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But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but tha

But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but that conjecture couldn't be proven under the given set of axioms... then in that case wouldn't it be that there were some truths about the universe that aren't possible to prove? I don't think it's irrational or anti-science to say or think that. It just seems like it might be a possibility, which sure, why the hell not? "There's this thing we can't prove under the given axioms"..."Ok, well just extend the axioms"..."Right but there is still this other thing we can't prove under that set of axioms."..."Ok, so rinse and repeat?" How, if you can't prove it, do you figure out if it is an axiom in the first place?

joe_the_user 8y ago
Questions, question... I invite to learn some mathematical logic but I can give points. But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but that conjecture couldn't be proven under the given set of axioms... then in that case wouldn't it be that there were some truths about the universe that aren't possible to prove? - Well, the axioms of a given model generally can't be proven. Every model one builds begins with "unprovable things". However, the "truth about the universe" would have to be demonstrated by experimentation and mathematics would only provide the model. So every theory about the universe is using something unprovable (the axioms of its model). But Godel's theorem in particular (technically Godel's 1st incompleteness theorem) is still just a statement about proof processes work and how truth-assignments work. Whether are ineffable truths or not is a different question. "There's this thing we can't prove under the given axioms"..."Ok, well just extend the axioms"..."Right but there is still this other thing we can't prove under that set of axioms."..."Ok, so rinse and repeat?" - Yes, you can do that, well you can talk about doing it. In fact, Godel's "completeness theorem" is based on this procedure, more or less. The thing is you start with a set of axioms, add a proposition that can't be proven either true or false under the axioms, and decide arbitrarily whether to make it true or false. Continue forever (or transfinitely, depending how huge your system is) and you get a complete truth-assignment. Of course, there's no algorithm for finding all these undecidable propositions so this procedure is very theoretical. You or I or an AI couldn't do this to arrive pocessing this complete set but mathematically you can say "I hereby choose all the unprovable axioms and string them together thus (one of those weird "axiom of choice" things). So this approach is used, it's just it doesn't us to escape unprovable things.
- james_s_tayler 8y ago
  Thanks for this answer. You helped me think about things a little deeper. I also read through the recent guide on how to self-study logic and learned a couple more things on the subject. If I'm understanding things right it's that proving things with logic means you are showing that something is consistent within a system and that is all you are definitely able to say from that. Not whether something is 'true' in a wider sense outside of that system. And all axioms are assumptions, so all our proofs are saying is that 'assuming that these things are the case then we can show the following result is commensurate with those assumptions and completely internally consistent'.

joe_the_user 8y ago

Questions, question... I invite to learn some mathematical logic but I can give points. But what's say there was some conjecture that if proved to be the case proved a unifying theory of everything in Physics that relied on that conjecture, but that conjecture couldn't be proven under the given set of axioms... then in that case wouldn't it be that there were some truths about the universe that aren't possible to prove? - Well, the axioms of a given model generally can't be proven. Every model one builds begins with "unprovable things". However, the "truth about the universe" would have to be demonstrated by experimentation and mathematics would only provide the model. So every theory about the universe is using something unprovable (the axioms of its model). But Godel's theorem in particular (technically Godel's 1st incompleteness theorem) is still just a statement about proof processes work and how truth-assignments work. Whether are ineffable truths or not is a different question. "There's this thing we can't prove under the given axioms"..."Ok, well just extend the axioms"..."Right but there is still this other thing we can't prove under that set of axioms."..."Ok, so rinse and repeat?" - Yes, you can do that, well you can talk about doing it. In fact, Godel's "completeness theorem" is based on this procedure, more or less. The thing is you start with a set of axioms, add a proposition that can't be proven either true or false under the axioms, and decide arbitrarily whether to make it true or false. Continue forever (or transfinitely, depending how huge your system is) and you get a complete truth-assignment. Of course, there's no algorithm for finding all these undecidable propositions so this procedure is very theoretical. You or I or an AI couldn't do this to arrive pocessing this complete set but mathematically you can say "I hereby choose all the unprovable axioms and string them together thus (one of those weird "axiom of choice" things). So this approach is used, it's just it doesn't us to escape unprovable things.

james_s_tayler 8y ago
Thanks for this answer. You helped me think about things a little deeper. I also read through the recent guide on how to self-study logic and learned a couple more things on the subject. If I'm understanding things right it's that proving things with logic means you are showing that something is consistent within a system and that is all you are definitely able to say from that. Not whether something is 'true' in a wider sense outside of that system. And all axioms are assumptions, so all our proofs are saying is that 'assuming that these things are the case then we can show the following result is commensurate with those assumptions and completely internally consistent'.