So in set theory (a part of math) there is this concept called a well-ordered set. The idea is pretty straightforward, just think of about any collection of items that you want and you can order those items so that there is a "first" item in that set. This makes plenty of sense when you don't question the normal workings in life, like for example how you count.
The thing is that there are sets of numbers that you can't count. For example take all the decimals (real numbers in math terminology) between zero and 1. now you could easily say that the "first" number is zero and the "last" number is 1, but what is the second number. I mean if you took zero out of your set, and tried to look at the first number, what would it be? Would it be .0000001? How about .000000000000000001? Maybe you could choose .5?
See that is the thing about "uncountable sets" I could go on all day picking elements from the decimals between zero and one, and even if I took forever doing it, I couldn't pick them all out.
To get an idea, try this game with a friend.
Draw a board of four squares by four squares. Then draw a line of four squares in a row. Player 1 gets the board and player 2 gets the line. Player 1 goes first and writes a series of X's and O's in the first row. Then player 2 gets to put an X or an O in their first box. Keep going this way until all the boxes are filled. Then player 1 wins if player 2's sequence of X's and O's is one of the ones written in one of the columns, and player two wins if it isn't.
If you can take this same game and "play" it on an infinite square in your head, then you'll have some idea of what i am talking about by uncountable. Even if you put all these things in a row from "first" to "last" you could still find an X and O line that was NOT in the set you made.
Hopefully this helps you see what I mean by having an uncountable set. And even if not, well at least you have some idea of the weird world mathematicians live in. We make sets you cannot count and then insist there must be a way of putting it so that you can have a "first" element of that set!
for some more reading try:
If you get through all those... let me know. You might have the makings of a great mathematician!! :)
Oh my nerdy math professor....I love you!
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