Transfinite Numbers

[Anonymous No.4749228]

I'm pretty new here, I've lurked a little. Now I have a question. Can/will anyone explain transfinite numbers to me? (Note: I understand functions, some set theory, but haven't had Calculus.)

[Anonymous No.4749255]

Lets say we want to order a group of things.
If we have, say, three of them, the easiest way to order them would be to assign the numbers 1,2,3 respectively to the things. So we can order the alphabet, for example, by assigning 1 to a, 2 to b, 3 to c, etc. Then you can compare them with < and > and have it make sense. For example a < c, and e > d.


Now lets say you have an infinite number of things. For example, all possible words where letters are strictly decreasing, so you can have an 'a' after a 'b' but not a 'b' after an 'a'. Then valid words are dddddccba, ba, a, bbbbbbbbbba, zbba, and other things like that. Then, if you sort them in alphabetical order, you can compare them like in the single letter case. It's obvious that, for example, a < aa. And that aaa < aaaaaaa. And that aaaaaaaaaaaaaaaaaa < b.

[Anonymous No.4749268]

>>4749255
The problem is, you can't just assign a natural number to each word in this case, even though it is a well ordered set. This is because we can have arbitrarily long strings of all 'a's, and no matter how long it is it is still less than "b".

So, since we want to order it, we assign 1 to a, 2 to aa, 3 to aaa, 4 to aaaa, etc, and then a new symbol <span class="math">\omega[/spoiler] to b, and continue by assigning <span class="math">\omega+1[/spoiler] to ba, <span class="math"> \omega + 2[/spoiler] to baa, etc. After assigning these to all strings baaaaa..., we need something for bb (which is the next thing in alphabetical order). So we choose <span class="math">2 \omega[/spoiler] to assign to bb. These are the transfinite ordinals.