
Re: Two Finite Arithmetics
Posted:
Apr 7, 2014 12:06 PM


On Sunday, April 6, 2014 4:54:57 AM UTC4, William Elliot wrote: > Naive Finite Arithmetic > > Let N be a set, 0,m two elements and SN > 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 >
Addressed by Seymour.
> 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 Download my DC Proof 2.0 software at http://www.dcproof.com Visit my new math blog at http://www.dcproof.wordpress.com

