Subtraction is not closed over positive integers, which is untidy. The point of Tarski’s conjecture was to propose a minimal number of axioms and operations, AFAICS they define the standard semiring of positive integers (with the natural definition of exponentiation added).
reply(Edit: positive integers aren’t exactly a semiring because 0 is excluded, although some authors do define a semiring without the requirement of an additive identity element.)
Well, yes, but negative numbers are also well known to every high school math student.
replySure. But "High School Algebra (Excluding Subtraction) Problem" isn’t as catchy a name.
replyThey subtracted the subtraction exclusion in the name of simplicity?
replyBecause subtraction is not a total operation on positive integers. Negative numbers leave the domain.
replyWhy was it important to Tarski to limit the domain to positive integers? That seems pointlessly arbitrary.
replyIf you allow negative integers, you get negative exponents. To accomodate negative exponents, you can expand the domain again, to rational numbers. If you allow rational numbers, then you have to allow rational exponents, which means you can take arbitrary nth roots. Which you could do, but now you're most of the way to the complex numbers (plus roots aren't unique anymore).
replyTruly, you could say that about many conjectures, especially more “fun” classical ones.
replyI'm not sure, but maybe it is due to that the expression a - b can be replaced as a + (-b)?
replySimilarly, I think a * b and a / b can be replaced with the same trick, but then I realized it may not work on non-abelian, or where multiplicative inverse is not available...
We’re in the semiring of positive integers, so there are no additive (or multiplicative) inverses.
replyThe subtraction point is interesting but I don't think it makes the result disappointing. The whole point of Tarski's problem is what follows from that very restricted set of elementary identities so finding the exact minimum countermodel under those rules still seems like a pretty satisfying result.
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