# Is a subset of infinity still infinity?

In looking at the post here which really got me thinking what infinity and how it is notated. As the question states, is a subset of infinity still infinity? Also, what area of the Math world covers this abstract thought. Thanks.

As a side note, I really tried hard to look for a similar post. If theres a similar post, you are a better man than I. Feel free to downvote and vote close ;)

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On the question whether the subset of "infinity" is still "infinity", I think you have a the freedom to choose what your subset is, so, you could choose that well to be a singleton or empty set which is a subset of anything. And, the branch of set theory in Math deals with notions of infinity, –  user21436 Dec 29 '11 at 22:31
As to "what area of the Math world covers this abstract thought", I would say all of it. –  Alex Becker Dec 29 '11 at 22:40

Infinity, as often argued, is a concept in real analysis.

If a set is infinite, it means that it has a non-finite number of elements. There is an accurate description of this in set theory. However this is irrelevant to your question.

Consider $\mathbb N$, which is surely an infinite set. $\{42\}$ is a finite subset of $\mathbb N$. However, not all subsets are finite, there are infinite subsets such as $\{n\in\mathbb N\mid 13<n\}$.

Some subsets are not only infinite, but so is their complement. For example $\{2\cdot n\mid n\in\mathbb N\}$ - the set of all even numbers.

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You could have said that either a subset of an infinte set is infinite or its complement subset is (or both are, as you did say) –  Henry Dec 29 '11 at 23:53

This, presumably, is in the realm of set theory, or if you are wondering more about what this "means" maybe mathematical philosophy.

If the answer to your question is "is every subset of an infinite set infinite" the answer is surely no. Consider that $\varnothing$ is a subset of EVERY set and has size zero!

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Ah, beat me to it :) –  Zev Chonoles Dec 29 '11 at 22:32
But, surely the empty set is defined to have size zero is arbitrary. Also, in away empty set is normally an axiom. –  simplicity Dec 29 '11 at 22:36

Under whatever interpretation of "infinity" as a set you choose, the empty set will necessarily be a subset of it, and the empty set is certainly not "infinity".

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A point on grammar (because language constrains thought, it is useful to get this right!). The noun "infinity" is rarely used to refer to something for which it would make sense to ask if it has a "subset". "Infinity" is usually used in a context to refer to a particular sort of quantity -- e.g. the extended real number $+\infty$ of real analysis.

I don't know of any naming scheme that would use the word "infinity" to refer to a particular set -- and if it did, it would follow the usual rules of set theory: any proper subset of that set would be a different set.

What you want, I think, is the adjective "infinite" as applied to sets or to cardinal numbers. i.e. "Is a subset of an infinite set still infinite?" In this case, it depends in the particular subset -- an infinite set has many subsets, some of finite size, and some of infinite size.

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That said, I can think of at least one case where it might make sense to ask for a subset of infinity -- e.g. asking for a region that is a subregion of the "line at infinity" in projective geometry. In that case, any subset would still be "at infinity", but I still don't think it really makes grammatical sense to ask if that subset "is infinity". –  Hurkyl Dec 29 '11 at 23:06