Is it ever possible to say that $\infty = \infty$?
For example, does the number of odd numbers ($\infty$) equal the number of even numbers ($\infty$)? Does does the number of odd numbers ($\infty$) even equal the number of odd numbers ($\infty$)? Does does the number of odd numbers ($\infty$) equal the number of real numbers ($\infty$)?
If not, what can we say about comparing these infinities?
Additionally, how is $\infty$ handled outside of set notation?
There are different sizes of infinity (more precisely, one infinite set can be decidedly larger than another), so we need to employ more than just a simple "$\infty$" symbol.
Two sets have the same size if there is a bijection (one-to-one and onto) function betwen the two, also called a one-to-one correspondence (which pairs every element of the first set with exactly one element of the second set, in such a way that there are no elements left unpaired). This allows us invent cardinal numbers that describe the size of any given set. (One can define them formally as equivalence classes of sets under bijection, or use specific representatives of these classes - namely, the ordinals.) The smallest infinity is $\aleph_0$ which describes the size of the set of naturals $\Bbb N$.
If a set $X$ can be bijected with $\Bbb N$, equivalently if it can be exhaustively listed out in a sequence such as $x_1,x_2,\cdots$ then it is called countable. (Finite sets are also called countable too.)
Moreover we can define when one set is larger than another. If there is a one-to-one map $A\to B$, this means that $A$ can "fit inside" $B$ and so we write $|A|\le|B|$. Or if there is an onto map $B\to A$ then this means $B$ can "cover" $A$, and in that equivalent case $|A|\le|B|$ too. If $|A|\le|B|$ but $|A|\ne|B|$, then we can say that $|A|<|B|$, so $B$ is strictly larger than $A$.
(If one discards a standard nonconstructive assumption of mathematics called the Axiom of Choice, which many thrill-seeking set theorists, model theorists and logicians do to explore the foundations of mathematics and ascertain how constructive different parts of math are, then it's not always the case that any two sets are comparable. That is, without AC, there may be sets $A$ and $B$ in which neither is larger than or smaller than or equal in size to the other.)
For example, does the number of odd numbers equal the number of even numbers?
Yes. The odd integers can be listed exhaustively out as $1,-1,3,-3,5,-5,\cdots$ which is essentially a bijection between them and $\Bbb N$, hence they have the same size as $\Bbb N$, namely $\aleph_0$. Similarly, so too can the even integers be listed exhaustively out: $0,2,-2,4,-4,\cdots$, thus they also have the size $\aleph_0$.
There is an explicit bijection $\{\textrm{odd integers}\}\xrightarrow{\sim}\{\textrm{even integers}\}$ given by $x\mapsto x+1$. This shows the even integers and odd integers have the same size as sets directly, without referencing $\Bbb N$.
Does does the number of odd numbers even equal the number of odd numbers?
Yes. Any set is equinumerous with itself, i.e. $|A|=|A|$ for any set $A$.
Does does the number of odd number equal the number of real numbers?
Nope, there is no bijection between odd numbers and real numbers. The real numbers are strictly greater in cardinality. Since odd numbers are countable and both are infinite sets, it suffices to show the reals are uncountable, or equivalently that no sequence of reals contains every real (this is done via proof by contradiction). There are many questions on this site addressing this The standard argument you can look up is called Cantor's diagonal argument.
For more on cardinal numbers, see Wikipedia's article.
