary function F such that: F (x 1,..., x n ) = { y 1,..., y k ϕ}
11 The Comprehension Axioms of ZF The addition of each of the comprehension axioms of ZF to RST is equivalent to adding to the definition of Rud a certain syntactic condition: Separation: ϕ for every formula ϕ Replacement: yϕ y(ϕ ψ) X if ψ X, and X F v(ϕ) =. Powerset: y(y x ϕ) (X {y}) {x} if ϕ X, y X, and x X. The Other Axioms of ZF Foundations: ( x( y(y x ϕ{y/x}) ϕ)) xϕ Choice: xϕ ϕ{εxϕ/x}
12 Predicative Set Theory P ZF Terms and Axioms: Like in RST. Formulas: If ϕ is a formula, t and s are terms, and x and y are distinct variables then (T C x,y ϕ)(t, s) is a formula F v((t C x,y ϕ)(t, s)) = (F v(ϕ) {x, y}) F v(t) F v(s) The Safety Relation P ZF : (T C x,y ϕ)(x, y) P ZF X provided ϕ P ZF X, and {x, y} X. Logic: T CLogic An explanation: (T C x,y ϕ)(x, y) is intuitively equivalent to the infinitary disjunction: ϕ(x, y) w 1.ϕ(x, w 1 ) ϕ(w 1, y) w 1 w 2.ϕ(x, w 1 ) ϕ(w 1, w 2 ) ϕ(w 2, y)... where all w 1, w 2,..., are new variables.
13 A General Framework A d.i.signature is a pair (σ, F ), where σ is an ordinary firstorder signature, and F is a function which assigns to every nary symbol s from σ (other than equality) a subset of P({1,..., n}). Let S 1 and S 2 be two structures for σ s.t. S 1 S 2. S 2 is called a (σ, F ) extension of S 1 if the following conditions are satisfied: If p σ is a predicate symbol of arity n, I F (p), and a 1,..., a n are elements of S 2 such that a i S 1 in case i I, then S 2 = p(a 1,..., a n ) iff a i S 1 for all i, and S 1 = p(a 1,..., a n ). If f σ is a function symbol of arity n, a 1,..., a n S 1, and b is the value of f(a 1,..., a n ) in S 2, then b S 1, and b is the value of f(a 1,..., a n ) in S 1. Moreover: if I F (f), and a 1,..., a n are elements of S 2 such that a i S 1 in case i I, then S 2 = b = f(a 1,..., a n ) iff a i S 1 for all i, and S 1 = b = f(a 1,..., a n ).
14 A General Framework (II) A formula ϕ of σ is called (σ, F ) d.i. w.r.t. X (ϕ di (σ,f ) X) if whenever S 2 is a (σ, F ) extension of S 1, and ϕ results from ϕ by substituting values from S 1 for the free variables of ϕ that are not in X, then the sets of tuples which satisfy ϕ in S 1 and in S 2 are identical. A formula ϕ of σ is called (σ, F ) d.i. if ϕ di (σ,f ) (σ, F ) absolute if ϕ di (σ,f ). F v(ϕ), and Examples: Let σ be a signature which includes P, and has no function symbols. Assume that the arity of P i is n i, and define F (P i ) = {{1,..., n i }}. Then ϕ is (σ, F ) d.i. w.r.t. X iff it is P d.i. w.r.t. X. Let σ ZF = { } and let F ZF ( ) = {{1}}. In this case the universe V is a (σ ZF, F ZF ) extension of the transitive sets and classes. Therefore a formula is σ ZF  absolute iff it is absolute in the usual sense of set theory.
15 A General Framework (III) The relation (σ,f ) is inductively defined as follows: 0. If ϕ (σ,f ) X and Z X, then ϕ (σ,f ) Z. 1a. If p is an nary predicate symbol of σ; x 1,..., x n are n distinct variables, and {i 1,..., i k } is in F (p), then p(x 1,..., x n ) (σ,f ) {x i1,..., x ik }. 1b. If f is an nary function symbol of σ; y, x 1,..., x n are n + 1 distinct variables, and {i 1,..., i k } F (f), then y = f(x 1,..., x n ) (σ,f ) {x i1,..., x ik }. 2. ϕ (σ,f ) {x} if ϕ {x x, x = t, t = x}, and x F v(t). 3. ϕ (σ,f ) if ϕ (σ,f ). 4. ϕ ψ (σ,f ) X if ϕ (σ,f ) X and ψ (σ,f ) X. 5. ϕ ψ (σ,f ) X Y if ϕ (σ,f ) X, ψ (σ,f ) Y, and Y F v(ϕ) = (or X F v(ψ) = ). 6. yϕ (σ,f ) X {y} if y X and ϕ (σ,f ) X. Conjecture. A formula is upward (σ, F )absolute iff it is logically equivalent to a formula of the y 1,..., y n ψ, where ψ (σ,f )
16 Computability in N The d.i. signature (σ N, F N ) is defined as follows: σ N is the firstorder signature which includes the constants 0 and 1, the binary predicate <, and the ternary relations P + and P. F N (<) = {{1}}, F N (P + ) = F N (P ) = { }. Obviously, N is a (σ N, F N )extension of a structure for σ N iff its domain is an initial segment of N. Theorem. The following conditions are equivalent for a relation R on N: 1. R is semidecidable. 2. R is definable by a formula of the form y 1,..., y n ψ, where ψ (σn,f N ). 3. R is definable by a formula of the form y 1,..., y n ψ, where the formula ψ is (σ N, F N )absolute.
17 A Characterization of (σn,f N ) Let N = (σn,f N ). Then N can be defined as follows: 1. ϕ N if ϕ is atomic. 2. ϕ N {x} if ϕ {x x, x = t, t = x, x < t}, and x F v(t). 3. ϕ N if ϕ N. 4. ϕ ψ N X if ϕ N X and ψ N X. 5. ϕ ψ N X Y if ϕ N X, ψ N Y, and either Y F v(ϕ) = or X F v(ψ) =. 6. yϕ N X {y} if y X and ϕ N X. Note. It is easy to see that the set of formulas ϕ such that ϕ N is a straightforward extension of Smullyan s set of bounded formulas.
18 Transitive Closure Again Let V 0 be the smallest set including 0 and closed under the operation of pairing. Then a subset S of V 0 is recursively enumerable iff there exists a formula ϕ(x) of PT C + such that S = {x V 0 ϕ(x)}, where the language PT C + is defined as follows: Terms of PT C + 1. The constant 0 is a term. 2. Every (individual) variable is a term. 3. If t and s are terms then so is (t, s). Formulas of PT C + 1. If t and s are terms then t = s is a formula. 2. If ϕ and ψ are formulas then so are ϕ ψ and ϕ ψ. 3. If ϕ is a formula, x, y are two different variables, and t, s are terms, then (T C x,y ϕ)(t, s) is a formula.
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