# Shutte's Consistency of Arithmetic

Lukasz T. Stepien sfstepie at cyf-kr.edu.pl
Wed Jun 3 13:43:34 EDT 2020

``` By the way, I recommend the paper: T. J. Stępień, Ł. T. Stępień, „On
the Consistency of the Arithmetic System", _Journal of Mathematics and
System Science_, vol. 7, 43 (2017), arXiv:1803.11072. There a proof of
the consistency of the Arithmetic System was published. This proof was
done within this Arithmetic System. The abstract related to this paper:
T. J. Stepien and L. T. Stepien, "On the consistency of Peano's
Arithmetic System" , _Bull. Symb. Logic_ 16, No. 1, 132 (2010).
http://www.math.ucla.edu/~asl/bsl/1601-toc.htm .

Łukasz T. Stępień

---

Lukasz T. Stepien

The Pedagogical University of Cracow
Institute of Computer Science,
ul. Podchorazych 2
30-084 Krakow
Poland

tel. +48 12 662-78-54, +48 12 662-78-44

The URL  http://www.ltstepien.up.krakow.pl

On 2020-06-03 02:45, Daniel Schwartz wrote:

> As requested by Sandy Lemberg, a scanned image of the appendix to
> Mendelson's first edition of Intro to Math Logic can be obtained at:
>
> http://www.cs.fsu.edu/~schwartz/mendelson.pdf
>
> I had tried posting the pdf to the FOM list, but the file it too large.
>
> Dan Schwartz
>
> Dear FOM group,
>
> In 1951 Kurt Schutte published a proof of the consistency of Arithmetic.
> This proof hass ben reiterated in the appendix to the first edition of
> Elliott Mendelson's "Introduction to Mathematical Logic".  The proof goes as
> follows.
>
> Starting with a first-order theory S of Arithmetic, he constructs a system
> S_infinity whose axiomatization employs the Cut Rule.  He shows that
> anything provable in S is also provable in S_infinity.  Then, using a system
> of ordinal notations for proofs, he shows by transfinite induction that
> anything provable in S_infinity is provable without using Cut.  He then
> notes that, by inspecting the inference rules other than Cut, the formula
> 0!=0 could not be derivable unless it were already an axiom, which it isn't.
> Thus, since it is not derivable in S_infinity without Cut, it is not in any
> way derivable in S_infinity, so it is not derivable in S, and S is
> consistent.
>
> I'm wondering is Shutte or anyone else has published any similar proofs.
>
> Specifically, I'm interested in other proofs that use this same approach for
> establishing the underivability of other, more complex, formulas in S or
> S_infinity.  I would like to know if there is a general strategy for showing
> that a formula cannot be derived in S_infinity without Cut.
>
> Thanks and best regards,
>
> Dan Schwartz
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