

Does Cantor’s Diagonalization Proof Cheat? - bdr
http://rjlipton.wordpress.com/2010/06/11/does-cantors-diagonalization-proof-cheat/

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mjtokelly
Good Math, Bad Math has a series of posts about attempts to disprove the
uncountability of reals:

<http://scienceblogs.com/goodmath/bad_math/cantor_crankery/>

Especially frustrating is the credibility Knol can provide to crackpots like
this.

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WilliamLP
Where I part ways with mathematicians is the intuition we should ascribe to
Cantor's argument. They say that "almost all" real numbers are irrational.
What Cantor's argument says is that "at least one" is irrational.

I know the arguments about measure and probability and such, but, there is
also the fact that for every two real numbers, there is an infinity of
rational numbers in between! So that makes me (personally, not mathematically)
not like the "almost all" intuition, instead of an intuition of there always
being at least one more, in a particularly slippery way.

~~~
ihodes
Ah, but Cantor's proof doesn't just prove that there's "at least one", it
proves that there are infinite irrationals.

It's just the way this proof is often presented that leads people to believe
that it only proves that at least one is irrational; but once you accept that
there is one irrational number, you can "index" it and add it to your
denumerable list (pretend it's rational, say) and then run the diagonlization
argument again, yielding yet another irrational number. Do this ad infinitum.
Or, rather, stop: this will not halt ;)

Other proofs prove that the rationals are not equinumerous to the reals, but
that there _is_ a one-to-one function from the rationals to the reals… but
that's another story.

(EDIT: as I'm getting a lot of hate and downvotes for this, I suppose I could
delete the comment. It was intended to try to clear matters up for the parent
of this thread. It wasn't intended as a deep explanation of anything at all.
Forgive my triviality.)

~~~
hugh3
It doesn't just prove that there's infinitely many irrationals -- that is
trivially easy to prove. For instance we know pi is irrational, therefore 2pi,
3pi, 4pi, 5pi,... etc are all irrational as well.

Cantor's proof proves that there's no one-to-one correspondence between reals
and integers (whereas there is a one-to-one correspondence between integers
and rationals), and therefore the infinite set of reals is "bigger" than the
infinite set of rationals.

That there is an infinite number of irrationals should follow as a trivial
consequence of this. (Left as an exercise for the reader...)

~~~
ihodes
I understand this: I was attempting to show, intuitively, how his theory could
demonstrate that there is more than one irrational number.

Apologies if I wasn't clear: this was something that popped into my head to
help clear this up!

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ggchappell
This is an interesting article. But I think he's being too mysterious about
this notion of "cheating".

Consider a far smaller set: {1, 2, 3}. Bob wants to prove that there are at
most 3 elements in this set. (Yes, I know this is trivial.) His strategy: name
3 elements in the set (1, 2, 3), and no matter what Alice picks from the set,
he will name her choice. Obviously, this strategy works.

Now suppose that Bob wants to prove that there is only 1 element in the set
{1, 2, 3}. His strategy is similar: name 1 element in the set (2, say), and --
he hopes -- no matter what Alice picks from the set, he will name her choice.
Of course, this doesn't work. However, in order to be _sure_ that Bob does not
name her choice: _Alice must know his strategy_. Does that mean she is
"cheating"? Hardly.

The above "cheating" issue is the same as that in the article. So we're not
talking about some mysterious property of infinite sets.

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jerf
The "real" problem here is that the POV the post argues against appears to
include the idea that mathematical proofs must be isomorphic to some game
between two (or perhaps more) people that a human brain would consider "fair".
This is not a generally useful way to approach proofs.

(The word "generally" is not extraneous in this post.)

