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Context:

I was watching Derek Muller's (Veritasium) latest video The Man who broke Maths. In it he discussed how Georg Cantor well ordered integers by writing them down as

0,1,-1,2,-2,3,-3.........

Problem:

He later proceeded to say that the size of Integers and Natural Numbers is same as we can may the above written rearrangement with 1 Natural Number. This was obviously very counter intuitive as my thought was that we can also map the Positive Integers to Natural Number and there will still be numbers left. So, how are there two ways that seem right but still give completely different answers.

What am I doing doing wrong and which approach is correct. I am a high school student so haven't learnt a lot about infinite sets apart from Youtube Videos like mentioned above.
Akazo
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2 Answers2

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Each approach is correct when you use the definition of "size" appropriate in each case. Those definitions are different.

One is that set $A$ is the same size as $B$ if you get the same answer when you count them. That works for sets that are finite. It makes no sense for the set of integers since you can't count them.

Cantor's revolution defines "$A$ is the same size as $B$" if you can match the elements of $A$ to those of $B$ one to one with nothing left over.

Then that definition leads to a definition of an infinite set - one that is the same size as a subset of itself. Cantor has turned the contradiction into a powerful tool.

If you rewatch the video and think about what it says you will probably find this explanation there.

(I will delete this answer if someone finds the duplicate.)

Ethan Bolker
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It's precisely because the mapping must not forget any number.

This is called surjectivity. Also, we can't map to the same number twice, this is called injectivity. When the mapping is injective and surjective we call it bijective. When then say that two sets have the same size (in the cardinal sense) if there exist a bijective mapping between them.

Namely just one is necessary, and we don't care about other possible injective or surjective mappings that are not bijective, like the one you mentionned that is only injective but not surjective (the negatives integers are forgotten by the mapping).

benito pepito
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    This answer is essentially correct, but it's not likely to help the OP since it explains things using vocabulary they won't encounter for several more years. – Ethan Bolker Apr 03 '25 at 15:59
  • kind of true, it doesn't explain to the OP the intuition of infinite sizes, but at least it answers the precise question he asked, on the second mapping he talked about. I didn't want to re-explain intuition of infinite sizes when he has just saw a whole video on it, but wanted to stay factual and recall the conditions. – benito pepito Apr 04 '25 at 08:53