infinity.

• H

HowlingWolf500

I read that the notion of an actual number of infinite things leads to contradictory results. Is that true?

• O

Olleus

It depends on what you mean by paradox and if you consider infinity in the way mathematicians do.

As a limit, it's perfectly well defined. You can even deal with systems in infinite dimensions (that is, with an infinite degrees of freedom) with little more than undergraduate maths. For example, the set of functions f(x) defined inside the interval x=[0,1] lives in an infinite dimensional vector space. You probably shouldn't consider infinity as a number like 2, -0.123, pi, 1 + 4i and the like however because it doesn't fit into the standard algebraic structures of groups and rings and all of that. However it is accepted parlance generally to say something like "consider infinite cats" even though that is technically undefined mathematically and what is really meant is "consider n cats in the limit n->infinity".

Now about paradoxes. If you take infinity in the non-precise way of just being a number, then you can get paradoxes. But that's simply because you are assuming infinity has all the properties of a normal number (when it doesn't), so if you start with incompatible falsehoods it's perfectly natural that you would "deduce" paradoxical nonsense. If you however do everything mathematically rigorously with a more precise notion of infinite, you do get some pretty weird results out (see Hilbert's Grand Hotel). However, they are weird, but not paradoxes. No contradictions come out of them, just a personal unease at how weirdly infinite things behave. But lots of maths is weird, and you eventually get used to it or ignore it.

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