Recent content by dcpo

You'll never win a Fields medal, and you'll never be equal to people like Grothendieck. I'm sorry, but that's something you're just going to have deal with. You, and the vast majority of professional mathematicians working at universities around the world. There are two particularly troubling...

Given hyperbola \frac{x^2}{a^2}-\frac{y^2}{b^2}=1 , and tangent equation y=mx+c the relationship c^2+b^2=a^2m^2 holds. The derivation of this is explained here. This can be combined with what we know about the relationship between m and c by setting x=2 and y=\sqrt{3} to give the required equation.

With regard the OP, remember that the existence of non-principal ultrafilters does not follow from ZF, so an ultraproduct construction is non-constructive in your sense (or trivial).

The powerset proof proceeds by, given f, showing that there can be no x with f(x) equal to the set I defined in my previous post (similar to Russell's paradox, x will be in f(x) iff it is not). A discussion of bijections between the reals and the powerset of the natural numbers can be found e.g...

A few people have a problem with the infinite list idea, I'm not entirely sure why, but it can be thought of as an illustration. The Cantor argument doesn't need it, nor does it need to be a contradiction. In essence the diagonal argument says that any function from the natural numbers to the...

Good answer.

I forgot about them being the last two batsmen. Is there definitely a positive resolution to this? I'm struggling to see how it could be done.

I included a hyperlink in my post, does it not open for you? Your link worked for me this time, though it doesn't appear to contradict my claim. It says the batsmen ran 3 while the ball was in the air, not that 3 were awarded to the batsman after the catch had been taken. It's common practice...

I must say that seems a rather flimsy reason to doubt the consistency of modal logic, especially considering that it has been rigorously studied for 100 years, and has its own well developed proof and model theories. Maybe you mean something non-standard when you talk about consistency. p.s...

According to the current laws of cricket I am correct. The relevant rule is http://www.lords.org/laws-and-spirit/laws-of-cricket/laws/law-18-scoring-runs,44,AR.html. Maybe things were different in the past. I can't comment on your link as it didn't open for me.

No, if you're caught you don't get runs, but the batsman could be run out taking a 5th run. That way he'd get 4 and the 2nd would be on strike.

^^ Why would you assume no modal logic could be consistent?

Well, what you need to say depends on how formal you have to be, and on what deduction system you have to use, but what you have is the basis for a rigorous 'everyday' proof, so long as you make explicit the role of the \forall A(G(A,A)) hypothesis. If you haven't been given an explicit formal...

How you formally work this through depends on what system you're asked to work with, but a strategy would be to let B=P(A).

Transfinite induction allows you to use induction proofs over ordinals greater than \omega, ordinary induction only allows you to prove things over \omega. The key difference is that transfinite induction involves a limit case to take into account limit ordinals i.e. if the property holds for...