So in Metric Spaces today we were shown the Banach-Mazur game (http://en.wikipedia.org/wiki/Banach-Mazur_game). By far the most interesting aspect of it is its application in existence proofs, and in the lecture he demonstrated the existence of irrational numbers.

Basically it really struck a chord with me. SO I did a bit of research and found this thing called determinacy (http://en.wikipedia.org/wiki/Determinacy) which basically studies games like this.

Has anyone come across determinacy? I'm looking for a couple of good book recommendations, or maybe some nice topics to study.

I did not understand how that was a proof.

It was an AWESOME proof

I understood that it was a proof but I don't like it that much. Plus because of excursions I've already done the function which is everywhere cts and nowhere diff, and it's a lot easier just stating a function then proving, than making a winning strategy for this weird "game".

Also i think any proof using that method could be reworded into a proper proof which would be much easier to understand

This is just my opinion and i'm open to being wrong here.

If, like you said, its just your opinion, how exactly are you open to being wrong?

It's pro-theory man, deal with it.

But no, that determinacy stuff is awesome- apparently, according to the colloquium we went to last term, logicians are coming close to a consensus on ZFC because of it: there is, I hear, a theorem which says that ZF + a weak version of the axiom of determinacy (that is an axiom that says a certain subset of 2-person baire type games are determined) is complete (for example- the previously independent CH is false in "ZFA") modulo Godel sentences. Which is a pretty bloody spiffy result...

I cant remember if the form he presented was consistent with C though- regular AD isn't: it says all lebesgue sets are measurable, which is a bare faced lie (see http://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox or http://en.wikipedia.org/wiki/Vitali_set)- for a more elementary proof try http://en.wikipedia.org/wiki/Axiom_of_determinacy...

But yeah... other than casual browsing + the talk I went to, no ideas on books- I'll have a look around.

I'd pay you real money to teach me all this, Tom ;__; By far the stuff I'm most interested in in the subject

I have to say I don't really know it myself, just bits and bobs I've gleaned from random reading...

Tell you what.. I'll borrow Jamie's copy of Set Theory by Jech and learn it- then I'll do a Dg in a couple of weeks-

I think it's doable... :p

Have been doing some research and it looks like an eminently doable topic... though I have been known to be over ambitious at ...

fingers crossed should be on for next thurs. :D