While Anton was waxing about Church & Turing, I figured that Occam's Razor would be the type of proof one would postulate when giving the nod to Lambda Calculus over Universal Turing Machines. This leads inexorably to the question of what is the smallest (as measured in binary bits) Turing Machine that can possibly be constructed. John Tromp provides an answer to this question in his always fun Lambda Calculus and Combinatory Logic Playground:

Pictured you can see the 210 bit binary lambda calculus self-interpreter, and the 272 bit binary combinatory logic self-interpreter. Both are explained in detail in my latest paper available in PostScript and PDF. This design of a minimalistic universal computer was motivated by my desire to come up with a concrete definition of Kolmogorov Complexity, which studies randomness of individual objects. All ideas in the paper have been implemented in the the wonderfully elegant Haskell language, which is basically pure typed lambda calculus with lots of syntactic sugar on top.

Interestingly, the version based on the Lambda Calculus is smaller than the one on Combinators. A statement I found of interest in the paper about PL's:

Although information content may seem to be highly dependent on choice of programming language, the notion is actually invariant up to an additive constant.

Not sure if that statement means that PL research is ultimately doomed. :-)