Infinity and Logic

This month, I have been taking a real interest in infinity and its effects on maths.  It started out with me pondering the idea of the area approaching a limit, and the fact that you can have a finite area, split into infinite parts, represented by an infinite amount of real numbers.

From this, I picked up a copy of Godel's Proof, a book that tries to describe Godel's incompleteness theorem, in about 200 pages.  And, I then discovered that Godel, among others, proposed the fact that if you identified sets of numbers, that were mind bogglingly large, you could just build new levels of infinity, on top of underlying numbers and then you could use these sets, to prove some of the maths problems that exist underneath.

It was then, that I stumbled on an article in the New Scientist periodical, this week.  The piece was boldly entitled 'Ultimate Logic - so powerful it could wipe out mathematics as we know it.'

In a nutshell, the piece describes how a mathematician (Hugh Woodin), believes he has solved the continuum hypothesis (Is there an infinite set that sits between the countable infinity, such as counting the integers from 1 towards infinity and the 'continuous' infinity, such as when you split a sphere into infinite sections, when the whole is finite.)

The difference with his claim, is that he states that he has solved this maths problem, by using a new type of logic language and structure called 'ultimate L'.  This method would allow extra steps of sets to be bolted on to the top of infinite sets, filling in gaps below sufficiently, to allow any lower mathematical problem to be solved.  He makes the bold statement, that this new theory, allows him to provide 'a definitive account of the spectrum of subsets of real numbers and thus, proves Cantor's continuum hypothesis, as true; ruling out anything between the countable infinity and the continuum'!

Woodin, even claims, that this may overturn major parts of Godels incompleteness theorem and be a tool that actually, artificially, allows us to root out unsolvability in number theory.  He doesn't go as far as saying that Godel's theories would be dead - but that you could 'chase it as far as you pleased up the staircase into the infinite attic of mathematics'.

This idea seems similar in theme, to the idea of calculus and limits.  In that, yes, you might never be able to say what is actually happening at the limit its self, but you can get so close to it, that your results have the same effect as if you were actually at the limit.

 This all reminded me of  the amusing anecdote relating to Zeno's paradox.  It is said, that when it was once explained to a student, that if you were trying to reach a girl on the other side of a room, that you would never actually get there, if you travel half of the previous distance travelled in discrete steps.  The student's retort was thus;

 'Well, I might not ever reach the girl, but I could get plenty close enough for all 'practical' purposes!  It seems, that perhaps Woodin is saying; 'why try and reach infinity? when you can just get so close to it, that your results are almost the same, as if you were actually there?'  He seems to believe, that he has created a new language for mathematicians, to solve these issue.

Could it be, that if we do end up with a radical departure from existing mathematical logic and ideas, to prove almost everything; that we will end up with two tiers of mathematics, such as happened in physics with 'classical' and 'quantum' theories?