Metalogic: An Introduction to the Metatheory of Standard First Order LogicThis work makes available to readers without specialized training in mathematics complete proofs of the fundamental metatheorems of standard (i.e., basically truth-functional) first order logic. Included is a complete proof, accessible to non-mathematicians, of the undecidability of first order logic, the most important fact about logic to emerge from the work of the last half-century. Hunter explains concepts of mathematics and set theory along the way for the benefit of non-mathematicians. He also provides ample exercises with comprehensive answers. |
Contents
Deductive apparatuses Formal systems Proof theory | 3 |
Syntactic Semantic | 9 |
Decidable sets | 16 |
Some theorems about infinite sets | 26 |
15 | 40 |
Truth functions | 48 |
A formal language for truthfunctional propositional | 54 |
Some truths about Fp The Interpolation Theorem for | 61 |
Extended sense of interpretation of P Finite weak models and finite strong models 116 118 | 120 |
Proof of the independence of the three axiomschemata of PS | 122 |
the system AB | 125 |
First Order Predicate Logic Consistency and Completeness | 135 |
the language Q The languages | 137 |
Semantics for Q and Q+ Definitions of interpretation of Q Q+ satisfaction of a formula by a denumerable sequence of objects satisfiable simultaneous... | 141 |
Some modeltheoretic metatheorems for Q and Q+ 137 | 152 |
141 | 156 |
Ps powers of expression Adequate sets of connectives | 62 |
the formal system PS De finitions of proof in PS theorem of PS derivation in PS syntactic consequence in PS prooftheoretically consistent set of | 71 |
Some truths about | 77 |
Concepts of consistency | 78 |
Proof of the consistency of | 79 |
The Deduction Theorem for | 84 |
Note on proofs by mathematical induction | 88 |
Some modeltheoretic metatheorems about | 91 |
Concepts of semantic completeness Importance for logic of a proof of the adequacy and semantic completeness of a formal system of truthfunctional... | 92 |
Outline of Posts proof of the semantic completeness of a formal system of truthfunctional propositional logic | 95 |
Proof of the semantic completeness of PS by Kalmárs method 71 77 78 79 84 88 91 92 95 | 96 |
Proof of the semantic completeness of PS by Henkins method | 105 |
0096 | 116 |
Proof of the decidability of PS Decidable system and de cidable formula Definition of effective proof procedure | 118 |
the formal system | 166 |
First order theories | 173 |
Proof of the semantic completeness of | 195 |
Isomorphism of models Categoricity Nonstandard models | 201 |
Some results about undecidability | 219 |
Churchs Thesis 1935 Churchs Theorem 1936 | 230 |
54 | 234 |
the system | 236 |
57 | 249 |
1 Logical validity and the empty | 255 |
| 262 | |
152 | 274 |
| 275 | |
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Metalogic: An Introduction to the Metatheory of Standard First Order Logic Geoffrey Hunter No preview available - 1971 |
Common terms and phrases
A₁ argument assigned axiom of QS axiom-schema cardinal number closed term closed wff consistent first order consistent set countable d₁ Deduction Theorem definition denumerable model denumerable sequence derivation domain effective method effectively enumerable false finite subset formal system formulas of Q free occurrences function symbols Gödel number immediate consequence individual constants induction hypothesis intended interpretation interpretation of Q kth term Lemma logical axiom mathematical induction metatheorems Modus Ponens natural numbers negation-complete normal interpretation normal model numbers to natural occurs free order theory p-consistent predicate symbol prenex normal form proof proper axioms propositional logic propositional symbols prove quantifiers real numbers recursive function recursive set rule of inference S₁ semantic completeness sequences of symbols set of formulas set of natural Suppose t₁ tautology theorem of QS true truth function truth table truth values truth-functional uncountably undecidable V₁ ɅvA wff of Q