```Date: Apr 6, 2014 4:54 AM
Author: William Elliot
Subject: Two Finite Arithmetics

Naive Finite ArithmeticLet N be a set, 0,m two elements and S|N -> N a function.Axioms for naive finite arithmetic:0, m in N;  Sm = mfor all x, Sx in Nfor all x, x /= S0for 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 ADefinition of addition by induction.	0 + y = yDefinition of mulplication by induction.	0 * y = 0	Sx * y = x*y + y	Sx + y = x + SyIs this a consistent set of axioms with the model of a finiteset of integerss { 0,1,.. m } and addition defined by	a + b = max{ m, a+b }?Are the axioms sufficient strong to prove most or all of the expected theorems?  Are there any infinite models?---Modular Finite ArithmeticLet N be a set, 0,m two elements and S|N -> N a function.Axioms for modular finite arithmetic:0, m in N;  Sm = 0for all x, Sx in Nfor all x,y. (Sx = Sy implies x = y)Induction.  For all A subset N, if 	0 in A, (for all x in A implies Sx in A)then N subset ADefinition of addition by induction.	0 + y = y	Sx + y = x + SyDefinition of mulplication by induction.	0 * y = 0	Sx * y = x*y + yIs this a consistent set of axioms with a model of integers modulus m-1?Are these axioms sufficient for modular arithmetic?Are there any infinite models?---
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