In a recent conversation, a smart and mathy friend of mine revealed they didn't understand Cantor's diagonal argument. Further questionning revealed that she was using the wrong concept of infinity. I thought I'd pass on my explanations from there.
What is infinity? Does one plus infinity make sense? What about one over infinity? Well, it all depends what concept of infinity you're using. Roughly speaking, infinity happens when you take a finite concept and add a "and then that goes on for ever and ever" at the end. There are four major examples:
- If you take the concept of "size" and push that to infinity, you get the infinite cardinals.
- If you take the concept of "ordered set" and push that to infinity, you get the infinite ordinals.
- If you take the concept of "increasing function" and push that to infinity, you get the poles and limits of continuous functions - a major part of real analysis.
- And if you take normal algebra and push that to infinity, you get the hyperreals.
To add to the confusion, there are normally only two symbols for infinity to go around - ω and ∞ - even though there are at least four very different concepts (honourable mention also goes to the infinities of complex analysis and algebraic geometry, which are like more limited versions of the real analysis one).
So, does ω+1 make sense (as something different from ω)? It does, for the ordinals and hyperreals only. What about ω-1, similarly? That only makes sense for the hyperreals. And -ω? That works for the hyperreals and real analysis. 1/ω? This is well defined for the hyperreals (where it is not equal to 0), and is arguably well-defined for real analysis, where it would be equal to 0.
Is ω times 2 the same as ω? "Yes", say real analysis and cardinals; "No", say ordinals and hyperreals. How about there being many different infinities of different sizes? That makes sense for all of these, except real analysis.
So when talking about "infinity", be careful that you're using the right concept of infinity, and don't mix one with another, or import the intuitions of one area into an arena where it doesn't make sense.
EDIT: See also Scott Garrabrant's comment:
Starting with the natural numbers:
- Cardinals: How many are there?
- Ordinals: What comes next?
- Limits: Where are they going?
- Hyperreals: What is bigger than all of them?