Edit by Natural ChemE (2015 09 16 0457 EST) wrote:This thread has been generated by splitting a prior thread, Infinity is a trivially simple concept. The prior thread was about seeing infinity as a simple concept.

The first post in this thread was by someguy1 who was arguing that Cantor's Continuum Hypothesis was complex, so infinity was complex (my interpretation of his position; not necessarily correct). His point was entirely appropriate to the prior thread, however it also led into a very interesting discussion on stuff like the Continuum Hypothesis.

This thread focuses on that interesting tangent. Discussion on the complexity of infinity should go into the prior thread.

So you've trivially solved the Continuum Hypothesis (CH)? There'a probably a Fields medal in it for you. Either that, or I'd argue that you haven't shown that infinity is trivial at all.

https://en.wikipedia.org/wiki/Continuum_hypothesis

What we know:

* There's an elaborate mathematical theory of infinity that's been in common use for about a century, give or take (depending on where you put your historical starting point: Cantor's paper in 1874, or Zermelo's axiomatization of set theory in 1922).

* CH is one of the most famous unsolved (and perhaps unsolvable) problems in math. It's still an active research area in set theory.

* The basic theory (countable and uncountable infinities, ordinal and cardinal numbers) is the starting point for the conversation about whether infinity is trivial. But it's difficult (for me) to see how anyone familiar with this material could regard infinity as trivial. On the contrary, the mathematical theory of infinity involves questions in logic and the philosophy of math, in addition to being deep and interesting math itself.

Here is the question that nobody knows the answer to.

We start from the intuition of the infinitude of the counting numbers 1, 2, 3, 4, 5, ... (some people like to start from 0, it makes no difference here).

How many subsets of the natural numbers are there?

That's the simple question that's bedeviled and confounded some of the greatest minds in mathematics for the last 140 years. The Continuum Hypothesis says that the number of subsets is the smallest possible infinity larger than that of the natural numbers themselves.

Eternal fame awaits the solver.

(ps) Let me get ahead of a couple of potential (or even actual!) objections to what I wrote. @Natural ChemE claimed that there is no actual infinity. If that is the case, then infinity is trivial in the exact same sense that purple unicorns are trivial. They're trivial because they don't exist.

Yet the novel Moby Dick is a work of fiction, and it's far from trivial. So I'd reject this line of thought. Works of fiction may be nontrivial.

On the other hand, suppose we do admit the (conceptual, mental, abstract, fictional) existence of completed infinite sets such as the natural numbers. Then the questions raised by transfinite set theory, such as CH, become meaningful and incredibly subtle and difficult questions. Again, far from trivial.

So I don't see any sense at all in which infinity is trival.