/sci/ - Science & Math

>>15049333
Yes I'd say so.

Again, the main thing that's different, I'd say, is that something like "{x in A | P}" can come from a far more richer class of things than classically.
E.g. basically in most theories, the set
S_CH := {0 in {0} | CH}
will be a subset of {0}, and so part of P({0}), but it can not be proven equal to either {} or {0}.
P({0}) is rich, you can't know it's cardinality, because (if your theory is a subtheory of ZF) it must remain true that new axioms (such as LEM) can collapse |P({0})| down to =2.

Similarly, S_CH is not a countable set. If you want to keep on using the classical language, then S_CH is a subset of {0} but it's also uncountable.
At the same time IZF still permits that N^N is in the surjective image of a subset of N.

>>15049384
There's a book draft online by Rathjen, just google and take the first ref.
If you want to do category theory you can just do topos theory I think, since the former is kinda taken over by type theory. Basically, I think extensionality makes constructive set theory quite weird, I'd argue.
The reason I like it and have researched it previously is that it's both conservative (not many assumptions) while still being directly compatible with ZFC.

>>15049333
Yes I'd say so.

Again, the main thing that's different, I'd say, is that something like "{x in A | P}" can come from a far more richer class of things than classically.
E.g. basically in most theories, the set
S_CH := {0 in {1} | CH}
will be a subset of {1}, and so part of P({1}), but it can not be proven equal to either {} or {1}.
P({1}) is rich, you can't know it's cardinality, because (if your theory is a subtheory of ZF) it must remain true that new axioms (such as LEM) can collapse |P({1})| down to =2.

Similarly, S_CH is not a countable set. If you want to keep on using the classical language, then S_CH is a subset of {0} but it's also uncountable.
At the same time IZF still permits that N^N is in the surjective image of a subset of N.

>>15049384
There's a book draft online by Rathjen, just google and take the first ref.
If you want to do category theory you can just do topos theory I think, since the former is kinda taken over by type theory. Basically, I think extensionality makes constructive set theory quite weird, I'd argue.
The reason I like it and have researched it previously is that it's both conservative (not many assumptions) while still being directly compatible with ZFC.