Gödel's incompleteness theorem. A total game changer in the XX century.

I'd pick Rice's theorem about the undecidability of any nontrivial properties of languages. http://en.wikipedia.org/wiki/Rice's_theorem I like it because it is a powerful theorem that characterizes a large and important set in one broad stroke.

My favurite theorem is the Weierstrass Approximation Theorem that shows every continuous function on a closed interval can be uniformly approximation by a polynomial to any given accuracy.

www.math.harvard.edu/~waffle/wapproxt.pdf

Hilbert spaces are homeomorphic to their dual spaces (Riesz representation theorem). This is why linear applications are represented by matrices !

Gödel's incompleteness theorem. A total game changer in the XX century.

My favourite  theorem is Kuratowski's Theorem in graph theory. The Four color theorem is also very much interesting.

  One of the simplest expressions of the fundamental theorem of algebra is:

  Every polynomial in one variable z with complex coefficients is the product of a complex constant and polynomials of the form z + a with a complex.

      Other more sophisticated version of what is called fundamental theorem of algebra is:

     The field of complex numbers is the algebraic closure of the field of real numbers.

There are several beautiful theorems that can be very easy to formulate, but very difficult to prove as is the case with the Four Color Theorem or Fermat's Last Theorem. In combinatorial mathematics, there is a problem which is, by itself, very interesting and whose solution will allow solving other fundamental problems in enumerative combinatorics, graph theory, and theoretical computer science. I would like to draw your attention to the question of whether there exists an infinite number of symmetric (n, k, lambda)-designs for any fixed lambda >= 2, which remains unresolved. If lambda = 2, the design is called "biplane" and the problem is known as "the problem of existence for biplanes." For the interested reader, the following article describes at the end the general picture of the problem.

Article ON THE POSSIBILITY OF A CONNECTION BETWEEN THE CONSTRUCTION ...

Gödel's Second Incompleteness Theorem. ;) ;)

Matrix-Tree Theorem

According to me, Fundamental Theorem of Galois Theory is the favourite for me to teach .

The compactness Theorem in Mathematical Logic which says a set of first order sentences has a model if and only if every finite subset of it has a model. It nearly appears across the Model Theory. One can find a version of it in Propositional Calculus too. The theorem is related to the finite intersection property in topological spaces and..

My favorite theorem, one that pretty much changed everything in the mathematics of its time, is the Pythagorean theorem. It transformed the concept of magnitude, ushering in the notion that there can be two lengths which cannot both be whole number multiples of a single length unit. Aside from its historical interest, it's key in setting up the euclidean distance formula, basic to the teaching of multi-variable calculus. I'll stop there, but it's difficult to overstate the importance of this fundamental result to modern mathematics.