How do you create the smallest possible field? The one with only two members, {0,1}? By specifying a new axiom whose sole purpose is to state that there will be no more axioms? No. The smallest possible field consists of solely the axioms themselves.
But, specifically I took that set of axioms to be a proof system. One where it’s meaningful to ask if something is provable/unprovable. The axioms by themselves do form a complete proof system; the field with two members.
And specifically, I filled in the truth table for that proof system using the axioms. The last couple steps were completed from theorems created from the axioms, and won’t be true of every possible field. (e.g. in a field with more than two members, I couldn’t have done the 1+1=0 step the same, since there would be another possible way to make an additive inverse.)
I see now that he wasn’t taking the feld axioms as a proof system. So, his statement that it was provable in one proof system based on this, but not provable in another system based on this, doesn’t mean that they are both incomplete systems. They are separate systems.
Yes, I did. Because he had said that there was no absolute provable/unprovable, only provable/unprovable within a system, I took it to mean that talking about the axioms was talking about a proof system.
It feels to me like he’s claiming it’s a single proof system to show incompleteness, (since that only has meaning in a single system,) but using every possible extension to “prove” that incompleteness.
No, it doesn’t. but, I see how you thought I was saying that. Once again a conflict between what is and isn’t a proof system.
proving something only has meaning in a proof system, right? From the axioms alone as their own proof system, there is such an x.
I wasn’t talking about what was true for every field since that would be talking about multiple proof systems. I didn’t think we were talking about multiple proof systems. Since claiming incompleteness based on multiple systems doesn’t have meaning.
Claiming that the field axioms are incomplete and this shows incompleteness even without godel’s result, is claiming it as a single proof system. x*x = 1+1 is not a question that shows incompleteness in any possible field. (and that’s the only place incompleteness has meaning, within a field. “can I answer the question in this field? Yes. Can I answer it in that other field? Yes. But I came up with two different answers. Does that make them inconsistant? No, you take each field separately for questions of completeness and consistancy.”)