Talk:Non-standard model of arithmetic

Attention of expert needed

I've added a note saying that the attention of an expert is needed for the section: From the incompleteness theorems . I've been involved in a discussion (on facebook) with other mathematicians. I did postgraduate research into the logical foundations of mathematics, and non standard models but that was twenty years ago, and am rusty and can no longer count myself as an expert on the subject. This is my version of the section: http://en.wiki.x.io/w/index.php?title=Non-standard_model_of_arithmetic&oldid=500958615#From_the_incompleteness_theorems

This is the version as it is today, as edited by the other mathematicians I've been discussing it with: http://en.wiki.x.io/w/index.php?title=Non-standard_model_of_arithmetic&oldid=501533392#From_the_incompleteness_theorems

Which of us is right? Or is it something between the two? Could someone who is up to date and researching into this subject please edit and clarify. Thanks, Robert Walker (talk) 17:25, 11 July 2012 (UTC)

After further discussion (on facebook, sorry), looks as if we may be able to fix the section ourselves, though attention of an expert will be helpful once done to check it is okay. Will see how it goes and update here. Robert Walker (talk) 08:29, 12 July 2012 (UTC)

Here is the result (for further discussion if needed): http://en.wiki.x.io/w/index.php?title=Non-standard_model_of_arithmetic&oldid=503351363#From_the_incompleteness_theorems Robert Walker (talk) 22:15, 20 July 2012 (UTC)

The current (short) version seems OK to me, so I have removed the expert tag. But please feel free to expand it if there is additional material you think would help the article. — Carl (CBM · talk) 01:17, 9 January 2013 (UTC)

A Question about proofs coded up by Godel numbers

Here is a question we have, if anyone knows the answer, it's about the "proofs" coded up by non standard godel numbers. It might help to have a better understanding of what is going on.

If g is a non standard godel number in PA, then just like standard finite numbers, it has unique factorization, largest prime factor, etc, so you can unpack it to make a long string of symbols of all the symbols coded up by the exponents of its prime factors in the order in which they occur.

That's similar to the way you can have a non standard length decimal for a hyper-real, e.g. pi = 3.14159... (infinitely many digits) ...3143...

Let's use "proof" in quotes to refer to one of these long strings of symbols unpacked from a non standard size godel number.

It's godel number has all the same properties in PA as a godel number of a standard length proof. You can't distinguish it from the godel number of a normal finite proof in PA.

So for instance, each statement of such a "proof" follows from two previous statements plus a deduction rule. Also though obviously you couldn't write out the full proof in practise, any fragmentary part of it of standard finite length looks like a fragment of a normal standard finite length proof.

So the question is, can a "proof" of this form, of non standard length, in any non standard model, ever have a last statement inconsistent with PA. E.g. can its last statement be "0=1"?

Or, even in a non standard model, can such "proof"s only prove results that are consistent with PA?

Note that this isn't Infinitary logic. The "proof" just uses ordinary deduction rules of PA. In all respects it is just like a normal finite proof in PA, except for its non standard length.

It's well ordered too, so long as you restrict yourself to predicates definable in PA (e.g. if you define a subset of the proof as a set of the symbols indexed by a set of numbers n satisfying some predicate in PA). Of course the proof is no longer well ordered if you define subsets with the extra predicate StandardFinite(n).

I doesn't matter if you think of them as some kind of generalised type of proof, or something else, the thing is, they are well defined whatever you call them. So, this is a purely mathematical question, whether such a "proof" can have a last statement such as "0=1" or any other statement inconsistent with PA.

Robert Walker (talk) 16:44, 11 July 2012 (UTC)

A non-standard "proof" is not necessarily only of non-standard length, but also the "formulas" of which it consists may be non-standard sequences of symbols, including non-standard "axioms". There are, indeed, models of PA containing a "proof" of the statement "0=1". That is a well-known theorem. As far as I know, there is no such "proof" consisting entirely of standard formulas in any model of PA, but don't quote me on that.

Taneli Huuskonen (talk) 17:01, 9 August 2015 (UTC)

Removed a long section about non standard models of ~G

This section was incorrect in some respects e.g. said you can have a godel number that encodes a proof of 0=1, and also gave the impression that non standard models are constructed using ~G.

You can make models with G true of any cardinality as well, so there is nothing special in that respect about ~G.

It's true that adding ~G as an axiom is one way to force a model to be non standard (while adding G doesn't force this). But because of the other issues involved, you use other axioms to make them non standard that simply assert existence of a non standard number in various ways.

Replaced it with a shorter section that makes the connection, points out some of the issues with ~G in a non standard model, and refers back to the main article on Godel's Theorems for more details. Robert Walker(talk)

What is number theory?

What do you mean by "number theory" in "nonstandard model of arithmetic is a model of all of number theory". If it is the set of all sentences which are true in natural numbers (standard model) then the fact "while the nonstandard model satisfies all of standard number theory, it also satisfies new sentences" is wrong because "number theory" in this sense is complete theory. But I dont know what else could it be - Peano arithmetic? Thanks for answer. Glivi 11:42, 4 April 2007 (UTC)

I think some kind of clarification needs to be made. There is a study of non-standard models of Peano arithmetic. There are also non-standard models of true arithmetic, i.e. models of the theory Th(N). These are different and i think the article does not distinguish these two things sufficiently. --DesolateReality 02:23, 13 June 2007 (UTC)

New truths

"(e.g. one could construct a model of number theory in which the twin prime conjecture holds)." I believe this statement is not known to be true. This is only possible if the twin prime conjecture were not provable from PA, which, as far as I am aware, is not known (yet :-). A better example would be that you could construct a model of arithmetic in which Goodstein's theorem does not hold, since it is known not to be provable by PA. I'm modifying the page to say that. Luqui 08:07, 22 September 2007 (UTC)

Hypernatural numbers?

Section 3 suddenly starts talking about the Galaxy of a hypernatural number, but the connection with the previous 2 sections is very unclear, especially as the term hypernatural has not been previously defined. HugoBarnaby (talk) 14:48, 17 June 2009 (UTC)

comment on goodstein

The result "there are models of Peano arithmetic in which Goodstein's theorem fails" cannot be used in order to prove the existence of a non-standard model of Peano arithmetic, because the proof of this statement uses such a non-standard model. — Preceding unsigned comment added by 88.163.61.169 (talk) 00:33, 30 March 2018 (UTC)

It would be helpful to clarify the comment on goodstein's theorem here. Tkuvho (talk) 09:00, 24 October 2010 (UTC)

More specifically, I agree with the comment by DesolateReality (hope he is feeling better) to the effect that one is not sufficiently distinguishing between two separate types of "non-standardness". The non-standard model in the sense of Skolem will certainly satisfy goodstein's theorem as there is no shortage of infinite ordinals in that theory. the non-standardness in the sense of Goodstein seems to involve tinkering with the foundations at a deeper level. Tkuvho (talk) 10:29, 24 October 2010 (UTC)

Moving this article to a different title

In the article, the spellings "nonstandard", "non-standard", and "non standard" are all used. This is quite annoying/distracting while reading. Since "nonstandard" seems to be the standard spelling (according to various dictionaries), I propose to change all spellings of said word to "nonstandard" and move the article to title "Nonstandard model of arithmetic". Does anyone have any good reason why I should not carry out said plan of action? JamesMazur22 (talk) 16:47, 20 December 2012 (UTC)

Our articles Non-standard analysis and non-standard calculus use the dash following Abraham Robinson's original book from 1966. Certainly "non standard" should be weeded out. Tkuvho (talk) 08:26, 24 December 2012 (UTC)

Title maybe Plural / MacDowell Specker

The title should maybe read "non-standard models of arithmetic". Other notable non-standard models are due to MacDowell-Specker, which are an end extension. See also:

[MS61] R. MacDowell and E. Specker, Modelle der Arithmetik, in: Infinitistic Methods (Proc. Sympos. Foundations of Math., Warsaw, 1959) Pergamon, Oxford, Warsaw, 1961, pp. 257–263.

Jan Burse (talk) 21:53, 14 April 2018 (UTC)