Originally posted by PinkFloyd
As a novice, is this something that only comes into play with calculus and other higher mathematics? I don't recall algebra class dealing much with this problem 🙂
In pure mathematics there is a field called "algebra", which I fear is very different from the school-taught concept of algebra. Essentially, it deals with a set (for instance, the set of integers) under an operation (for instance, addition). This, I believe, deals with the concept of infinity much more than calculus does. 😉
Anyway, Infinity. The concept of infinity, I believe, has been looked at in it's purist form by set theorists. They introduced the "aleph" notation for sets of cardinality (size) infinity. Now, here is the wied bit... There is more than one infinity.
Without getting into too much detail, as I learned this during honours mathematics courses and only really started to understand it when I came across it at masters-level, but basically there are infinetly many infinities. The natural numbers is the smallest set of cardinality infinity, and we say this has cardinallity "aleph null". Curiously, this had the same cardinality (size) as the integers and also of the the rational numbers. This is because aleph_0 * aleph_0=aleph_0 (and of course the rational numbers are the set of all pairs of integers)*.
Now, it can be shown that aleph_0 /= aleph_0 ^ aleph_0=2^aleph_0 = the cardinality of the real numbers = the cardinality of the powerset of the natural numbers. Infact, you can always take the power set of a set to get a set of size strictly greater than the original set. This is why there are infinetly many infinities.
Now, I know this is all confusing, and explained badly, but I shall confuse you some more. Because it's an interesting result. Now, there are more than one "levels" of infinity. They go, "aleph_0, aleph_1, aleph_2,...". Gregor cantor hypothesised in his "continuum hypothesis" that the cardinality of the real numbers was equal to aleph_1. That it is the next level of infinity. In 1939, Kurt Godel showed that this could not be
disproved. In 1963, Paul Cohen showed that it could not be proved. In short, it is a result outside of our general axioms of set theory (called ZF, or, more controversially, ZFC).
And for the icing on the cake, and a vague explanation of the ZF/ZFC controversial ataement? http://en.wikipedia.org/wiki/Axiom_of_choice
NOTE: /= is does not equal.
*Actually, just to confuse you more, this is the wrong way round. aleph_0 * aleph_0 = aleph_0
because the rational numbers is the set of all pairs of integers, and because we can find a bijection between the set of pairs of integers and the set of integers.
http://en.wikipedia.org/wiki/Continuum_hypothesis
http://en.wikipedia.org/wiki/Power_set
Infinity
22 Jun '08 22:36
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