Re: A question about Caontor's proof of the uncountability of the reals

Michael Olea wrote:
What's wrong with this argument? This is an honest question, not a troll -
I've never studied mathematical logic.
What am I missing?
(snip)


First, if one follows your conclusion, any proof by contradiction is no
longer allowed, since proving that some proposition is false no longer
means it is true.

In fact, if you *did* manage to prove the contradiction of some
proposition P, you're done. P is false, and therefore not P is true.
However, if you only manage to prove that the assumption that P is true
does *not* lead to contradiction, you cannot yet conclude that P is
true, because there may be a case that assuming that P is false does
not lead to contradiction either, in which case P is undecidable. I
hope you see the difference between two cases.

Second, in relation to Cantor's diagonal proof, to you and all others
who claim something like "I don't under Cantor's proof, therefore the
hypothesis that R is uncountable is not true" - you might be surprised,
but there are other proofs, which in some respect more solid, since
they do not rely on the fact that the real numbers are represented as
some infinite decimals. There is a proof that shows that reals as a
special case of a linear continuum cannot have a bijection to the set
N. But then you know a few things about topology.

.

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