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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