Hi Harvey,
Thanks for the reply.
I didn't say that the term 'true' had no mathematical meaning. Meaning is possible by distinguishing true results from false results (i.e., in two-value logic), by following rules of inference that lead to only true results, etc. This is not a definition per se. A definition is a reduction, but you cannot reduce the word 'true' in mathematics. The reason is that any reduction would entail use of axioms (etc) that are considered 'true'.
I'm sorry but I have no idea what you are talking about there. Can you be less elusive and more specific? For one thing, truth has nothing to do with meaning, I don't know why you brought meaning up. A statement can be true even if you don't know what it means. For instance, this statement:
"if A includes B, and B includes C, then A includes C"
The truth of that statement is far clearer than its meaning. Which is the whole point of mathematics, to discover what is true regardless of meaning.
Axioms are statements that tell you under which certain conditions a mathematical statement is to be considered 'true'.
But that's the whole point. You can't have truths without axioms, but the truths of math are no less true just because the axioms can't be proven true. Whether explicitly or not, all mathematical truths are expressed in the form "if A then B". Math never says "things are such and such", all it says is "if things are such and such, then such and such necessarily follow".
With that in mind, it doesn't even make sense to talk about the truth of an axiom. |