Relatedly, intuitionists reject bivalence - the view that every proposition is either true or false. This means that they can’t straightforwardly accept the t-schema of p is true iff p, since in cases where a proposition is neither true nor false, we have ‘p is not true’ and ‘not not p.’
Slight correction RE intuitionism. Intuitionists don’t accept that there are cases where neither p nor not p. That would entail contradiction. Rather they don’t affirm p or not p in certain cases. But they will always affirm the double negation of p or not p and so will always reject ‘neither p nor not p.’ I know this was intended as an introduction and so the explanations were kept simple, but I felt a sudden urge to be the ‘well acktually’ guy.
43:30 I'm not sure what it means for God to know all the propositions like 1 = 1 and 2 = 2 and 3 = 3 etc. or what the difference is from when a human thinks about that. I would say that I also know those things, even if I can never actual write that infinite number of propositions down. Of course you could also reduce those propositions to a single proposition with a quantifier over N but I think it's also reasonable to see them as an infinite number of propositions that all people with a basic understanding of mathematics know. Because of this I'm not sure if characterizing omniscience as knowing an infinite number of propositions is very useful.
small mistake: rational numbers are the quotient of integers, not just natural numbers.
Relatedly, intuitionists reject bivalence - the view that every proposition is either true or false. This means that they can’t straightforwardly accept the t-schema of p is true iff p, since in cases where a proposition is neither true nor false, we have ‘p is not true’ and ‘not not p.’
Slight correction RE intuitionism. Intuitionists don’t accept that there are cases where neither p nor not p. That would entail contradiction. Rather they don’t affirm p or not p in certain cases. But they will always affirm the double negation of p or not p and so will always reject ‘neither p nor not p.’ I know this was intended as an introduction and so the explanations were kept simple, but I felt a sudden urge to be the ‘well acktually’ guy.
43:30 I'm not sure what it means for God to know all the propositions like 1 = 1 and 2 = 2 and 3 = 3 etc. or what the difference is from when a human thinks about that. I would say that I also know those things, even if I can never actual write that infinite number of propositions down.
Of course you could also reduce those propositions to a single proposition with a quantifier over N but I think it's also reasonable to see them as an infinite number of propositions that all people with a basic understanding of mathematics know. Because of this I'm not sure if characterizing omniscience as knowing an infinite number of propositions is very useful.