When you get into infinity, you have two notions of "number" that diverge. Mathematical operations on them do different things. (For example, cardinal "exponentiation" is the power set; ordinal "exponentiation" is something different and smaller.) One is size, but proper subsets can have the same size at infinity (integers, even numbers, rationals). That's where Aleph-0 (cardinality of the integers) and "c" (cardinality of the reals) come from. With cardinal infinities, you can't really do meaningful arithmetic because the field properties don't apply. "Infinity" violates the mathematical fact that x+1 != x, for example.
The other notion comes from the concept of a well-ordered set, which also maps nicely to "indexes" into possibly infinite lists. With this foundation, you have more options in terms of mathematical manipulations: you can add ordinals (but not always subtract them) and, because they pertain to list operations, the traditional "field" properties aren't always commutative. That's where we get ω, ω+1, ω^2, ω^ω, ε_0 and so on. Those all have rigorous definitions. For example, ω^2 is the order type of ordered pairs of numbers with lexicographic comparison:
Where things get messy is that the relationship between cardinal and ordinal numbers (more formally, what ordinal number has the same cardinality as the reals, or the continuum?) is, in fact, formally undecidable. (Continuum Hypothesis). That doesn't mean no one has solved it. It means there's no mathematical way to refute or prove it from ZFC, the Zermelo-Frankel set axioms plus the Axiom of Choice. The CH is neither true nor false, insofar as one can have valid mathematics with or without it.
To put the above more succinctly, we know that the countable ordinals are a well-ordered set (totally ordered with a minimum) and since no set contains itself, that set is uncountable. It is, in fact, the smallest countable set (the ordinal numbers are totally ordered by the subset relation). That's called ω_1. Intuitively, we might hope that that's also the same "size" as the real numbers (we don't know of any smaller uncountable infinities, and we can't construct any). But there is no way to prove or refute whether that is true. Mathematics is valid either way; it has to "fork".
It's not "nonsensical". What it is is formal. It may or may not map to the real world. You can't actually perform Banach-Tarski (Axiom of Choice hack) on an orange, nor can you store a complete Hamel basis on your hard drive. But these concepts are still useful in defining our notion of what a "set", precisely, is.
Well put; a little quibble: these are the two notions of infinity that most interest set theorists, but there are many other notions of infinity in mathematics, e.g.,
1. Representation of geometric entities "at infinity" in, e.g., the point at infinity from the projective sphere that allows straight lines to be treated as circles;
2. Infinitesimals;
3. Game-theoretic constructions of infinite numbers, e.g., in Conway numbers. Incidentally, the set-theoretic cardinals are equivalent to a special case of these;
4. Definition of numbers as equivalence classes of functions under their speed of growth as they tend to infinity, e.g., Hardy's logarithmico-exponential functions. Incidentally, the computable set-theoretic ordinals are equivalent to a special case of these.
Sorry to give a minor correction to a little quibble, but it is the ordinals, not the cardinals, that are a special case of Conway numbers. The cardinals are equivalence classes of these of the form [א_a,א_(a+1))
(Also you can get infinitesimals from Conway's construction as well)
A quibble of my own: a lot of the "infinity" constructions in mathematics only use infinity as a name. Projective geometry (1) is a good example of that. The formalization doesn't actually appeal to any sort of infinite quantities.
Projective geometry: even in the simple case I outlined, you have lines being circles of infinite diameter. That infinity is just an additional closure point on the plane for shapes (and you can think of the similar projective line providing the complementary notion of displacement we can use to measure the diameter of infinite circles) doesn't stop the geometry from representing shapes with infinite attributes.
It is the case that all of this can be finitely represented. But this is true of a quite large part of large cardinal set theory as well, which can be represented in constructive type theory - mathematicians make it their business to transform the infinitary into the finitary.
Conway's surreal numbers are awesome. Combinatorial game theory seems silly at first (why are we analyzing Hackenbush?) if you expect it to be like "regular" game theory but is mind-blowing when you actually get it in all its glory.
When you get into infinity, you have two notions of "number" that diverge. Mathematical operations on them do different things. (For example, cardinal "exponentiation" is the power set; ordinal "exponentiation" is something different and smaller.) One is size, but proper subsets can have the same size at infinity (integers, even numbers, rationals). That's where Aleph-0 (cardinality of the integers) and "c" (cardinality of the reals) come from. With cardinal infinities, you can't really do meaningful arithmetic because the field properties don't apply. "Infinity" violates the mathematical fact that x+1 != x, for example.
The other notion comes from the concept of a well-ordered set, which also maps nicely to "indexes" into possibly infinite lists. With this foundation, you have more options in terms of mathematical manipulations: you can add ordinals (but not always subtract them) and, because they pertain to list operations, the traditional "field" properties aren't always commutative. That's where we get ω, ω+1, ω^2, ω^ω, ε_0 and so on. Those all have rigorous definitions. For example, ω^2 is the order type of ordered pairs of numbers with lexicographic comparison:
... and ω^ω is the order type of formal natural-number polynomials in one variable with lexicographic comparison: Where things get messy is that the relationship between cardinal and ordinal numbers (more formally, what ordinal number has the same cardinality as the reals, or the continuum?) is, in fact, formally undecidable. (Continuum Hypothesis). That doesn't mean no one has solved it. It means there's no mathematical way to refute or prove it from ZFC, the Zermelo-Frankel set axioms plus the Axiom of Choice. The CH is neither true nor false, insofar as one can have valid mathematics with or without it.To put the above more succinctly, we know that the countable ordinals are a well-ordered set (totally ordered with a minimum) and since no set contains itself, that set is uncountable. It is, in fact, the smallest countable set (the ordinal numbers are totally ordered by the subset relation). That's called ω_1. Intuitively, we might hope that that's also the same "size" as the real numbers (we don't know of any smaller uncountable infinities, and we can't construct any). But there is no way to prove or refute whether that is true. Mathematics is valid either way; it has to "fork".
It's not "nonsensical". What it is is formal. It may or may not map to the real world. You can't actually perform Banach-Tarski (Axiom of Choice hack) on an orange, nor can you store a complete Hamel basis on your hard drive. But these concepts are still useful in defining our notion of what a "set", precisely, is.