4 ms·
But Galileo's objection would seem to imply that there should be more infinities, not fewer.
by dllthomas 4y ago
But Galileo's objection would seem to imply that there should be more infinities, not fewer.
- cubefox 4y agoA definition like "F and G are the same size if there is a one-to-one correspondence, except when F is a proper subset of G, then F is smaller" wouldn't work, because that isn't an equivalence relation. While "is the same size" seems in fact to be such an equivalence relation. For example, there might be a set H, such that H would be "the same size as" F and as G (due to one-to-one correspondence), but F would be smaller than G (due to proper subsethood). This would violate transitivity.
- dllthomas 4y agoI'm not saying there's a definition that works down that path (I don't know); I'm saying that "we only have multiple infinities because of this definition, which people objected to because it didn't give us enough infinities" seems a little muddled.
- cubefox 4y agoThe argument seems to be rather that intuitions for size and subsethood and bijection aren't compatible anymore, which is a reductio argument against assuming that size comparisons make any sense for infinities at all.