More on proofs

[ciphergoth]

Some really interesting answers on what proofs you know. The most common proofs people mention are the ones I name, plus Cantor's diagonalisation argument that |R| > |N| - cool. More specific comments follow.

wildbadger -- product of two compact spaces is compact

A strong opening! I looked it up and found Tychonoff's Theorem - looks interesting.

cryptodragon -- A number of them from crypto stuff

Interesting, name us a favourite?

simple_epiphany -- As a third-year maths student, I'm required to know quite a lot of them, but the one for the Bolzano-Weierstrass theorem is quite nice.

Bolzano–Weierstrass theorem on Wikipedia. I think I could remember that proof. Cool, thanks!

aegidian -- Cantor's Diagonalisation, proving there are as many rational numbers as there are integers.

Ah, the proof that |Q| = |N| rather than the proof that |R| > |N|?

olethros -- Maybe I lied. I can prove (by recursion) that all marbles in the world are the same colour.

I know that proof :-) Oh go on, there must be a *valid* proof you like!

keirf -- Bolzano-Weierstrass theorem - every bounded sequence in R{n} has a convergent subsequence

A second showing for this theorem!

ajva -- that 0.999...=1

Don't you need to get into the construction of the real numbers to explain this one?

ergotia -- The infinite number of primes/hotel at the end of the universe one

Two proofs for the price of one :-)

nikolasco -- irrationality of sqrt(2) (fundamental theorem of arithmetic, even/odd, well-ordered)

What proofs are you referring to with "even/odd, well-ordered"?

Thanks all, please keep commenting :-)

Comments

[elsmi.livejournal.com]

> product of two compact spaces is compact
> A strong opening! I looked it up and found Tychonoff's Theorem - looks interesting.

Beware! Those are not the same! The former is elementary, the latter is equivalent to the axiom of choice!

Because... that might... be really important. Someday. That you not mix those up. You know.

The diagonalization proof of the Halting Problem that someone mentioned up above is truly beautiful, much better than the classic diagonalization results *or* Goedel's first incompleteness theorem. I mostly put that generalized diagonalization theorem as my answer because it has some personal significance.

It might be fun to talk about favorite results, too. Of the top of my head, Stone-Weierstrass comes to mind (the proof is boring and technical, but the result is so neat! I hadn't thought of it in years but the Bolzano-Weierstrass stuff reminded me), or the curious embedding properties of the Cantor set. (That dude just keeps popping up...)

[ciphergoth.livejournal.com]

Ah, yes, the Wikipedia article cautions that product-of-two is a lot more straightforward than general product.