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.
In the end there is no deeper truth here just the results of various completely arbitrary definitions.
Your post doesn’t make sense mathematically. As a function from the reals to the reals the function f(x) = x+1 doesn’t “include negative infinity” (have no idea what that is supposed to mean). Negative infinity is not a real number and is not in the domain of the function or its range. If you think “infinity” (which one?) is not a number then do you believe there are no infinite ordinals? How do you justify the view that countably infinite ordinals don’t exist?
There are well defined models of the real numbers. You don’t know enough mathematics to comment intelligently on this topic. What you wrote above makes no sense.
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.
You don’t know enough set theory and model theory. Of course there are different models of the real numbers. There are non isomorphic models of the first 4 axioms of Euclidean geometry even. A given set of axioms can have lots of different models.
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.
Non standard models of arithmetic really takes me back. 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.
Please consider that the poster telling you that you do not know enough to comment intelligently about this topic may be correct.
> 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.
The negative reals are an infinite set, that doesn't make "negative infinity" a real number though. Negativy infinity is a concept, that can be used in some math frameworks, but it is not defined within R.
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'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.
In the end there is no deeper truth here just the results of various completely arbitrary definitions.