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

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 William Elliot Posts: 2,564 Registered: 1/8/12
Re: Two Finite Arithmetics
Posted: Apr 7, 2014 11:51 PM

On Mon, 7 Apr 2014, Dan Christensen wrote:

> 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?

Naive finite arithmetic is not your finite arithmetic.
S is a full function to banish the pests of your system.

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

for all x, Sx /= 0.

> > 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

Sx + y = x + Sy

> 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.

That definition is a binary function over N^2.
That it works for double induction is common knowledge.

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

No. That's a mess for the partiallity is difficult to describe.
This way, once you get to m, then adding more is not more.

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

Hey, that's a better idea, avoiding double induction
as now the inductive definition is the add y function.

> > 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 }?

For the model work, I should have carified that + is usual addition
++ the naive finite arithmetic and write, by definition
a ++ b = max( m, a+b }
> Again, you probably want a partial function. The way I see it, m + 1 should
> be undefined.

No way!. You have been able to finish your system.
Mine is done and complete, corrected and imporved.

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

The max ceiling prevents cancelation
a + b = a + c implies b = c
At best, if a + b = a + c /= m, then b = c.

Also it not very useful. For very large m, it can
suffice for most mathematics, but the needed m seems to
keep increasing with time. Would m = googleplex suffice?