Is a subset of infinity still infinity?

Question

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 ;)

Answer 1

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.

Answer 2

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!

Answer 3

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".

Answer 4

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.