Greatest mathematicians ever!

I got a question today in aardvark - who are greatest 20th century mathematicians?

I answered Turing, Godel and Von Neumann.

Then I searched in google about who are greatest mathematicians. I saw many lists and I decided to create my own list based on my own knowledge - of course it need not be objective as it is my own list!

My criteria are a bit harsh - If two people invented the same thing, the person who invented it first or Who is more influential in other respects are given the credit. One good example is evolution. We always credit it to Darwin although Wallace came up with the idea around the same time. So credit for calculus goes to Newton not his opponent. Also I give more credit for a great foundational discovery than breadth of output - Best example is Euler for the latter - but Godel beats him with his incompleteness theorem.
1. Sir Isaac Newton
Greatest by a long distance to me. His invention of Calculus alone would have placed him here. But he discovered binomial theorem, Newton's method for square roots etc.
2. Carl Frederich Gauss
Greatest pure mathematician [or at least one mainly known for mathematics]. He discovered many things like Non Euclidean geometry, Gaussian curvature etc.
3. Kurt Godel
He found incompleteness theorem, which shattered the hope for a theorem proving machine which will find all theorems.
4. Archimedes
From his writings I can see he almost invented integral calculus. Particularly have a look at things like his approximation for PI.
5. Riemann
Invented things which were precursor to General Relativity [Riemann curvature] and a rigorous definition for Integral calculus and contributed to the theory of prime numbers [Riemann zeta function]
6. George Cantor
He proved real numbers aren't countable - so vastly bigger than natural numbers using his famous diagonal slash argument.
7. Alan Turing
He discovered the theory of computability with his Turing machines. It provided the formal basis for computers and proved there are things in Mathematics which are un computable.
8. Euclid
I think he is more of a chronicler than discoverer. But since I don't know who actually discovered his results I might as well give him the credit. Of course he gets into the list due to Elements.
9. Euler
I have to include Euler for the sheer amount of his work. On complex numbers like Euler's formula or his discovery of the sum of 1/1^2+1/2^2 etc. Although in my personal opinion none of them are as path breaking as others in the list.
10. Dedekind
I am including Dedekind for his rigorous definition of real numbers [which are anything but real!] - Although we should probably mention people like Cauchy and Weirstrass, I still think single most important was that of Dedekind.