PeteOlcott wrote: ↑Fri Apr 05, 2019 1:11 am
Ah, I see that my prior reply was incorrect.

This one also does not address my question. You be dancin' when you should be clear and direct.

Originally you said:

PeteOlcott wrote: ↑Thu Apr 04, 2019 4:02 am
It turns out that axioms are the ultimate foundation of Truth

So I put to you this situation. There are two mathematicians. Following the custom of cryptographers we will call them Alice and Bob. Alice is pro-choice (bad pun ok). She accepts the axiom of choice. In her world you can pick an element from each of a collection of nonempty sets. Every vector space has a basis. A set is infinite if and only if it's Dedekind-infinite. The

Banach-Tarski paradox is a theorem.

Bob, on the other hand, is no-choice. He rejects the axiom of choice. In his world, there's a collection of nonempty sets without a choice function. There's a vector space without a basis. There's an infinite set that's not Dedekind-infinite. The Banach-Tarski paradox is NOT a theorem.

Now you say that axioms determine truth. So what is the truth of the axiom of choice?

Now to be fair I used an example only familiar to people who have studied some math. I used the axiom of choice because it's abstract and it's clear that it has a definite truth value only to a Platonist.

But I could just as easily have put to you the same question about the parallel postulate. We have a consistent geometry in which there's exactly one parallel to a line through a given point not on the line. We have consistent geometries where there are no such parallels, and consistent geometries where there are many such parallels.

Which geometry is true? It's a question of physics; so unlike with the axiom of choice,

*there is a definite truth of the matter*. However, the axioms of geometry do not resolve the issue. To determine the truth of the parallel postulate, one must examine physical reality using scientific experiment.

*This is an active research area of cosmology*. Real life physicists and astronomers go to work each day and attempt to discover the true geometry of the universe.

*But they cannot rely on axioms*. They must make physical observations.

These two examples, one abstract and the other physical, falsify your claim that axioms determine Truth (your capitalization).