It depends on what you think a number is. Most people who think about it agree a number is the name that we give to equinumerate sets, so two eggs and two ducks are equinumerate, we can give a bijection between these sets; they have the same property, "twoness" and we abstract twoness to the number 2. If so, then infinity is a number, it this the number-ness of the set {1, 2, 3, ... }
There is no bijection between the natural numbers and the power set of the natural numbers. When talking about bijections to study the sizes of sets, there is clearly not just one infinity. Also consider the extended real line, this makes infinity a perfectly good number as well in a different way.
My point is that infinity is certainly not an individual number. It represents quite a few concepts, all of which have some number like qualities.
Indeed, but when people say "infinity is not a number" they generally mean ℵ₀, and that is (in my view) as much a (cardinal) number as 2. Once you have that established, then you go on to discuss the higher ℵs and the continuum hypothesis ...
> when people say "infinity is not a number" they generally mean aleph-0
Sure, maybe. I think saying "infinity is a number" is wrong regardless. I'm happy with aleph-0 being a number, and I'm happy with it having a label of infinity. I'm just not happy with treating infinity as a single thing. We're totally on the same page in every way that matters though. I'm also certainly with you on questioning Retric's comment for a number of reasons, haha.
edit: Actually...I'm not so sure they mean aleph-0. I think frequently they mean the infinity of the extended real line. Maybe it depends on the context of the conversation.
It's actually a great ice-breaker for first-year maths undergrads, when talking about limits or whatever drop in "... to infinity and beyond!", laughter follows, then say "You think I'm joking? Put down your pens ..." and informally outline the aleph hierarchy, then things get rather quiet.
Ahh I wish I had gotten the chance to interact with new math majors more often. In grad school most of my students were engineering majors. I'm not super likely to end up back in academia. My published research is...sparse. I have post-doc opportunity but it's not the greatest, and the route to life stability is not so clear. Are you a math professor?
No, a developer; I spent several years as applied maths researcher though, and UK universities love to use researchers as cheap lecturers (not that I'm complaining, rather enjoyed it).
Cool. I have a second interview for a software developer position at a quantum computing place this week. Hopefully I'll be following you into the field!
Let’s not mix up teminology and semantics. What a number here is is quite clear, we’re all talking about relatively standard math, the standard axiom of choice, and not some opinion on numbers. Sure, you can also say a planet is a mammal »depending on what you think a mammal is«, but that doesn’t lead to any sort of fruitful discussion.
It depends on what you think a number is. Most people who think about it agree a number is the name that we give to equinumerate sets, so two eggs and two ducks are equinumerate, we can give a bijection between these sets; they have the same property, "twoness" and we abstract twoness to the number 2. If so, then infinity is a number, it this the number-ness of the set {1, 2, 3, ... }