100 Greatest Theorems of All Time

[TomCollins]

web page

[BZ_Zorro]

#47 Sklansky IS God

[mockingbird]

Great web page, thanks.

[sirio11]

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Great web page, thanks.

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[gumpzilla]

Thanks for the link. Kind of a silly idea, but fun nonetheless.

I decided to scroll down the list until I saw the first thing I didn't recognize. The winner was 25. Schroeder-Bernstein Theorem. I have a vague recollection of maybe seeing this in a set theory book, but I'm not entirely sure. Somebody want to fill me in on what this is?

[jason_t]

[ QUOTE ]
Thanks for the link. Kind of a silly idea, but fun nonetheless.

I decided to scroll down the list until I saw the first thing I didn't recognize. The winner was 25. Schroeder-Bernstein Theorem. I have a vague recollection of maybe seeing this in a set theory book, but I'm not entirely sure. Somebody want to fill me in on what this is?

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The Schroeder-Bertstein Theorem states that if A and B are sets, there is a map f:A->B and a map g:B->A such that f(x) = f(y) implies x = y and g(x) = g(y) implies x = y (i.e., fa and g are injections) then there is a map h:A->B such that h(x) = h(y) implies x = y and if b is in B there is an a in A such that h(a) = b (h is a bijection).

[gumpzilla]

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The Schroeder-Bertstein Theorem states that if A and B are sets, there is a map f:A->B and a map g:B->A such that f(x) = f(y) implies x = y and g(x) = g(y) implies x = y (i.e., fa and g are injections) then there is a map h:A->B such that h(x) = h(y) implies x = y and if b is in B there is an a in A such that h(a) = b (h is a bijection).

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Thanks. I like bijection just fine, but I've always preferred 1-1 and onto instead of injection/surjection.

EDIT: I assume this theorem is non-trivial only when we start talking about infinite sets? It seems to me that saying there are injections from A to B and vice versa means that A cannot have more elements than B and B cannot have more elements than A, so they're the same size and then it's kind of obvious. With infinite sets I know this reasoning is kind of problematic.

[ACPlayer]

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he Schroeder-Bertstein Theorem states that if A and B are sets, there is a map f:A->B and a map g:B->A such that f(x) = f(y) implies x = y and g(x) = g(y) implies x = y (i.e., fa and g are injections) then there is a map h:A->B such that h(x) = h(y) implies x = y and if b is in B there is an a in A such that h(a) = b (h is a bijection).

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Thanks so much. This reminds me why I decided to abandon my Ph D (applications of category theory to strongly typed computer languages was the general area) and instead pursue life in the private sector. Grubby work, but much more rewarding. [img]/images/graemlins/smile.gif[/img]

[fishsauce]

Notable omissions:
The Riesz Representation Theorem
The Lax-Milgram Theorem
Poincare-Freidrich Inequality
Sobolev Embedding Theorem


And yes, I am biased. At least Green's Theorem and Brouwer's Fixed Point Theorem made the list, though Brouwer's should be higher than 36.

BTW, the FTC should be #1.

[Rev. Good Will]

I personally, like Clarkmeister's theorem best