[chalst]

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.

[dborg]

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)

[j2kun]

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.

[chalst]

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.

[michaelochurch]

Excellent point.

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.