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> That there are infinities of various sizes follows if you accept Hume's Principle, which says "the number of things with the property F equal the number of th
by bmacho 4y ago
> That there are infinities of various sizes follows if you accept Hume's Principle, which says "the number of things with the property F equal the number of things with the property G if and only if there is a one-to-one correspondence between those that are F and those that are G."
I don't think that accepting(?) principles(?) is the right way to think about it.
This ordering on this family of infinities is as much of a definition as everything else. You don't accept principles when you talk about matrices, or circles or rings.
- cubefox 4y agoMatrices and rings are technical terms without preexisting intuitions, but "the same size" seems to be an intuitive concept, like "natural number" or "circle". So presumably any definition should be compatible with our existing concepts. Although Galileo argues against size having an intuitive sense for infinities.
- bmacho 4y ago> So presumably any definition should be compatible with our existing concepts. Yes, this is the wrong way (IMO) to think about it. The right way is to notice that it is a definition, and to think about it like every other definition. It has the name "same size", because it has some properties common with "same size" of finite cardinalities, but not all properties common. And this is the proper way to think about every other definitions and objects as well, even if they are technical terms, or borrowed from common English. (BTW I believe that rings are called rings, because of some similarities with rings.)