> To be fair, axiomatizing integer arithmetic has been attempted (unsuccessfully) by intellectual titans like Godel, Russel, and Whitehead, and is nowadays cons

> To be fair, axiomatizing integer arithmetic has been attempted (unsuccessfully) by intellectual titans like Godel, Russel, and Whitehead, and is nowadays considered "too hard" by research mathematicians That is just not true in any interesting sense. The Peano axioms[1] have been around since the 1880s. Gentzen showed in the 1930s how to prove their consistency given transfinite induction up to a sufficient height. There are certainly hard open problems in the areas of mathematics surrounding theories of arithmetic, but to say it's a problem that mathematicians have thrown up their hands and given up on is false. [1] http://en.wikipedia.org/wiki/Peano_arithmetic http://en.wikipedia.org/wiki/Peano_arithmetic