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).