Statement of The Problem
Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school:
- x + y = y + x
- (x + y) + z = x + (y + z)
- x · 1 = x
- x · y = y · x
- (x · y) · z = x · (y · z)
- x · (y + z) = x · y + x ·z
- 1x = 1
- x1 = x
- xy + z = xy · xz
- (x · y)z = xz · yz
- (xy)z = xy · z.
These eleven axioms, sometimes called the high school identities, are related to the axioms of an exponential ring. Tarski's problem then becomes: are there identities involving only addition, multiplication, and exponentiation, that are true for all positive integers, but that cannot be proved using only the axioms 1–11?
Read more about this topic: Tarski's High School Algebra Problem
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