## An Introduction to Gödel's TheoremsIn 1931, the young Kurt Gödel published his First Incompleteness Theorem, which tells us that, for any sufficiently rich theory of arithmetic, there are some arithmetical truths the theory cannot prove. This remarkable result is among the most intriguing (and most misunderstood) in logic. Gödel also outlined an equally significant Second Incompleteness Theorem. How are these Theorems established, and why do they matter? Peter Smith answers these questions by presenting an unusual variety of proofs for the First Theorem, showing how to prove the Second Theorem, and exploring a family of related results (including some not easily available elsewhere). The formal explanations are interwoven with discussions of the wider significance of the two Theorems. This book will be accessible to philosophy students with a limited formal background. It is equally suitable for mathematics students taking a first course in mathematical logic. |

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### Contents

1 | |

8 | |

3 Axiomatized formal theories | 17 |

4 Capturing numerical properties | 28 |

5 The truths of arithmetic | 37 |

6 Sufficiently strong arithmetics | 43 |

Taking stock | 47 |

8 Two formalized arithmetics | 51 |

21 Using the Diagonalization Lemma | 175 |

22 Secondorder arithmetics | 186 |

Incompleteness and | 199 |

24 Gbdels Second Theorem for PA | 212 |

25 The derivability conditions | 222 |

26 Deriving the derivability conditions | 232 |

27 Reflections | 240 |

About the Second Theorem | 252 |

9 What Q can prove | 58 |

93 Adding g to Q | 61 |

10 Firstorder Peano Arithmetic | 71 |

11 Primitive recursive functions | 83 |

expat 1H s 4min A | 98 |

12 Capturing pr functions | 99 |

13 Q is pr adequate | 106 |

A very little about Principia | 118 |

15 The arithmetization of syntax | 124 |

16 PA is incomplete | 138 |

17 G6dels First Theorem | 147 |

About the First Theorem | 153 |

19 Strengthening the First Theorem | 162 |

20 The Diagonalization Lemma | 169 |

29 uRecursive functions | 265 |

30 Undecidability and incompleteness | 277 |

31 Turing machines | 287 |

32 Turing machines and recursiveness | 298 |

BEBE19 | 301 |

BBB | 302 |

33 Halting problems | 305 |

34 The ChurchTuring Thesis | 315 |

35 Proving the Thesis? | 324 |

36 Looking back | 342 |

Further reading | 344 |

Bibliography | 346 |

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### Common terms and phrases

A0 wff algorithm argument assuming assumption axioms basic arithmetic canonical captures chapter characteristic function Church’s Thesis claim computable functions consistent deﬁned deﬁnition derivability conditions Diagonalization Lemma effectively computable effectively decidable effectively enumerable Entscheidungsproblem equivalent example express fact ﬁnd ﬁnite ﬁrst ﬁrst-order logic ﬁxed point function f G6del sentence Gbdel given Godel halting problem halts Hence Here’s idea Incompleteness Theorem inﬁnite input interpretation intuitive language let’s loops mathematics natural numbers nice theory numerical property open wff p.r. adequate p.r. deﬁnition p.r. function PA’s predicate PrfT primitive recursive primitive recursive function Proof sketch provable prove quantiﬁers recursively solvable reﬂection result satisﬁes scanned cell Second Theorem second-order second-order arithmetic Section semantic sentence G sequence speciﬁcation successor suppose symbols theory of arithmetic total functions trivial true truths Turing machine Turing program u-recursive function undecidable unprovable variables w-consistent we’ll zero

### Popular passages

Page 3 - following it up right back to the primitive truths. If, in carrying out this process, we come only on general logical laws and on definitions, then the truth is an analytic one.