How can some infinities be bigger than others?

Re:How can some infinities be bigger than others? by anikan_vader on Thursday April 20, 2023 @03:06AM (#35637197)

Cantor’s arguments always receive a very negative reception from most of the HN crowd, just as they did from his peers at the time! He was nearly shunned for opening this can of worms, which is one of the reasons why he spent many years refining his arguments to be as elegant as possible (diagonalization came long after the original result).

Similarly, there are plenty of people who don’t believe in the square root of negative one. Others don’t believe in the existence of irrational numbers (e.g. the Greeks). Kids often argue against negative numbers, especially the idea that you can multiply two of them together. Personally I think it is abhorrent that mathematicians believe 0 exists (how can nothingness be a number?!)

But on the whole the rigorous mathematical arguments tend to win in the long run over the impassioned appeals to “common sense.” Today Cantor is regarded as a hero by nearly all mathematicians.

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Re:How can some infinities be bigger than others? by Ivoirians on Thursday April 20, 2023 @03:34AM (#35637344)

It doesn't matter if someone thinks Cantor "breaks the rules", or the square root of -1 "doesn't exist." The (majority of) mathematicians who take those as true have created a wealth of rigorous, interesting, and worthwhile results built on top of these concepts. I vehemently believe mathematics is more "abstract thought experiments" than "discoveries of universal truth".

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Re:How can some infinities be bigger than others? by cubefox on Friday April 21, 2023 @01:38PM (#35657307)

I doubt that any results which rested on the assumption of Hume's Principle are worthwhile. Indeed, set theory seems to have been useless for most mathematicians except those who are interested in "foundations", since it is mostly ignored.

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Re:How can some infinities be bigger than others? by atoav on Thursday April 20, 2023 @03:18AM (#35637254)

I think all of this has to do with our intuition about numbers and what we think they are.

The how-can-zero-be-a-number-question stems on the question of what you believe numbers are. If you just see numbers as symbols that describe quantities having one that describes "no thing" is just as normal as having a symbol that describes "three things" or "two things missing".

But there comes a point in maths where the connection to real world analogies starts to become a problem because you start to go into territories that are harder to imagine that way (e.g. complex numbers). Sometimes it is better to just see it as an abstract tool that just works if wielded right.

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