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Re: Two Finite Arithmetics
Posted:
Apr 8, 2014 11:57 PM
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On Tuesday, April 8, 2014 11:47:13 PM UTC-4, William Elliot wrote:
> On Mon, 7 Apr 2014, Dan Christensen wrote:
>
> > > > > Naive Finite Arithmetic
>
> > > > > Let N be a set, 0,m two elements and S|N -> N a function.
>
> > > > > Axioms for naive finite arithmetic:
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> > > > > 0, m in N; Sm = m
>
> > > > m should have no successor. Perhaps a partial function S on N?
>
> > > Naive finite arithmetic is not your finite arithmetic.
>
> > > S is a full function to banish the pests of your system.
>
> > Inventing a successor for the maximum number seems a bit weird to me.
>
> It fits much better than not having one.
>
We have a finite succession of objects with a beginning and an end. The first element, by definition has no predecessor. The last element, by definition has no successor. I'd say your work-around isn't up to spec.
Dan
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