Python cryptography, Rust, and Gentoo

Infinity is neither an integer nor a real number (when both terms are defined in the mathematical sense rather than the computational sense). The real numbers observe something called the "Archimedean property," which states that there are no infinities or infinitesimals (except that zero is infinitely smaller than all non-zero values).

Why do real numbers have this limitation? Well, the blunt fact is, there's only one totally ordered metrically complete field, and it's the real numbers.[1] If you want to introduce an infinity, you have to give up one of the following:

1. The field axioms (which, broadly speaking, say that you can add, subtract, multiply, and divide real numbers, and these operations behave in sensible and familiar ways).
2. The total ordering of the reals (i.e. for any two reals a and b, either a > b, a < b, or a = b, where = is given its ordinary interpretation of "are literally the same mathematical object" rather than something which the ordering is allowed to define).
3. Two additional axioms about how the ordering interacts with the field operations (basically, you can always add the same number to both sides of an inequality without invalidating it, and the product of two positive numbers is always positive).
4. The reals are Dedekind-complete (in simple terms, "you can take limits" - in more precise terms, every non-empty subset of the reals that has an upper bound, has a least upper bound).

For example, IEEE 754:

1. Everything is non-associative, which is not allowed under the field axioms. Also, NaN and ±inf don't have additive inverses.
2. Since 1.0 / 0.0 != 1.0 / -0.0, we cannot have 0.0 and -0.0 be "the same value" (because you get different answers when you try to use them in the same expression). Neither number is greater than the other according to IEEE 754, and so they violate total ordering. Also, all of the NaNs violate total ordering, too.
3. There are cases for which x < y, but x + z == y == y + z (because x is the largest value with exponent n and y is the smallest value with exponent n+1). Also, you can trivially break this with ±inf.
4. Almost satisfied: The set of negative floats has two upper bounds which are incomparable (0.0 and -0.0), so we cannot say which is the "least" upper bound. But I'm pretty sure this is the only counterexample (ignoring trivial alterations such as "negative floats greater than -1.0," etc.) because I can't think of a way to construct a counterexample out of NaN or inf.

Or the extended real numbers, which IEEE 754 is intended to mimic:

1. inf - inf is not defined (rather than giving an NaN), which is not allowed under the field axioms. Also, ±inf don't have additive inverses.
2. Satisfied if you assume that -inf < all reals < inf.
3. Not satisfied because, for finite numbers x and y with x < y, x + inf = y + inf = inf.
4. The extended reals are compact, which in this context is an even stronger property than completeness.

Or the hyperreal numbers, which are explicitly designed to "follow all of the usual rules" (for use in nonstandard analysis):

1. Satisfied by the transfer principle.
2. Satisfied by the transfer principle.
3. Satisfied by the transfer principle.
4. Not satisfied: Consider the set of infinitesimals. This is clearly bounded, but it cannot have a *least* upper bound, or else you could derive contradictions by doubling this least upper bound (which must give you a non-infinitesimal) and reasoning about the relationship between the resulting number and the set of infinitesimals.

TL;DR: If you like doing calculus etc. in the usual way, then you can't have infinities or infinitesimals.

[1]: Every totally ordered metrically complete field is isomorphic to the real numbers. So they're all, effectively, "the same field" with different names for their elements. We need this caveat because, if you really wanted to, you could just start calling 2 ** 128 "infinity." But it would still be 2 ** 128. You could still multiply two by itself 128 times to get to your "infinity." "A rose by any other name" and all that.