Closures

can someone explain them to me.. I'm thinken they might be simple but my pee sized brain can't comprehend most definitions off of google

>>119
If you ignore Cantor's work, you can say that a sequence of numbers that does not end is infinite. For example, there is no largest natural number - if you say you found one, just take its successfor to show otherwise. That's the simplest kind of infinity. A much stranger type of infinity is that of real numbers - any number has an infinite amount of digits (there is no last digit).

If you take Cantor's work into consideration, he basically takes the set theory (a bunch of axioms that define it, look at http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory), and 2 axioms of particular importance (which are rather artificial, but are needed to define 'infinity' as a concrete value as far as ordinals are concerned), the axiom of infinity ( http://en.wikipedia.org/wiki/Axiom_of_infinity ) and the axiom of power set ( http://en.wikipedia.org/wiki/Axiom_of_power_set ). Within set theory, you can find the cardinality of some set (the amount of elements in the set). He then defines the cardinality of N (or of the infinite set defined in that axiom before this), and that value is denoted by N₀, or aleph-null. What he did here is say that the cardinality of an infinite set is defined as a concrete value, the ordinal N₀, instead of merely left as some undefined infinite. That is what I meant by accepting the concrete infinity - you can say that the number of natural numbers is potentially infinite, but what Cantor did was consider this infinite value something concrete, after that he shows that the cardinality of R (real numbers) is greater than that of natural numbers (N₀), and this greater value is N₁. By the axiom of power set, he then shows that for any such ordinal (N_x), there is always a set whose cardinality is a greater infinite ordinal. He also set forth a conjecture which states that the next ordinal after N₀ is N₁ and there is nothing inbetween, this is called the continuum hypothesis. The very strange thing about this hypothesis is that it was proven that it's impossible to prove and disprove, which is a very weird result which likely shows that set theory as defined by ZF+Infinity+PowerSet might be inconsistent in some ways (along with some other paradoxes that plague it). Regardless of what you think of wether real numbers exist or not, or wether you can treat infinities as something concrete like he does, it does show you want happens when you do, and math is done by taking some axiomatic system and running with it to obtain results - if the results show that it's inconsistent, it was still a worthy exercise, I do know that I learned something by reading some of his work.