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, ... }
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.
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.
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.
Note that this view is common but not universal: https://cs.nyu.edu/pipermail/fom/2023-January/023710.html
See this exchange for details
https://math.stackexchange.com/questions/2812355/does-infini...
EDIT: There is no least real number under the standard ordering. There is a least real number for a given well ordering. If you are asking me to define a well ordering on the reals then your request question should have been stated differently.
I’ve seen some truly bizarre definitions of the reals which are internally consistent and include things like the lowest possible number.
The widely accepted first order axioms of the natural numbers have nonstandard models. Of course each such model has an isomorphic initial segment. There are two things going on. There is the set of axioms and there are models for those axioms. I think in your usage: “define real number” corresponds to “give the axioms for the real numbers”. I’m not sure though.
> Anyway, axiom/models is the common terminology though I recall a rather long rant that postulate is better once you start talking about multiple incompatible systems.
Model has a very specific definition in this area of mathematics. This sentence is nonsense.
It's not a real number, true. But that has no bearing at all on the validity of f(x) = x + 1, since that function is well-defined with both its domain and its range as all real numbers and no others. You don't need infinity to be a number to define that function.
It’s easy to pretend math is some universal language but it’s really several mutually exclusive languages.
Infinity is not a number. You can pretend it is and there are several ways to build a mathematical system around such a choice, but it isn’t inherently correct. So in your example f(x) may no longer include negative infinity and there is no paradox, it may include multiple infinite numbers and the lowest of them are now excluded, or you can say x and f(x) are the same etc.