Typical statements (first order) of the Peano Axioms puzzle me. Neither a mathematician nor logician, I find myself thinking the following: One would hope that arithmetic is consistent with the world as it is. So the axioms of arithmetic should be true in a domain containing the items that populate reality, e.g., a domain containing this keyboard upon which I now type. But this keyboard is neither identical to zero nor is it the successor (or predecessor) of any whole non-negative number. So what's with, e.g., (Ax)((x = 0 v (Ey)(x = Sy))? On what would think its intended interpretation, the axiom (theorem in some versions) seems false "of reality." And some other typical items of (first order) expositions seem either false or at least meaningless, e.g., (Ax)(Ay)(x + Sy = S(x + y)). What could be meant by "the sum of this keyboard and the successor of 6 is equal to the successor of the sum of this keyboard and the positive integer 6?
Unless one has already limited the domain to exclude typical non-arithmetic items, then stating the (first order) Peano Axioms with leading universal quantifiers seems to produce false and false or meaningless statements. So how would one try to change/complicate the (first order) axioms to avoid this? I recall reading somewhere that in some of his work Tarski would use a predicate for non-negative integers to limit the scope, something like "for all x, if x is a member of the non-negative numbers, then...." But how else might I think about this? Thanks for helping un-confuse me. Or don't we care if the Peano Axioms are not true of the world we live in? Wayne W.
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