micromass

I have not been a mentor for very long, but I can see pretty clearly that we get quite a lot of complaints of people who think the PF-rules are too strict or that we should be more lax in enforcing the rules. It may not surprise any of you that I disagree with this. But let me try to explain my motives behind my disagreement.

First of all, there is only one person in this entire forum who decides what the rules are. That person is Greg. It is his forum. He is the one who spends a...

Whenever I need to learn a new subject in mathematics, I try to be very careful about which book I learn it from. Usually, I look on the internet and I see what other recommend. Very often, this gives very good results and gives textbooks that are real good. However, there are two books that everybody seems to adore, but which I absolutely hate. The purpose of this blog is a bit to rant about this books.

The first book which I think is far too overrated is the rubbish that...

Welcome to my final blog on the axiom of choice. This time I'm going to tell you my personal opinion on the matter. The question is a philosophical one, since Godel and Cohen proved the independence of the axiom of choice, and had thus shown that there is no mathematical problem with either accepting or refusing the axiom of choice. And the problem with philosophical questions, is that everybody is correct!

Firstly, I'm not against the axiom of choice at all, I have used the axiom...

Right now, I may have given you all the impression that the axiom of choice is an evil thing and that mathematicians are stupid people for accepting it. This is of course not so. There are many beautiful things that happen with the axiom of choice and that we really need the axiom of choice for. Let me give some nice results that are equivalent(!) with the axiom of choice:

• Every surjective function has a section. That is, for every surjection f:X--> Y, there exists a function g:Y-->X

...

Now we have had an entire discussion on the meaning of the axiom of choice and of it's history, but we've not yet told why it is controversial. The most encouraging reason not to take on the axiom of choice is the infamous Banach-Tarski paradox, this states

A version of the Banach-Tarski paradox which I like better...