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You're right there!Intention or not, you've been hoist by your own petard on this one.

But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals.

He has misquoted me on a

**number**of occasions (and you have too, once). Now he assigns implications to me that I have NOT made and heads off on his flight of 'post hoc' fancy, trying to imagine my thought processes. I have acknowledged valid criticisms ... please refer to my earlier posts.

Paul.

PS.

You are forgetting your own advice. According to you, I have to prove every statement I make ...

... but you don't have to!Your claim that:

" The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't."

is not fine on it its own since you have not proven that the reals are not countable, and that the finite union of two countable sets is countable.

Best make up your mind, Matt.Though I've not proven you may list the rationals, and not list the reals, though they are proofs found in many places.