8 ms·
Great question, I have no answer. The article explained forcing in such a way as to simply restate what I thought we already knew: given a real, there is no "n
by finnh 5y ago
Great question, I have no answer.
The article explained forcing in such a way as to simply restate what I thought we already knew: given a real, there is no "next" real. (ie, there are a non-countable-infinite number of reals between any two reals).
I don't see the newness that forcing brings to this.
- deleted 5y ago[deleted]
- dwohnitmok 5y agoForcing is about a completely different question. The question is not where there are reals between reals, it's a question of the size of sets. In particular is there a set strictly larger than the natural numbers, but strictly smaller than the reals? Forcing allows us to construct such a set in ZFC.
- CamperBob2 5y agoNot a mathematician, so this probably has an obvious answer, but who says that "size" is a necessary universal property of a set? It doesn't seem any more rational to speak of size as a required property of a set than it would be to treat color or weight that way. Some sets have those attributes, but not all. It seems perfectly reasonable to say that the set of real numbers is one of those sets that doesn't have a "size" property.
- dwohnitmok 5y agoSize is a hand-wavy way of saying it. What we're really asking is given two sets A and B, is there a function from A to B such that no two elements from A are mapped to a single element in B? It turns out if you use ZFC (in particular choice) then one direction must hold. Either such a function exists from A to B or from B to A. This means that we can order all sets using the existence of these functions, and so in that sense size is a "universal property."