Apparently Tarski and Mostowski proved this, but intuitively I'm not seeing the difference between statements in a language of non-negative integers with equality, addition, and multiplication vs Presburger Arithmetic, which has only equality and addition. It would seem to me that multiplication with non-negative integers can be represented as repeated addition, so what is it about multiplication that makes statements undecidable? Or is it some other subtlety that I'm missing?
2026-03-27 01:47:40.1774576060
Why is the set of all true first-order statements about non-negative integers in the language with only equality, $+$ and $\times$ undecidable?
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Multiplication cannot be represented in Presburger arithmetic as repeated addition. To say that $ab=a+a+\cdots +a$ ($b$ times) would use a "sentence" of variable length, and that is not how a sentence is defined. For a proof that it cannot be done, combine the Tarski-Mostowski result with the result that Presburger arithmetic is decidable.