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this post was submitted on 27 Feb 2024
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There are infinite numbers between 1 and 2, none of which are 3.
If there are infinite numbers, then there's 3 in there somewhere. If 3 is not there then it's not infinite.
Oh okay.
No, this is not true. Just because you have infinitely many numbers in some collection, doesn't mean one of the numbers in your collection has to be 3.
Look at the number line. There are infinitely many numbers on the number line between 1 and 2. For example 1+1/2, 1+1/4, 1+1/8, ... are in there (among many others). But all of the numbers between 1 and 2 are strictly smaller than 3, so none of them can be 3.
Alternatively, there are infinitely many numbers strictly smaller than 3, none of which are 3 either.
Well consider the set of numbers 3+1, 3+2, 3+3, 3+4, ... (the set of integer numbers strictly larger than 3). This set of numbers is also infinite and does not contain 3. So a set being infinite doesn't imply it must contain the number 3.
Ah, thank you for the explanation. That makes sense now.