Tarski's High School Algebra Problem - Statement of The Problem

Statement of The Problem

Tarski considered the following eleven axioms about addition ('+'), multiplication ('·'), and exponentiation to be standard axioms taught in high school:

  1. x + y = y + x
  2. (x + y) + z = x + (y + z)
  3. x · 1 = x
  4. x · y = y · x
  5. (x · y) · z = x · (y · z)
  6. x · (y + z) = x · y + x ·z
  7. 1x = 1
  8. x1 = x
  9. xy + z = xy · xz
  10. (x · y)z = xz · yz
  11. (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?

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