4 ms·
You’ve given excellent explanations on this topic. It’s rare that someone writes about the meaning of the Incompleteness theorems correctly. I think it’s wort
by hykh 6y ago
You’ve given excellent explanations on this topic. It’s rare that someone writes about the meaning of the Incompleteness theorems correctly.
I think it’s worth pointing out or adding that when talking about The Natural Numbers most people are implicitly talking about the standard model with the first order Peano axioms. There are statements in this model that are true (have no counterexample) but which can’t be proven by this set of axioms.
I think many people making claims like,
“There are true statements about The Natural Numbers that aren’t provable.”
don’t realize exactly the nuances you’ve pointed out. Assuming The Natural Numbers are consistent then all true statements are provable in some axiom system. Just take the collection of all true statements as the axiom system. Now every true statement is trivially provable.
I hope what I’ve written doesn’t muddy your excellent explanations!
- abjKT26nO8 6y agoThank you. > Assuming The Natural Numbers are consistent then all true statements are provable in some axiom system. Just take the collection of all true statements as the axiom system. Now every true statement is trivially provable. That axiom system wouldn't be particularly useful to humans though. When we talk about sets of axioms, we almost always talk about finite sets of axioms. This is what makes them useful to us, allows us to use them for describing things. But you do have the right intuition here. The next step is using the compactness theorem[1]. [1]: https://en.wikipedia.org/wiki/Compactness_theorem https://en.wikipedia.org/wiki/Compactness_theorem
- hykh 6y agoIt doesn’t have to be a finite set of axioms just recursively enumerable. The first order Peano axioms are not finite in number. One of them is an axiom schema. (I believe.)
- abjKT26nO8 6y agoYou are correct.