4 ms·

Math on it's own, with concepts like this, can get a bit abstract. I like connecting the concept with something concrete. For example the infinity of time. I

Math on it's own, with concepts like this, can get a bit abstract. I like connecting the concept with something concrete. For example the infinity of time. I once watched a fascinating documentary on the end of the universe, that talked about what would happen if the universe were to keep expanding forever and ever. first all the stars burn out, then a bunch of blackholes form, thenthe blackholes dissipate, etc,etc. they mentioned that actual protons would eventually breakdown after many 10 to the 10s etc.

zikduruqe 4y ago
This? https://www.youtube.com/watch?v=uD4izuDMUQA https://www.youtube.com/watch?v=uD4izuDMUQA
mym1990 4y ago
I watched 'A Trip to Infinity' last night on Netflix and it touched on this part as well! Everything eventually spreads out so far away from everything else, that things burn out/die out in a very final kind of way. Its kind of sad, but also mostly irrelevant since we will long be gone by then(well, as far as we know!)
yamrzou 4y ago
Related: Is Infinity Real? — https://www.quantamagazine.org/the-infinity-puzzle-solution-20160630 https://www.quantamagazine.org/the-infinity-puzzle-solution-...
daveslash 4y ago
"God made the integers; all the rest is the work of Man." ~ Leopold Kronecker, when criticizing Cantor's work [0]. See also Stephen Hawking's Anthology by a similar name. [1] [0] https://en.wikipedia.org/wiki/Leopold_Kronecker https://en.wikipedia.org/wiki/Leopold_Kronecker [1] https://en.wikipedia.org/wiki/God_Created_the_Integers https://en.wikipedia.org/wiki/God_Created_the_Integers
- bmacho 4y ago
  I'd argue that the positive real numbers are existing/physical/God-made, while the negative numbers (even the negative integers) exist only to help the computations.
  - bawolff 4y ago
    You think e exists but 0 is only to help computations?
  - daveslash 4y ago
    Yes. That's exactly the point! Same with imaginary numbers! And all sorts of other concepts. Such as different magnitudes of "infinity", which is what this post was about, what George Cantor studied, and the topic which prompted the quote! Everything other than the integers is just philosophy made up by consciousness.
bawolff 4y ago
I feel like that is a different concept of infinity than the one the article is talking about. Thinking about time as it approaches infinity feels more like an asymptotic notion of infinity instead of a cardinality one, i.e. similar to the type of infinity we mean when we say quicksort is O(n log n). That said, i dont know how to really make this any more concrete then what the article did. Perhaps an analogy of an infinitely precise ruler with every possible tick marked off (e.g. marking off 0.1 cm, 0.01, and so on for every real number), and how if in addition to labeling the ticks with lengths we also wanted to label them as first tick, second tick, third tick, it would be impossible. The basic gist of the proof from the article is given any proposal for a way to label the ticks on our ruler as first, second, third, ... covering all the ticks, there is a generic way to find a tick that is missed, so any such proposal must miss some ticks. So in a sense the real numbers are "bigger" than the integers, because if you try to match them up you will always have real numbers left over. Thus people say the real numbers are uncountable as there is no way to count them all. This is in contrast to something like the integers where you could say 0 is the first one, 1 is second, -1 is third, 2 is fourth and so forth, eventually getting to all of them given infinite time with nothing left over.

zikduruqe 4y ago

This? https://www.youtube.com/watch?v=uD4izuDMUQA https://www.youtube.com/watch?v=uD4izuDMUQA

mym1990 4y ago

I watched 'A Trip to Infinity' last night on Netflix and it touched on this part as well! Everything eventually spreads out so far away from everything else, that things burn out/die out in a very final kind of way. Its kind of sad, but also mostly irrelevant since we will long be gone by then(well, as far as we know!)

yamrzou 4y ago

Related: Is Infinity Real? — https://www.quantamagazine.org/the-infinity-puzzle-solution-20160630 https://www.quantamagazine.org/the-infinity-puzzle-solution-...

daveslash 4y ago

"God made the integers; all the rest is the work of Man." ~ Leopold Kronecker, when criticizing Cantor's work [0]. See also Stephen Hawking's Anthology by a similar name. [1] [0] https://en.wikipedia.org/wiki/Leopold_Kronecker https://en.wikipedia.org/wiki/Leopold_Kronecker [1] https://en.wikipedia.org/wiki/God_Created_the_Integers https://en.wikipedia.org/wiki/God_Created_the_Integers

bmacho 4y ago
I'd argue that the positive real numbers are existing/physical/God-made, while the negative numbers (even the negative integers) exist only to help the computations.
- bawolff 4y ago
  You think e exists but 0 is only to help computations?
- daveslash 4y ago
  Yes. That's exactly the point! Same with imaginary numbers! And all sorts of other concepts. Such as different magnitudes of "infinity", which is what this post was about, what George Cantor studied, and the topic which prompted the quote! Everything other than the integers is just philosophy made up by consciousness.

bawolff 4y ago

I feel like that is a different concept of infinity than the one the article is talking about. Thinking about time as it approaches infinity feels more like an asymptotic notion of infinity instead of a cardinality one, i.e. similar to the type of infinity we mean when we say quicksort is O(n log n). That said, i dont know how to really make this any more concrete then what the article did. Perhaps an analogy of an infinitely precise ruler with every possible tick marked off (e.g. marking off 0.1 cm, 0.01, and so on for every real number), and how if in addition to labeling the ticks with lengths we also wanted to label them as first tick, second tick, third tick, it would be impossible. The basic gist of the proof from the article is given any proposal for a way to label the ticks on our ruler as first, second, third, ... covering all the ticks, there is a generic way to find a tick that is missed, so any such proposal must miss some ticks. So in a sense the real numbers are "bigger" than the integers, because if you try to match them up you will always have real numbers left over. Thus people say the real numbers are uncountable as there is no way to count them all. This is in contrast to something like the integers where you could say 0 is the first one, 1 is second, -1 is third, 2 is fourth and so forth, eventually getting to all of them given infinite time with nothing left over.