Why is the set of all true first-order statements about non-negative integers in the language with only equality, $+$ and $\times$ undecidable? 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?

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.
