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1.4 Theories

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1.4 Theories 1.4 Theories Before we start to work with one of the most central notions of mathematical logic – that of a theory – it will be useful to make some observations about countable sets. Lemma 1.3. 1. A set X is countable iff there is a surjection ψ : N X. 2. If X is countable and Y X then Y is countable. 3. If X and Y are countable, then X Y is countable. 4. If X and Y are countable, then X ¢ Y is countable. k 5. If X is countable and k 1 then X is countable. ä 6. If Xi is countable for each i È N, then Xi is countable. iÈN 7. If X is countable, then ä Xk is countable. k ÈN Ø Ù Proof. 1. If X is countably infinite note that every bijection is a surjection. If X x0,...,xn Õ Ô Õ is finite, let ψ Ôi xi for 0 i n and ψ j xn for j n. For the other direction, Ô Õ Ô Õ Ô Õ Ù let ψ be a surjection, then X is Øψ 0 , ψ 1 , ψ 2 ,... and removing duplicates from the Ô ÕÕ Ô N sequence ψ i i¥0 yields a bijective ϕ : X. È 2. Obvious if Y is finite. Otherwise, let ϕ : N X be a bijection and let y0 Y . Define " Õ Ô Õ È ϕÔn if ϕ n Y ψ : N Y,n y0 otherwise which is a surjection. 3. Let ϕ : N X and ψ : N Y be bijections. We can obtain a bijection χ : N X Y by setting " Õ ϕÔk if n 2k Õ χÔn : Õ ψ Ôk if n 2k 1 We can picture χ in a diagram as ­ ­ ­ ¤ ¤ ¤ X ­ ? ? ? ­ ­ ­ ¤ ¤ ¤ Y ­ where the first line represents X in the order given by ϕ and the second line Y in the order given by ψ and the bijection χ is a path through the infinite graph X Y that touches every vertex exactly once. 11 ¢ 4. We can write X ¢ Y and a bijection χ : N X Y as a diagram ­ ­ ­ ­ ¤ ¤ ¤ ­ / ? / ? ­ ­ ­ ­ ¤ ¤ ¤ ­ ? ? ­ ­ ­ ­ ¤ ¤ ¤ ­ ? ­ ­ ­ ­ ¤ ¤ ¤ ­ ? ­ ­ ­ ­ ¤ ¤ ¤ ­ . .. k 1 k ¢ 5. By induction on k. If k 1 it is obvious. For k 1 observe that X X X, by assumption X is countable and by induction hypothesis Xk is countable, so by 4 also Xk 1 is countable. È ¢ 6. By assumption, there are bijections ϕi : N Xi for every i N. Define ψ : N N ä Õ Ô Õ ¢ Xi by ψ Ôi, n ϕi n , then ψ is a surjection. By 4 there is a bijection χ : N N N, iÈN ä ä so ψ ¥ χ : N Xi is a surjection hence Xi is countable. È iÈN i N 7. by 5 and 6. Given a language L we can assign a number to each symbol that might appear in a formula of L and so we can identify a formula (as well as a term) in L containing k symbols with a k-tuple of natural numbers. By the above result the set of terms as well as the set of formulas of a given language (and any subset of it, in particular the set of sentences) is countable. Õ Definition 1.8. Let Γ be a set of sentences. The deductive closure ClÔΓ of Γ is defined as ° Ù ØA sentence Γ A . Õ A set of sentences is called theory if it is deductively closed, i.e. ClÔT T . A theory T in a language L is called Henkin theory if for each sentence x A in L there is a Ö Þ × constant cA in L s.t. T ° x A A x cA . ½ ½ ½ ½ A theory T in a language L is an extension of a theory T in a language L if L L and T T . A theory T ½ is a conservative extension of a theory T in a language L if it is an extension and ½ ° if for every L-sentence A: T ° A implies T A. A theory T ½ is a Henkin extension of a theory T if it is an extension and a Henkin theory. Our first task will be to construct Henkin theories. These will turn out to be very useful for the Ö Þ × proof of the completeness theorem. Sentences of the form x A A x cA are called Henkin axioms, the cA are called Henkin constants. Let T be a theory in a language L. A first naive attempt to obtaining an Henkin extension of T might be to define ¦ Ø Ù L : L cA x A sentence in L , and ¦ Ô Ø Ö Þ ×Ù ÙÕ T : Cl T x A A x cA x A sentence in L Note that L¦ is countable because L as well as the set of L-sentences are. 12 Lemma 1.4. If π : Γ ° A, c is a constant and x a variable which does not appear in π, then Þ × Ö Þ × ° Ö Þ × π Öc x :Γ c x A c x . Proof. left as exercise (cf. Lemma 1.2). Lemma 1.5. T ¦ is a conservative extension of T . ¦ Proof. It is obviously an extension. For conservativity suppose T ° A with A in L. Then ° there are Γ T , Henkin axioms H1,...,Hn s.t. Γ,H1,...,Hn A. We proceed by induction ° Ö Þ × on n. For n 0, T A. For n 0, let Hn xB B x cB . Then ° Ô Ö Þ ×Õ Γ,H1,...,Hn¡1 xB B x cB A. By Lemma 1.4 ° Ô Ö Þ ×Õ Γ,H1,...,Hn¡1 xB B x y A so ° ÔÔ Ö Þ ×Õ Õ Γ,H1,...,Hn¡1 y xB B x y A and using quantifier shiftings ° Ô Ö Þ ×Õ Γ,H1,...,Hn¡1 xB yB x y A hence ° Γ,H1,...,Hn¡1 A. We obtain T ° A by induction hypothesis. ä ¦ Lemma 1.6. Let T be a theory, define T0 : T , Tn 1 : Tn and Tω : Ti. Then Tω is a i¥0 conservative Henkin extension of T . ä ä Ô Õ Ô Õ Proof. Note that Ti Ti 1 and hence Cl Ti Cl Ti (see exercises). Therefore ¥ i¥0 i 0 ¤ ¤ ¤ Ô Õ Ô Õ Ô Õ Tω Ti Cl Ti Cl Ti Cl Tω ¥ ¥ i¥0 i 0 i 0 ° is a theory. It is conservative because Tω ° A implies that there is a k s.t. Tk A and Tk is conservative over T by induction using Lemma 1.5. Tω is a Henkin theory because for x A ° Ö Þ × ω A A A ω in L , x A is in some Lk so c is in Lk 1 and Tk 1 x A A x c so c is in L and Ö Þ × Tω ° x A A x cA (where Li is the language of Ti). Definition 1.9. A theory T in a language L is called complete if for every sentence A in L: ° T ° A or T A. à A theory T is called consistent if T ² . Lemma 1.7. Every consistent theory T has a consistent complete extension T ½ in the same language. Proof. Let L be the language of T and let Ai be an enumeration of all sentences in L. Define T0 : T " Ø ÙÕ Ô Ø ÙÕ ClÔTn An if Cl Tn An is consistent Tn 1 : Tn otherwise ¤ Tω : Ti i¥0 13 Ã È ° à Then Tω is consistent for suppose Tω ° , then there are A1,...,An Tω s.t. A1,...,An à and therefore there is a k s.t. Tk ° but all Tk are consistent. Furthermore Tω is complete: Let Ô Ø ÙÕ ° A be in L, then A Ak for some k. If Cl Tk Ak is consistent, then Tk 1 Ak. Otherwise Ø ÙÕ ° à ° à ° ClÔTk Ak is inconsistent, i.e. Tk, Ak , but then Tk Ak hence Tk Ak by I. 1.5 Completeness A central technical tool for the proof of the completeness theorem is the canonical structure of a theory, sometimes also called term model. Let T be a theory in a language L. We construct Ô Õ the canonical structure M M, Φ of T as follows. For variable-free terms t,s of T write ° Ö × t s for T t s. Note that is an equivalence relation, for a term t we write t for its -equivalence class. The elements of M are the equivalence classes of . We now define Φ by Õ Ö × 1. ΦÔc c for a constant symbol c, ÕÔÖ × Ö ×Õ Ö Ô Õ× 2. ΦÔf t1 ,..., tn f t1,...,tn for a function symbol f, and × Ö ×Õ È Ô Õ ° Ô Õ 3. ÔÖt1 ,..., tn Φ P iff T P t1,...,tn for a predicate symbol P . For the above items 2 and 3 to be well-defined we have to verify that the definition does not × depend on the choice of the representative ti of the equivalence class Öti . Suppose we had ° chosen different representatives s1,...,sn, then, as si ti also T si ti and by the equality Ô Õ Ô Õ Ô Õ Ô Õ rules of NK we have T ° f s1,...,sn f t1,...,tn hence f s1,...,sn f t1,...,tn and Ô Õ ° Ô Õ T ° P t1,...,tn iff T P s1,...,sn so Φ is well-defined. Lemma 1.8. Let T be a consistent complete Henkin theory, M be its canonical structure, A be ° a sentence.
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