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CronoDAS comments on Harry Potter and the Methods of Rationality discussion thread, part 8 - Less Wrong
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Yes, ZFC is quite enough to imply the existence of the first uncountable ordinal.
On the other hand, I don't see what's unbelievable about such a thing; it's just (the order type of) the set of all countable ordinals, and I don't see why it's unbelievable that there is such a set. (That is, if you're going to accept uncountable sets in the first place; and if you don't want that, then you can criticise ZFC on far more basic grounds than anything about ordinals.)
I had to look carefully in order to see that it doesn't necessarily contradict itself even though I should have known this from Gödel, Escher, Bach.
On reflection this ordinal probably represents something real -- a set of Gödel statements, which we'd regard as 'true' if we knew about them. Or rather, the fact that it seems meaningful to deny the existence of a general formula for producing these Gödel statements that will generate any given example if the process runs long enough. (To get an uncountable set of the right kind I might have to qualify this by saying something like "G-statements you could generate starting from a given system and a given method of Gödel numbering," but I can't tell how much of that we actually need.)