I meant “the set of all natural numbers”, IIRC, he’s explicitly said he’s not an ultrafinitist, so either he considers that as an acceptable infinite set, or he considers the natural numbers to exist, but not the set of them, or something.
I meant “if you accept countable infinities, where and how do you consider the whole cantor hierarchy to break down?”
I find it best not to speak about “existence”, but speak instead of logical models that work. For example, we don’t know if our concept of integers is consistent, but we have evolved a set of tools for reasoning about it that have been quite useful so far. Now we try to add new reasoning tools, new concepts, without breaking the system. For example, if we imagine “the set of all sets” and apply some common reasoning to it, we reach Russell’s paradox; but we can’t feed this paradox back into the integers to demonstrate their inconsistency, so we just throw the problematic concept away with no harm done.
I meant “the set of all natural numbers”, IIRC, he’s explicitly said he’s not an ultrafinitist, so either he considers that as an acceptable infinite set, or he considers the natural numbers to exist, but not the set of them, or something.
I meant “if you accept countable infinities, where and how do you consider the whole cantor hierarchy to break down?”
What would it even mean for the natural numbers (the entire infinity of them) to “exist”?
What makes a set acceptable or not?
This question sounds weird to me.
I find it best not to speak about “existence”, but speak instead of logical models that work. For example, we don’t know if our concept of integers is consistent, but we have evolved a set of tools for reasoning about it that have been quite useful so far. Now we try to add new reasoning tools, new concepts, without breaking the system. For example, if we imagine “the set of all sets” and apply some common reasoning to it, we reach Russell’s paradox; but we can’t feed this paradox back into the integers to demonstrate their inconsistency, so we just throw the problematic concept away with no harm done.
It sounds weird to me too, which is why I asked it—because Psy-Kosh said EY said something about integers, or the set of integers, existing or not.
secondary sources? bah!
LMGTFY (or the full experience)