The Contradictory Nature of Equality as a Foundational Axiom in Arithmetic

Properties of Equality

First, we define three basic properties as follows:

a = a

If a = b then b = a

If a = b and b = c, then a = c.

1. a = a requires, a=b and b=a; hence (a,b)=(a,b)

2. a=b and b=c, then a=c; hence b=(a,c) where subject to point 2 ((a,b)=(a,b))=c

3. with (a,b)=c necessitating c=c with c=d and d=c; hence (c,d)=(c,d)

4. repeat of point 2 through axioms of point 3.

5. Thus the property of equality is subject to the fallacy of circularity and infinite regress while being subject as an unjustified axiom in itself in accords to the Munchauseen Trillema.

6. This continual recursion necessitates equality as fundamentally "=" as fundamentally a state of indefiniteness where it is a foundation of contradiction.

## The Contradictory Nature of Equality as a Foundational Axiom in Arithmetic

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