I've been doing well with my private study - am making a proper go of learning Galois theory and boning up on set theory, which leads me to the title of this post.
The Axiom of Choice (AC). Introduced initially by Zermelo when formulating the axioms of set theory, so that they would cover all of Cantor's work on infinite sets and avoid paradoxes such as Russell's. It soon became apparent that mathematicians had been using AC for years without really knowing it.
I don't accept it. If accepting it means you can prove that a sphere can be cut into no more than 9 pieces which are then reassembled into 2 spheres the same size as the first one (Banach-Tarski paradox/theorem whichever way you look at it), then it either has to be wrong or its level of abstraction is so far removed from reality that it's irrelevant.
You need the axiom of choice to fully appreciate Cantor's infinite cardinals. Again, there's got to be something wrong here, and it must be with the priniciple that a bijection between infinite sets means they are equinumerous. How can the set of natural numbers be equinumerous with the set of even natural numbers? I know there's a bijection n -> 2n, but by any reasonable reckoning there are clearly half as many even natural numbers as there are natural numbers.
I've come to the conclusion that infinity is just infinity, which just means without bound. This works for epsilon-delta treatments in real analysis (but for sequential continuity you need AC). Complex analysts happily define the set and then divide by zero. I'm happy with that. I'm not happy with saying there are more irrational numbers than rationals on the one hand, and then saying that between any two rational numbers there is an irrational, and between any two irrationals there is a rational. Makes no sense. I follow the proof, just don't accept the philosophy.
I'll soon finish the set theory book. I'm then going to study the Banach-Tarski proof (which is on about page 2 of my Axiom of Choice book) just so I understand what I don't believe. Mohammed put it best when he said (probably in the Koran somewhere) that one should study witchcraft, but not practice it.
More on Galois theory anon - algebra gets more interesting, and so far seems devoid of some of the nonsense that seems to plague foundational mathematics such as set theory and logic.
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...but by any reasonable reckoning there are clearly half as many even natural numbers as there are natural numbers.
This is where infinity gets you, it isn't reasonable.
arb
nellie
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I've thought a little more about this today, in more physical terms. Black holes and the Big Bang are considered to be space-time singularities - where the laws of physics that apply in the rest of the universe break down. My limited understanding of these is that the gravitational field is effectively infinite. Physicists just accept that their laws don't work - why can't mathematicians?
I think all this sent Cantor mad. I'm going to take the complex analyst's view of infinity as something that is without bound, and stop worrying about whether one infinite set is bigger than another.
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"I'm not happy with saying there are more irrational numbers than rationals on the one hand, and then saying that between any two rational numbers there is an irrational, and between any two irrationals there is a rational. Makes no sense. I follow the proof, just don't accept the philosophy."
Rational numbers have infinite repeating or finite decimal expansions. Irrational numbers have infinite decimal expansions without repetition so intuitively there should be "more" irrationals than rationals. Also, there is a rational between any two irrationals (just terminate the expansion for the larger one) and vice versa.
The, lego-based comic, webpage http://www.irregularwebcomic.net/2339.html contains an outline of how to prove the Banach-Tarski theorem using sequences of irrational angled rotations about perpendicular axes. The article makes the comment that "it is extremely difficult to define the volume of an arbitrary mathematical object in such a way that is corresponds to our intuition of how "volume" should behave."
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Thanks Adnan. I was looking at the Banach-Tarski proof last night. Similar to your lego proof except that instead of rotations by an irrational number about two perpendicular axes it uses rotations of 180 degrees about 1 axis and +/-120 degrees about another to define the group of rotations - in this case the angle between the 2 axes is chosen so that you don't end up with the identity rotation by combining a finite number of the generator rotations.
I once saw an analogy of the Banach-Tarski proof. Imagine a dictionary containing all possible words. It would start A, AA, AAA, AAAA......AABA.... etc. By taking off the first letter of each word in your dictionary you would create 26 dictionaries identical to the first one.
This infinity stuff really is philosophy and not maths!