From what (little) I understand of this Wikipedia article, where $\omega$ denotes the ordinal "identified with" $\mathbb{N}$, and $\aleph_0$ is the cardinality of $\mathbb{N}$, and $\mathfrak{c}$ is the cardinality of $\mathbb{R}$:
$$\lim_{n\to\infty}n=\lim_{n\to\infty}an+b,\,\forall a,b\in\mathbb{N}$$
$$\aleph_0=a\aleph_0+b,\,\forall a,b\in\mathbb{N}$$
$$\mathfrak{c}=2^{\aleph_0}\gg\aleph_0$$
$$\omega=1+\omega\lt\omega+1\lt\omega+2\lt\dots$$
These are all infinite, or "transfinite" quantities. From algebra and calculus, I've learned to respect infinity as not a real number, and to treat it only in the sense of limits, where infinity $+,-,\times,\div$,[insert essentially any operation here] with anything else is still infinity, as otherwise paradoxes arise. I understand the bijection argument for why the cardinality of the reals is much greater than the cardinality of the naturals, and I understand the origin of the exponentiation base two, but that seems incongruous with the identities that show $\aleph_0$ remains $\aleph_0$ when operated on: e.g. $\aleph_0^2=\aleph_0\neq2^{\aleph_0}$, somehow (???) – how can we be certain that the power set has a different cardinality? – and I wonder if this is justified by the hyperreal or the surreal number system: is the $\omega$ ordinal the same $\omega$ seen in the hyperreals?
I feel like there is a formal underpinning here that I'm completely missing, that rigorously and logically treats these infinities, or these "trans"finities in different and sensible ways.
Additionally, in this article, $\omega+1>\omega=1+\omega$ is justified with by taking the union of two sets, say $S,T$, and defining this (seemingly arbitrarily?) as equivalent to finding the union of $\{0\}\times S$ and $\{1\}\times T$. I assume this is the Cartesian product, so assuming $S,T$ are copies of the naturals, T denoted by a dash, this would be the union of $\{\{0,0\},\{0,1\},\{0,2\}\dots\}$ and $\{\{1,0^{\prime}\},\{1,1^{\prime}\},\{1,2^{\prime}\}\dots\}$ which is apparently well-ordered as $\{\{0,0\},\{0,1\},\dots\{1,0^{\prime}\},\{1,1^{\prime}\}\dots\}$ and each subsequent element is strictly greater than the one before it, and so this has ordinal $\omega+\omega=\omega\cdot2>\omega$, which is apparently justified as $\{1,0^{\prime}\}$ has "no direct predecessor", implying the ordinals of the rest of $\mathbb{N}^{\prime}$ begin at $\omega$. What if we had defined $T$ as also being $\{0\}\times T$? How come the set product is valid, necessary and meaningful/non-arbitrary?
Best Answer
As it stands, your question is a bit difficult to answer because it spans a lot of subjects, and there is no way to do them all justice in an answer box. Let me try to give you some useful pointers, and then do my best to answer your questions concretely but very concisely.
First, the main thrust of your question: yes, all the equations/inequalities that you've listed in your question are true and rigorous when interpreted in the right context. The apparent inconsistencies stem from the fact that the same notation ($+, \times$, exponentiation) is used for different operations. There are various definitions for these operations that turn out to be the same on natural numbers (or: finite sets), but different on infinite sets. For example, we could define $n^m$ by saying $n^0 = 1$ and $n^{m+1} = n \times n^{m}$, or we could define $n^m$ by saying that it is the number of functions from $\{0,\ldots, m-1\}$ to $\{0, \ldots, n-1\}$. This gives the same result, but they do not generalize in the same way.
There are three "main" ways of talking about infinity:
Apart from these two, you mention the hyperreal and surreal numbers: these are interesting objects, but also unrelated to the three main ways of talking about infinity that I described above. As far as I know (but an expert may disagree), these are a bit closer to mathematical curiosities than to standard ways of thinking about infinity.
Indeed. These are alephs, so we are talking about set cardinalities. $\aleph_0^2 = \aleph_0$ essentially says that the sets $\mathbb N \times \mathbb N$ and $\mathbb N$ have the same number of elements. The proof for this is pretty similar to the standard proof that $\mathbb Q$ is countable. The inequality says that there are more functions $\mathbb N \to \{0, 1\}$ than there are elements of $\mathbb N$: this is Cantor's theorem.
This is not really arbitrary: when talking about ordinals, we thinking about orderings (specifically: well-orderings) of sets. It turns out to be a natural operation to "concatenate" two orders. (In fact, on the finite ordinals i.e. natural numbers, this is just addition.) The construction you mentioned just makes this formal.
The reason why we do $\{0\} \times$ and $\{1\} \times$ at all, is to avoid problems when $S, T$ have elements in common. If you already know that $S \cap T = \emptyset$, you might as well use $S \cup T$. (You would still give it the ordering where each element of $S$ comes before each element of $T$, though.)
The only real way to learn why a particular definition is meaningful and non-arbitrary is to learn more about it, and to understand that these definitions allow us to concisely ask interesting questions.