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

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
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 }?
Are the axioms sufficient strong to prove most or all of the
expected theorems? Are there any infinite models?

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Modular Finite Arithmetic
Let N be a set, 0,m two elements and S|N -> N a function.
Axioms for modular finite arithmetic:

0, m in N; Sm = 0
for all x, Sx in N
for 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 A

Definition of addition by induction.
0 + y = y
Sx + y = x + Sy
Definition of mulplication by induction.
0 * y = 0
Sx * y = x*y + y

Is 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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