✬ G¨ odeli meeldetuletus Tarmo Uustalu Teooriap¨ aevad Arulas, 3.–5.2.2003 ✫ 1
✬ • What is this about? (Rich) languages with a decided intended interpretation, (powerful) theories in such languages, axiomatized (powerful) theories languages. • Definition: A language L is a first-order logical language with equality denumerable amount of non-logical individual, function and predicate assume a fixed intended interpretation. This singles out a subset of set of true sentences. | = A means A is true in the intended interpretation. An L - theory T is a subset of all L -sentences, these sentences are called ⊢ T A means A is a T -theorem. An axiomatized L -theory is a L -theory generated by a p.r. subset of (called axioms ) and the inference rules of first-order logic. ✫ 2
✬ • Definition: Let T be a theory in a language L (with fixed intended olaline), if ⊢ T A implies �⊢ T ¬ – T is said to be consistent (koosk˜ more theorems than syntactically ok). – T is said to be sound (korrektne), if ⊢ T A implies | = A (there are theorems than semantically ok). – T is said to be syntactically complete (s¨ aielik), if �⊢ untaktiliselt t¨ ⊢ T ¬ A (there are no less theorems than syntactically ok). aielik), if �⊢ – T is said to be semantically complete (semantiliselt t¨ (there are no less theorems than semantically ok). ✫ 3
✬ • Observation: The semantic properties are stronger than the syntactic – soundness implies consistency, – and semantic completeness implies syntactic completeness. • Observation: The converses don’t hold in general, but: – consistency implies soundness under the assumption of semantic – and syntactic completeness implies semantic completeness under soundness. • T syntactically perfect, if it’s both consistent and syntactically complete, every sentence A , either ⊢ T A or ⊢ T ¬ A (which mimicks bivalence). • T is semantically perfect, if it’s both sound and semantically complete, theoremhood exactly captures truth. ✫ 4
✬ • Definition: A language L is rich if natural numbers, p.r. operations numbers and p.r. relations on natural numbers are effectively represented (faithfully wrt. the intended interpretation) in L by terms, schematic schematics sentences. Terms representing natural numbers are called numerals . • Definition: An L -theory T is powerful , if natural numbers, p.r. operations relations on them satisfy the following presentation conditions (esitlustingimused): – for f a p.r. operation, m n ] . ⊢ T ¯ f [ ¯ m 1 , . . . , ¯ = ¯ m iff f ( m 1 , . . . , m n ) = – for p a p.r. relation, ⊢ T ¯ p [ ¯ m 1 , . . . , ¯ m n ] iff p ( m 1 , . . . , m n ) ( ¯ m denotes the representation of m .) ✫ 5
✬ • Fact: The terms and sentences (and schematic terms and schematic rich language L (with denumerable signature) are effectively enumerable numbers so that all important syntatic operactions on them reduce to numbers ( G¨ odel numbers ). • Consequence: Because of the representability of natural numbers in and sentences of L therefore translate to L -numerals ( codes ). � m � denotes the code of m . In powerful L -theories, facts about important operations and relations codes are reflected quite well since the presentation conditions hold. • Convention: From now on, saying “language”, we always mean a saying “theory”, we always mean a powerful theory. ✫ 6
✬ • Diagonalization Lemma: Given a language L , one can for any schematic P effectively find a sentence S s.t. | = S ≡ P [ � S � ] and, for any L -theory ⊢ T S ≡ P [ � S � ] . • Proof: Instantiating schematic L -sentences with L -numerals is a p.r reduced to G¨ odel numbers thus a p.r. operation on numbers, hence representable Let subst be the schematic L -term representing it. Then | = subst [ � any schematic L -sentence Q and any numeral t . For an L -theory T ⊢ T subst [ � Q � , t ] . = � Q [ t ] � by the presentation conditions. Consider any schematic L -sentence P . Let D be the diagonal schematic given by D [ t ] := P [ subst [ t, t ]] . Set S := D [ � D � ] . Then | = S ≡ P [ � S � ] and ⊢ T S ≡ P [ � S � ] since by the definitions of S and D , S ≡ P [ � S � ] is identical to P [ subst [ � D � , � D � ]] ≡ P [ � D [ � D � ] � ] . ✫ 7
✬ • Tarski’s theorem about non-representability of truth. Given a language L -sentences is non-representable in L : there is no schematic L -sentence | = A iff | = True [ � A � ] • Proof. Suppose a schematic L -sentence True with the stated property Then, applying the Diagonalization Lemma to the schematic L -sentence can produce an L -sentence Tarski such that | = Tarski ≡ ¬ True [ � Ta the effect that | = Tarski iff �| = True [ � Tarski � ] , which, by our assumption happens iff �| = Tarski . Hence Tarski is a sentence stating its own falsity, a “liar”. Independent Tarski is true or false, it is true and false, which cannot be. ✫ 8
✬ • G¨ odel’s theorem about representability of theoremhood. Given theoremhood in an axiomatized L -theory T is effectively representable effectively find a schematic sentence Thm T in L s.t. ⊢ T A iff | = Thm T [ � A � ] • Proof: For an axiomatized L -theory T , the relation of a sequence of being a T -proof of a L -sentence is a p.r. relation, reduced to G¨ odel relation on numbers, thus effectively representable in L . Let Proof L -sentence representing it. Thm T is constructed by letting Thm T [ t ] := ∃ x. Nat [ x ] ∧ Proof T [ x, ✫ 9
✬ • Lemma (G¨ odel): Given a language L , each axiomatized L -theory following derivability conditions (tuletatavustingimused): D1 ⊢ T A implies ⊢ T Thm T [ � A � ] (the theory is positively introspecti D2 ⊢ T Thm T [ � A ⊃ B � ] ⊃ ( Thm T [ � A � ] ⊃ Thm T [ � B � ]) (the theory closed under modus ponens), D3 ⊢ T Thm T [ � A � ] ⊃ Thm T [ � Thm T [ � A � ] � ] (the theory knows it introspective). • Proof: Hard work (unrewarding). ✫ 10
✬ • Corollary: Given a language L , a sound axiomatized L -theory T is semantically incomplete (and hence because of the assumption of soundness syntactically incomplete). • Proof: If some L -theory T was both sound and semantically complete, T -theoremhood of L -sentences would be the same as truth. But one L -representable, the other is not. ✫ 11
✬ • G¨ odel’s first incompleteness theorem: Given a language L , for an L -theory T , one can effectively find an L -sentence Godel T s.t. – if T is consistent, then �⊢ T Godel T , but | = Godel T (so T is semantically incomplete), – if T is omega-consistent, then �⊢ T ¬ Godel T (so T is also syntactically • Proof: For an axiomatized L -theory T , we know that a schematic L exists s.t. ⊢ T A iff | = Thm T [ � A � ] . Using the Diagonalization Lemma, we construct Godel T as an L -sentence | = Godel T ≡ ¬ Thm T [ � Godel T � ] and ⊢ T Godel T ≡ ¬ Thm (so informally Godel T says it’s a non- T -theorem and that’s a T -theorem). Assume T is consistent. Suppose ⊢ T Godel T . Then, by D1, also ⊢ T Thm T [ � Godel T � ] . But then, by the construction of Godel T , ⊢ T contradicts consistency. Suppose �| = Godel T , then by the construction of Godel T , | = Thm T [ by the construction of Thm T , equivalent to ⊢ T Godel T , but we already ⊢ T ¬ Godel T , so again we are contradicting consistency. ✫ 12
✬ • Remark: Note that while Tarski is an antinomic sentence, it must not merely paradoxical, its existence looks potentially troublesome, but harmful about it. ✫ 13
✬ • G¨ odel’s second incompleteness theorem: Given a language L , for L -theory T , if T is consistent, than �⊢ T Cons T where Cons T := ¬ Thm T [ � ⊥ � ] (which says T is consistent). (So consistenc consitent axiomatized theory T is not a T -theorem.) • Proof: Assume T is a consistent axiomatized L -theory. By the construction have ⊢ T Godel T ⊃ ¬ Thm T [ � Godel T � ] From this, by D1, we get ⊢ T Thm T [ � Godel T � ⊃ ¬ Thm T [ � Godel T � ]] from where, by D2, we further get ⊢ T Thm T [ � Godel T � ] ⊃ Thm T [ � ¬ Thm T [ � Godel T But by D3 we also have ✫ ⊢ T Thm T [ � Godel T � ] ⊃ Thm T [ � Thm T [ � Godel T � 14
✬ Combining the last two using D2 and the construction of Cons T , we ⊢ T Thm T [ � Godel T � ] ⊃ ¬ Cons T which of course gives ⊢ T Cons T ⊃ ¬ Thm T [ � Godel T � ] Together with the construction of Godel T again (the second half of this yields ⊢ T Cons T ⊃ Godel T If now it were the case that ⊢ T Cons T , then also ⊢ T Godel T , but since the First Incompleteness Theorem tell us the that �⊢ T Godel T . ✫ 15
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