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According to metamath, the set-theoretic definitions of positive and negative infinity are $$+\infty=\mathscr{P}{\cup\mathbb{C}}$$ and $$-\infty=\mathscr{P}{+\infty}$$ respectively, i.e., positive infinity is defined as the power set of the union of the complex numbers and negative infinity is the power set of positive infinity. When I first came across these definitions, I was slightly puzzled by their arbitrariness. The descriptions state that such definitions are used to guarantee that $+\infty$ and $-\infty$ do not appear in $\mathbb{C}$ using Cantor's Theorem and that they are distinguishable without the Axiom of Regularity.

I can accept that the definitions of these infinities can be arbitrary so long as they are provably not in $\mathbb{C}$ and provably distinguishable from one another, but my question is how do we know whether or not these definitions of infinities do not accidentally conflict with set-theoretic definitions of other mathematical structures and sets? As the descriptions admit, these definitions are arbitrary and all we care about are their membership properties with respect to $\mathbb{C}$, so we don't care what $\mathscr{P}{\cup\mathbb{C}}$ actually is. But this is problematic because we haven't guarenteed that $\infty$ as a set itself is not different from other mathematical structures, e.g., $\mathscr{P}{\cup\mathbb{C}}$ may have coincidentally created the class of all groups or created the class of all models of ZF or created the equinumerosity relation. If infinity just so happened to equal one of these (or all of these), then it would be absurd to say that $\mathrm{Groups} = \infty$ or $15<\mathsf{ZF}$ or $\lim_{x\to\approx}e^x=\mathsf{ZF}$. And as more and more mathematical structures are defined, these definitions of infinity (and maybe other arbitrary mathematical structures) are potentially threatened.

So is there a way to prove their distinguishability from other mathematical structures or are we stuck with this? Are there any other set-theoretic definitions of infinity like these ones?

Don't get me wrong, I know infinity should be understood conceptually and is obviously distinguishable from things like groups and $\mathsf{ZF}$, but that doesn't excuse allowing absurdities like those stated above.

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The extended real numbers $\pm \infty$ are a red herring; even with the ordinary real numbers you have similar problems; e.g. is $\pi$ the set of all integers? Are integers the same thing as diagonal lines in the infinite 2-dimensional grid whose coordinates are indexed by natural numbers?

(the answer to that last question, incidentally, is yes under one common path of constructing things)

This problem you're facing is a general bug involved in doing mathematics in an untyped style; that is, all mathematical objects are lumped together as being the same kind of object.

This bug is dealt with by language; typical mathematical arguments attach a type judgment to each object when they're introduced (e.g. "let $x$ be a real number"), and within that scope, one only uses the objects in accordance to the attached type.

One can, incidentally, encode this practice directly into foundations via something like type theory. In type theory, if you have an extended real number $x$ and a variety of algebraic structures $V$, you can't ask whether $x = V$, since they are objects of different types.

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  • $\begingroup$ Downvote, because the answer is gibberish and completely uninformed. No, $\pi$ is a greek letter. With no further qualifications, $\pi$ is taken to be the name of a real number. No real number is the set of integers. No integer is a diagonal line in a grid. Yes, mathematical objects are the same kind of object, called 'mathematical object'. To state that the name $x$ will denote a real number is not a 'type judgment'. Everything is wrong in this 'answer'. $\endgroup$ – Tommy R. Jensen Aug 10 at 11:18

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