Headhurtiness!
Sep. 26th, 2004 11:02 pmAn apparent paradox:
Consider the class of all undefinable ordinals. Assume it is nonempty. It therefore has a least element. However, this least element can be defined as the least undefinable ordinal, and is thus definable! Therefore the class is empty, and all ordinals are definable. But, as the ordinals are very definitely not countable, there must be undefinable ordinals.
I came up with this a few days ago, and it hurt my head for quite a while. I asked Josh about this, naturally enough, and he was, in fact, able to resolve the paradox for me very quickly; I could post the resolution here, but I'd rather just hurt all your heads for a few days. :D (Well, those of you who can't figure it out, anyway.)
-Sniffnoy
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There are 2 possible numbers of sources for a quote: 0 and infinity.
Consider the class of all undefinable ordinals. Assume it is nonempty. It therefore has a least element. However, this least element can be defined as the least undefinable ordinal, and is thus definable! Therefore the class is empty, and all ordinals are definable. But, as the ordinals are very definitely not countable, there must be undefinable ordinals.
I came up with this a few days ago, and it hurt my head for quite a while. I asked Josh about this, naturally enough, and he was, in fact, able to resolve the paradox for me very quickly; I could post the resolution here, but I'd rather just hurt all your heads for a few days. :D (Well, those of you who can't figure it out, anyway.)
-Sniffnoy
--
There are 2 possible numbers of sources for a quote: 0 and infinity.
