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Topic: Two Finite Arithmetics
Replies: 19   Last Post: Apr 9, 2014 9:27 PM

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 Dan Christensen Posts: 2,394 Registered: 7/9/08
Re: Two Finite Arithmetics
Posted: Apr 7, 2014 12:06 PM

On Sunday, April 6, 2014 4:54:57 AM UTC-4, William Elliot wrote:
> Naive Finite Arithmetic
>
> Let N be a set, 0,m two elements and S|N -> N a function.
>
> Axioms for naive finite arithmetic:
>
>
>
> 0, m in N; Sm = m
>

m should have no successor. Perhaps a partial function S on N?

> for all x, Sx in N
>
> for all x, x /= S0
>

> for all x,y /= m. (Sx = Sy implies x = y)
>
>
>
> For all A subset N, if
>
> 0 in A, (for all x in A implies Sx in A)
>
> then N subset A
>
>
>
> Definition of addition by induction.
>
> 0 + y = y
>

Is this where the "naive" comes in? The use of the infix '+' really needs to be justified, i.e. the sum of a pair of a numbers should be formally proven to be unique, however you may define sums.

As with your function S, perhaps '+' should be a partially function.

Also, shouldn't you have something have something like Sx + y = S(x+y)?

> Definition of mulplication by induction.
>
> 0 * y = 0
>
> Sx * y = x*y + y
>
> Sx + y = x + Sy
>
>
>
> Is this a consistent set of axioms with the model of a finite
>
> set of integerss { 0,1,.. m } and addition defined by
>
> a + b = max{ m, a+b }?
>

Again, you probably want a partial function. The way I see it, m + 1 should be undefined.

There may be a good reason that finite arithmetic has never been successfully formalized. It may be impossible.

Dan