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Peano Arithmetic, Incompleteness

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Abstract

This chapter introduces Peano Arithmetic and discusses two respects in which it is incomplete.

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Notes

  1. 1.

    Although the ‘P’ in ‘PA’ commemorates the mathematician Giuseppe Peano (1858–1932), another mathematician, Richard Dedekind (1831–1916), deserves much of the credit. For some of the history, see Wang [11] (http://www.jstor.org/stable/2964176).

  2. 2.

    See Quine [9, pp. 33–37]. I will soon get sloppy and stop using the corner quotes (Absolute rigor does get a bit tedious). I thought, however, that you should know about them.

  3. 3.

    See Frege [2]; English translation in van Heijenoort [10, pp. 5–82]. If you have had a logic course, you have probably worked with a close cousin of Frege’s system.

  4. 4.

    More modestly, we want an interpretation that would make the first six axioms true if there were such things as the natural numbers. This sort of conditional claim is what I will generally intend when I talk about an interpretation making certain sentences true.

  5. 5.

    Gödel’s dissertation is reprinted, with an English translation, in Gödel [7, pp. 60–101].

  6. 6.

    Here are three equivalent definitions of inconsistency: an inconsistent theory is one that proves a logical absurdity such as ‘\(0\ne 0\)’; an inconsistent theory is one that proves a sentence \(\phi \) and its negation \(-\phi \); an inconsistent theory is one that proves every sentence in its language.

  7. 7.

    See Gödel [6]; English translation in Gödel [7, pp. 145–195], and van Heijenoort [10, pp. 596–616].

  8. 8.

    The corresponding PA-numeral consists of a single occurrence of ‘0’ preceded by 47,278,574,201,250 occurrences of ‘\(S\)’. If you were to produce a token of this numeral, it would be about 100 million km long. That is about two thirds the distance from the Earth to the Sun or about 2,500 times the circumference of the Earth. This suggests that it may be naive to think of PA-numerals as actual physical objects.

  9. 9.

    For a readable discussion of Gödel’s construction, see Nagel and Newman [8]. Another helpful resource on this and other issues of interest to us is George and Velleman [5]. It might help you wrap your brain around Gödel’s proof if you read \(\gamma (\mathbf {n})\) as “\(n\) does not code a PA-proof of G” where G is a certain extra-special sentence of PA. Then \(\forall x\;\gamma (x)\), the universal generalization of \(\gamma (\mathbf {n})\), says that no natural number codes a PA-proof of G. Now it so happens that \(\forall x\;\gamma (x)\) is G. So G says of itself that it is not provable in PA. A PA-proof of G would prove that G is not provable in PA: a strange situation, to say the least. Of course, all this is a bit sloppy. \(\gamma (\mathbf {n})\) does not say anything unless we interpret it. Furthermore, under the intended interpretation it does not say anything about PA-proofs: it only refers to natural numbers. But the road to clarity is sometimes paved with slop.

  10. 10.

    See Gentzen [3]; English translation in Gentzen [4, pp. 132–213].

  11. 11.

    See Feferman [1, p. 192].

References

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  1. Department of Philosophy and Religion, Truman State University, Kirksville, MO, USA

    Stephen Pollard

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  1. Stephen Pollard
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Correspondence to Stephen Pollard .

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Pollard, S. (2014). Peano Arithmetic, Incompleteness. In: A Mathematical Prelude to the Philosophy of Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-05816-0_2

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