How are some infinities bigger than others?

Question

I was watching Veritasium’s video about math having a fatal flaw. He explained that if we make a set of all the numbers in between 0 and 1, then added one to the first digit of the first number and added one to the second digit of the second number etc, we would always have a new number. He said this proved that there were more numbers in between 0 and 1 than natural numbers.

I was confused as to why you can’t do this with natural numbers, and how that proved one infinity was smaller than another.

In: Mathematics

Answer 1

reading your post I may be mistaken but I think you may have misunderstood something.

we make a set of all the numbers in between 0 and 1, then added one to
the first digit of the first number and added one to the second digit of
the second number etc, we would always have a new number. ”

what makes me think you may have misunderstood is where you say “new number”

if my understanding is correct the number itself is not “new” it may be clearer to say it is “unpaired”

the proof is a proof by contradiction and “proves” that one “infinity” is larger than others by

1. first assuming you can pair up (a bijection) the set of natural numbers and real numbers the set of both is infinite and it is assumed that it could be done.

2. it is proven that our assumption leads to a contradiction this is done by “finding” (proving there exists) a “unpaired” number after all the numbers were paired up (this was assumed)

3. the proof that the there is a unpaired number works works by applying a process to the infinite list where a number is shown to be different from every number on the list. this is done by adding some value to the infinite representation of a real number for every real number Importantly the number “created” exists and is a real number it would exist in a set of real numbers but it can’t exist on the set of paired numbers between natural and real numbers.

4. because we assumed we could pair up the all numbers of each and proved there was a unpaired number this implies that our assumption was wrong.

further because there is a unpaired real number and presumably all the natural numbers were paired then their are “more” real numbers ie its a larger set but both sets were infinite.