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Isn't this just a special case of one of Godel's incompleteness theorems? Roughly, the incompleteness theorems state that, in any sufficiently powerful, self-c
by csense 14y ago
Isn't this just a special case of one of Godel's incompleteness theorems?
Roughly, the incompleteness theorems state that, in any sufficiently powerful, self-consistent, decidable mathematical axiom system, there are statements which are true, but cannot be proved. And "sufficiently powerful" means basically "including addition and multiplication." The high school axioms look like they're at least that powerful, so wouldn't it just be a direct application of Godel that such statements exist?
- siwei 14y agoNo, the question here is not "are there statements in _this axiom system_ that cannot be proved by this axiom system" (what incompleteness theorem says), but "are there statements about _the positive integers_ that cannot be proved by this axiom system". In other words, is this system of axioms equivalent to the axioms that define positive integers.