Picking holes in mathematics: Finitely stated, undecidable propositions

Mathematicians have been dealing with this problem in concrete ways for longer than implied. For instance, the necessity of the parallel postulate for Euclidian geometry and the existence of non-Euclidean geometries where it doesn't hold were demonstrated in the 19th century. The question of how many and which axioms are required as the basis of a field of mathematics has long been known to be a valid, meaningful question.

What Gödel proved is that in the formal systems we use to model mathematical propositions and proofs, there is no finite number of axioms that enable you to prove or disprove all of the propositions that can be formulated. For someone who grew up in the post-Gödel world, this isn't shocking at all and looks like a natural progression from previous results. You need a certain number of postulates. To get Euclidean geometry, for example, you need the parallel postulate. It's natural to ask how many postulates you need to decide all interesting mathematical questions, and there's no reason to assume the answer to that question is finite. Apparently, that attitude reflects a lot of hindsight bias, but there is no way for us to experience the shock of Gödel's work the way it was experienced by mathematicians at the time, since the books we learn from now are written by mathematicians who have already digested Gödel's discoveries.

As he posted on the FOM mailing list earlier this month, the 'final' version of Friedman's book on boolean relation theory is now up on his website. [1] It clocks in at around 900 pages. Not for the faint of heart.

[1] http://www.cs.nyu.edu/pipermail/fom/2011-May/015383.html

tldr; Gödel proved the incompleteness of finite axiomatizations of arithmetic. Further more there is no maximum set of unprovable sentences in arithmetic. Therefore, there must be interesting and unprovable sentences in arithmetic.

The article discusses a couple such sentences but unfortunately does not actually state the sentences, which is the interesting part.

Edit: unsurprisingly: "The extent to which we have concrete incompleteness and what the wider implications are, will not be clear for hundreds of years, if not longer." -- Friedman

I'm curious: If you add a new axiom in order to make a undecidable proposition decidable, will this new axiom add more than one undecidable proposition, or is the number of new undecidable propositions unknown? Or are there infinitely undecidable propositions for every set of rules?

If the former is true, couldn't we theoretically just apply axioms in order to solve undecidable propositions which have a practical application if solved? Or will it practically be hard to find these axioms?

If you like this, you'll like that:

http://news.ycombinator.com/item?id=2567222