I absolutely love this kind of mathematics, it's my favourite stuff we've covered in my degree so far (which is strange because I'm not normally a fan of pure maths).
I didn't think that Cantor was crazy per se, but I do know that he suffered depression later in life because almost everybody else rejected his work as rubbish. Another mathematician at the time told him it was about 100 years to early for mathematicians to handle, which turned out to be pretty much true.
Basically, infinity is usually used to mean one of two completely distinct concepts.
First there is the infinity of analysis (which concerns itself with limits). An example of its usage would be
'As x tends to 0, 1/x^2 tends to infinity'.
Colloquially this is easy to see
but the colloquial interpretation is actually completely wrong, because in this context infinity is not a number; it's totally wrong to think of infinity as some huge thing arbitrarily high up the y axis which the function is heading towards.
In symbolic terms what 'as x tends to a, f(x) tends to infinity' actually means is this beastie:
∀ ε > 0 ∃ δ > 0 : x  a < δ ⇒ f(x) > ε
which in English means 'no matter how high a number you give me, I can give you a range of values around a so that the function is higher than your number'.
In other words, in analysis, infinity only really has meaning when it is used as a part of other expressions, and never means a value, but rather some kind of process (in this case unbounded increase).
In number theory however, which is what Cantor was concerned with, infinity has a totally different meaning. It's much more like a number than the infinity of analysis in the sense that it's kind of used for counting objects, but it's a bit more subtle than that.
Basically when we count things, we are pairing them off with elements of the set of natural numbers ('N'), which are
1, 2, 3, 4, ...
If you were counting 3 sheep for example, you pair sheep 1 with 1, sheep 2 with 2, sheep 3 with 3, and then you have no more sheep to pair up, so your number of sheep is 3.
When things get really interesting is when you try to count infinite sets.
An easy example to start with is the set {0, 1, 2, 3...}. Can we count this? Using the above, this question is equivalent to 'can we pair up every element in this set with an element in N?'.
It's quite easy to see how (N is on the left):
1  0
2  1
3  2
4  3
...
We can be confident with this set up that every element in N is going to be paired up with an element in our set, and importantly, that no elements in our set are going to be left out.
Therefore, we say, our set has the same size as N, which we call 'Aleph0'.
Surprisingly we can also do this with the rational numbers ('Q'), which is the set of all fractions. We do this in the following way:
Again, we've assigned every fraction a number in N, and we can be sure that we've 'hit' every fraction, hence the size of the set of fractions is Aleph0.
Now we try for the reals ('R'), which are colloquially the 'decimals'. This set is called the 'continuum' because there is a sort of smoothness about it; it forms a line with no gaps (unlike the naturals or rationals; for example the square root of 2, or pi, will be gaps in both of these sets).
Imagine that we've done it and we've got some system of pairing up, e.g.
1  1.1200394810...
2  0.0013223491...
3  2.2221340139...
4  1.0104910495...
The trick now is to construct some decimal which differs from the decimal paired to 1 in the first digit, the decimal paired to 2 in the second, and so on. We do this by changing 0s to 1s and everything else to 0s. In this case we will get:
0.101...
This is called the diagonal slash method. As our decimal differs from the nth listed decimal by the nth digit, we know that our list will always miss it. Hence, there are 'more' real numbers than natural numbers, and we have different 'sizes of infinity', which are denoted by different aleph numbers.
The continuum hypothesis is simply that the size of the real numbers is Aleph1, or the 'next infinity up' from the naturals' Aleph0, i.e. there are no sets with more elements than N but less than R.
This turns out to be impossible to decide with the normal rules of mathematics. What mathematicians have to do, then, if they require it for a proof, is simply accept it is true or accept it is false (and yes, this often gives mutually contradictory results). This is related to Godel's incompleteness theorem. It's not really unhealthy though, because with all mathematics you have to start with some base assumptions (or 'axioms'), for example 'a(b + c) = ab + ac' is no more provable than the continuum hypothesis, and must be assumed for all that follows.
Regarding the circle question: that seems to be a question relating to the infinity of analysis rather than Cantor's infinities, because it's regarding an unbounded limiting process (rather than counting things).
You are right that it is impossible to have an infinite number of angles which don't exist, and this touches on a point which is crucial to the whole of analysis: there is a distinction between the limit as you approach something, and the value when you actually hit it. If there were not, analysis would be redundant.
Take for example the previous function 1/x^2; the limit as x tends to 0 clearly exists, but there is no value AT x = 0.
In the same way a circle is the limit of the process you described, but the nonexistence of the object at the 'end point' of the process (which is of course a contradiction because the process is infinite) does not matter.
I think that covers most of the basic ideas... a good idea might be to try to count some of the following:
Z = {..., 2, 1, 0, 1, 2, ...}, the integers.
N^2 = {(a,b) such that a and b are in N}, the set of all pairs of numbers.
Q^2 = {(a,b) such that a and b are in Q}, the set of all pairs of fractions.
{nonoverlapping discs in a plane}; picture a dalmatian with circular spots. Remember there could be infinitely many spots (the dalmatian is grotesquely obese).
{nonintersecting figures of 8}; this is hard but awesome.
{set of all subsets of N}
It sounds like the guy was heavy into sacred geometry.
No, he wasn't. He was heavy into proper mathematics. Sacred geometry is a load of nonsense. It doesn't even mean anything.


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