Consistency of arithmetic Mod N

[Count Iblis]

"[URL consistency of ordinary arithmetic has not yet been satifactorily settled[/URL]. What is the upper limit for N such that arithmetic modulo N is known to be consistent?

 

[CRGreathouse]

I don't know of any nontrivial results. I suppose consistency mod 2 is equivalent to the consistency of predicate calculus, so it's fine there.

 

[torquil]

On the Wikipedia site you linked to it says "some feel that these results resolved the problem, while others feel that the problem is still open."

While any mathematician may guess as much as they want about a certain result being correct or not, at the end of the day that means nothing in mathematics. Either it is correct or not, and feelings will not make either result more probable, IMO

So I wonder why some "feel" that the problem is resolved, if a proof doesn't exist?

Torquil

 

[CRGreathouse]

So I wonder why some "feel" that the problem is resolved

There is a problem, phrased in natural language. There are proofs written in mathematical formalisms. Some feel that the problem described in natural language are solved by those proofs; some feel that the problem described in natural language is not addressed by the formal proofs.

The issue isn't the validity or existence of the proofs, but their applicability. It is fundamentally an issue of translation.

 

[Count Iblis]

2 is rather depressingly low value for N.

 

[torquil]

There is a problem, phrased in natural language. There are proofs written in mathematical formalisms. Some feel that the problem described in natural language are solved by those proofs; some feel that the problem described in natural language is not addressed by the formal proofs.

The issue isn't the validity or existence of the proofs, but their applicability. It is fundamentally an issue of translation.

Is it then just that the "problem described in natural language" has not been accurately and uniquely translated into mathematical terms? If so, it would be a case of people interpreting the concept of "consistency" in different ways?

Torquil

 

[CRGreathouse]

Is it then just that the "problem described in natural language" has not been accurately and uniquely translated into mathematical terms? If so, it would be a case of people interpreting the concept of "consistency" in different ways?

Right. My interpretation of the natural language is such that I consider the problem solved. But it's easy to interpret the problem more broadly, and in that sense it's still unresolved.

 

[Werg22]

I would think there is no upper-bound. If we think of the number 0, 1, ..., n-1 arranged in a clockwise manner on a circle, and interpret addition by 1 as one spot clockwise, then we have enough machinery to build a "physical" model of arithmetic modulo n, so it certainly has to be consistent.