
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 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? 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?

