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Topic: Peano's Integers Modulus n.
Replies: 12   Last Post: Mar 23, 2014 4:44 AM

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LudovicoVan

Posts: 3,201
From: London
Registered: 2/8/08
Re: Peano's Integers Modulus n.
Posted: Mar 23, 2014 1:19 AM
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"William Elliot" <marsh@panix.com> wrote in message
news:Pine.NEB.4.64.1403222106350.8893@panix1.panix.com...
> On Sun, 23 Mar 2014, Julio Di Egidio wrote:
>> "William Elliot" <marsh@panix.com> wrote in message
>
>> > > > Let N be a set and S:N -> N an injection and 0 & z be two elements
>> > > > of
>> > > > N

>
>> > > > Assume Sz = 0 and the schemata,
>> > > > if A subset N, 0 in A and for all n in A, Sn in A,
>> > > > then A = N.
>> > > >
>> > > > Are the integers modulus z a model for these axioms?

>
>> There is an error in that a model would be the integers modulo z+1, not
>> modulo
>> z. Then it seems trivial to check that all axioms hold (using the usual
>> successor function for the integers modulo z+1 as S), except for the
>> induction
>> axiom: which makes me wonder, in which theory is one supposed/allowed to
>> do
>> the proof?

>
> No. Let N = { 0,1,2 } be the integers modulus 3.
> Then z = 2 and Sz = 2 + 1 = 3 = 0.


No what?

No answer to the other questions?

Julio





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