G odeli meeldetuletus Tarmo Uustalu Teooriap aevad Arulas, - - PDF document

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G¨

• deli meeldetuletus

Tarmo Uustalu Teooriap¨ aevad Arulas, 3.–5.2.2003

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• 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.

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• Definition: Let T be a theory in a language L (with fixed intended

– T is said to be consistent (koosk˜

• laline), if ⊢T A implies ⊢T ¬

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¨ untaktiliselt t¨ aielik), if ⊢ ⊢T ¬A (there are no less theorems than syntactically ok). – T is said to be semantically complete (semantiliselt t¨ aielik), if ⊢ (there are no less theorems than semantically ok).

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• 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.

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• 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, ⊢T ¯ f[ ¯ m1, . . . , ¯ mn] . = ¯ m iff f(m1, . . . , mn) = – for p a p.r. relation, ⊢T ¯ p[ ¯ m1, . . . , ¯ mn] iff p(m1, . . . , mn) ( ¯ m denotes the representation of m.)

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• 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¨

• del 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.

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• 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¨

• del 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]].

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• 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.

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• G¨
• del’s theorem about representability of theoremhood. Given

theoremhood in an axiomatized L-theory T is effectively representable effectively find a schematic sentence ThmT in L s.t. ⊢T A iff | = ThmT [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¨

• del

relation on numbers, thus effectively representable in L. Let Proof L-sentence representing it. ThmT is constructed by letting ThmT [t] := ∃x. Nat[x] ∧ ProofT [x,

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• Lemma (G¨
• del): Given a language L, each axiomatized L-theory

following derivability conditions (tuletatavustingimused): D1 ⊢T A implies ⊢T ThmT [A] (the theory is positively introspecti D2 ⊢T ThmT [A ⊃ B] ⊃ (ThmT [A] ⊃ ThmT [B]) (the theory closed under modus ponens), D3 ⊢T ThmT [A] ⊃ ThmT [ThmT [A]] (the theory knows it introspective).

• Proof: Hard work (unrewarding).

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• 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.

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• G¨
• del’s first incompleteness theorem: Given a language L, for an

L-theory T, one can effectively find an L-sentence GodelT s.t. – if T is consistent, then ⊢T GodelT , but | = GodelT (so T is semantically incomplete), – if T is omega-consistent, then ⊢T ¬GodelT (so T is also syntactically

• Proof: For an axiomatized L-theory T, we know that a schematic L

exists s.t. ⊢T A iff | = ThmT [A]. Using the Diagonalization Lemma, we construct GodelT as an L-sentence | = GodelT ≡ ¬ThmT [GodelT ] and ⊢T GodelT ≡ ¬Thm (so informally GodelT says it’s a non-T-theorem and that’s a T-theorem). Assume T is consistent. Suppose ⊢T GodelT . Then, by D1, also ⊢T ThmT [GodelT ]. But then, by the construction of GodelT , ⊢T contradicts consistency. Suppose | = GodelT , then by the construction of GodelT , | = ThmT [ by the construction of ThmT , equivalent to ⊢T GodelT , but we already ⊢T ¬GodelT , so again we are contradicting consistency.

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• Remark: Note that while Tarski is an antinomic sentence, it must not

merely paradoxical, its existence looks potentially troublesome, but harmful about it.

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• G¨
• del’s second incompleteness theorem: Given a language L, for

L-theory T, if T is consistent, than ⊢T ConsT where ConsT := ¬ThmT [⊥] (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 GodelT ⊃ ¬ThmT [GodelT ] From this, by D1, we get ⊢T ThmT [GodelT ⊃ ¬ThmT [GodelT ]] from where, by D2, we further get ⊢T ThmT [GodelT ] ⊃ ThmT [¬ThmT [GodelT But by D3 we also have ⊢T ThmT [GodelT ] ⊃ ThmT [ThmT [GodelT

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✬ ✫ Combining the last two using D2 and the construction of ConsT , we ⊢T ThmT [GodelT ] ⊃ ¬ConsT which of course gives ⊢T ConsT ⊃ ¬ThmT [GodelT ] Together with the construction of GodelT again (the second half of this yields ⊢T ConsT ⊃ GodelT If now it were the case that ⊢T ConsT , then also ⊢T GodelT , but since the First Incompleteness Theorem tell us the that ⊢T GodelT .

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