
Re: Two Finite Arithmetics
Posted:
Apr 7, 2014 12:50 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 > > 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 > > 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 }? >
This assumes addition on infinite N.
Also, if m+1=m and m+2=m, then m+1=m+2 and 1=2.
Dan Download my DC Proof 2.0 software at http://www.dcproof.com Visit my new math blog at http://www.dcproof.wordpress.com

