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.

See this exchange for details

https://math.stackexchange.com/questions/2812355/does-infini...

I am trying to allude to transfinite numbers and annoy mathematicians without losing a general audience, feel free to ignore such things.

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 said models not model here, so you acknowledge there isn’t a single universal definition.

I’ve seen some truly bizarre definitions of the reals which are internally consistent and include things like the lowest possible number.

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 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.

I am describing an actual event with that sentence. Are you suggesting that someone can’t rant about such things?