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Thanks for the reply. Question though. in Cantor's argument we explicitly mapped the reals to aleph-0 so it makes sense that our conclusion decides that mapping

Thanks for the reply. Question though. in Cantor's argument we explicitly mapped the reals to aleph-0 so it makes sense that our conclusion decides that mapping to aleph-0 is too small so it's size must be larger. Where in the forcing process do we even "use" aleph-1? If we used aleph-1 then it could see the parallels and the argument would make sense - but all I see in the forcing process is "start with a set of all reals". Nothing about trying to map them to aleph-1. Maybe it's implicit, but couldn't I have just changed step 1 to instead be "start with a list of size aleph-90, then use forcing to prove that the reals are larger than aleph-90." In Cantor's argument it feels like we "used" the natural numbers. Here it seems more like we said take an arbitrary number of them and see this one isn't in it. But that arbitrary number of them (aleph-1) could've been any arbitrary number of them because we never really used any property of the set being aleph-1 vs aleph-90.

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
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- dwohnitmok 5y ago
  Forcing 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 ago
    Not 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 ago
      Size 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."

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
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dwohnitmok 5y ago
Forcing 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 ago
  Not 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 ago
    Size 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."