Posted by: Alexandre Borovik | May 19, 2011

Consistency of Peano Arithmetic

A dispute rages at the Foundation of Mathematics mailing list about consistency of Peano Arithmetic. I am an outsider in the foundations of mathematics domain, but I was struck by a quote from Edward Nelson’s Predicative Arithemtic:

```The reason for mistrusting the induction principle is that
it involves an impredicative concept of number. It is not correct
to argue that induction only involves the numbers 0 to n; the
property of n being established may be a formula with bound
variables that are thought of as ranging over all numbers.
That is, the induction principle assumes the natural number
system as given. A number is conceived to be an object satisfying
every inductive formula; for a particular inductive formula,
therefore, the bound variables are conceived to range over
objects satisfying every inductive formula, including the one
in question.```

This is something that has always made me uncomfortable with the principle of mathematical induction .

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Responses

1. Here is the link to a related thread in MO.

(BTW, I wonder since how long doubts on those consistency issues and their relevance circulate among e.g. IHES- or IAS circles. My layman’s impression is that the tradition among russian mathematicians of taking physics *really* serious (e.g. the way Drinfeld, Kontsevich and Manin enjoyed taking up concepts from th. physics like Feynman integrals, renormalization, deformation quantization,mirror sym. …), i.e. as inputs of “new platonic ideas” after those coming from everyday experience had been exhausted, and the permanent consistency troubles with quantum field theory caused since long the impression that consistency issues were of secondary relevance. But that impression may be entirely wrong.)

2. I think of the Peano-style generation of the integers as structural induction, a weaker technique than full induction over the integers. The rule only requires we assume the existence of the nth peano number to construct the n 1th.

3. 0 down vote

I would like to inform that I (commonly with Teodor Stepien) delivered a talk at the Conference “2009 European Summer Meeting of Association for Symbolic Logic, Logic Colloquium’09” (July 31 – August 5, 2009, Sofia, Bulgaria). In this talk, entitled “On consistency of Peano’s Arithmetic System”, we presented a sketch of the proof of the consistency of Peano’s Arithmetic System (of course, the full proof was constructed by us before the mentioned Conference “Logic Colloquium 2009”). This proof is ABSOLUTELY ELEMENTARY, i.e. there are used ONLY the axioms of first-order logic and the axioms of Peano’s Arithmetic System. Hence, from the construction of this proof, it follows that Gödel’s Second Incompleteness Theorem is INVALID. The asbtract of this talk was published in “The Bulletin of Symbolic Logic”: T. J. Stepien and L. T. Stepien, Bull. Symb. Logic 16, 132 (2010). This abstract is accessible under the following link http://www.math.ucla.edu/~asl/bsl/1601-toc.htm and after clicking on: “2009 European Summer Meeting of the Association for Symbolic Logic, Logic Colloquium ’09, Sofia, Bulgaria, July 31—August 5, 2009” (page 132).