100 Greatest Theorems of All Time

[TomCollins]

web page (http://personal.stevens.edu/~nkahl/Top100Theorems.html)

[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).

[/ QUOTE ]

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. /images/graemlins/smile.gif

[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

[jason_t]

[ QUOTE ]
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.

[/ QUOTE ]

You study partial differential equations?

[Rotating Rabbit]

I think FTOA should be #1, you cant do anything in maths without it.

[fishsauce]

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You study partial differential equations?

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Yes, mainly the numerical simulation of pdes. My MS work was in algebra and finite fields (what was I thinking?), but I am now in numerical analysis.

The list is ok, but is obviously heavily biased towards number theory. Apart from what's mentioned above, some other glaring omissions are: Atiyah-Singer's index theorem, Riemann-Roch's theorem, the classification of simple groups, Cauchy's integral formula, the Perron method for solving elliptic partial differential equations, the uniformization theorem, etc, etc.

Also, some of the things that are listed are rather silly, like #14 (evaluation of zeta(2)), #26 (Leibniz' basically useless formula for pi), #97 (Cramer's rule), just to mention a few.

[jason_t]

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The list is ok, but is obviously heavily biased towards number theory. Apart from what's mentioned above, some other glaring omissions are: Atiyah-Singer's index theorem, Riemann-Roch's theorem, the classification of simple groups, Cauchy's integral formula, the Perron method for solving elliptic partial differential equations, the uniformization theorem, etc, etc.

Also, some of the things that are listed are rather silly, like #14 (evaluation of zeta(2)), #26 (Leibniz' basically useless formula for pi), #97 (Cramer's rule), just to mention a few.

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Cauchy's integral formula: /images/graemlins/heart.gif.