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It's true that -we have statements about linear algebra for which (in ZFC) we can neither prove nor disprove-[1]. However, given that these statements simply ar

It's true that -we have statements about linear algebra for which (in ZFC) we can neither prove nor disprove-[1]. However, given that these statements simply aren't true, it doesn't tell us much: it's not that we can't know everything there is to know about linear algebra. It's that these things we can't know about linear algebra aren't facts about linear algebra to start with. [1]: On a second thought, let me rephrase that: we have statements about structures partially described by linear algebra which (in ZFC) we can neither prove nor disprove for all of them at the same time.

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 ago
  Thank 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 ago
    It 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 ago
      You are correct.

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 ago
Thank 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 ago
  It 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 ago
    You are correct.