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I had a conversation about infinity with my friend yesterday and what I left with was a theory about infinity.

HERE goes. I believe that there are different quantities of infinity because if you have two ranges, 0–1 and 0–2. Let's set this up, if you consider two sets of calculations that have the same calculation density; what I mean by calculation density is that say one sets up a computer program to begin finding/calculating—for example—an infinite value for every 0.1 on the two ranges. Now, with the same density of 0.1, the range 0–1 has 10 starting points for the calculation set; the range 0–2 has 20 starting points, how cool!

So, in my opinion, calculations of infinities differ in speed, "sets", and more. Now, please give some feedback, Stackers; tell me what I got wrong, what is strange etc., so that we can improve our ideas!

Kristian Berry
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Xebiq
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    See, e.g., https://en.wikipedia.org/wiki/Aleph_number So you're right in general about "different quantities of infinity", but all your examples have the same aleph-1 cardinality. – eigengrau Jan 09 '23 at 06:58
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    The issue has a very very long history... Already [Galileo](https://en.wikipedia.org/wiki/Galileo%27s_paradox#Galileo_on_infinite_sets) discovered that if we compare two infinite sets in terms of "inclusion" we have that one of them is "greater" than the other but if we compare them with a "counting procedure" based on element-to-element correspondence they are "equal". – Mauro ALLEGRANZA Jan 09 '23 at 07:24
  • See also [this post](https://math.stackexchange.com/questions/661855/is-it-faster-to-count-to-the-infinite-going-one-by-one-or-two-by-two). – Mauro ALLEGRANZA Jan 09 '23 at 07:25
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    I also found this in addition, and have to consider one of the comments: https://mathworld.wolfram.com/Aleph-1.html – Xebiq Jan 09 '23 at 08:32
  • Objections: 1) "two sets of calculations that have the same calculation density" If we need a rigorous proof we have to define what does it mean "calculation density". 2) "to set up a computer program to begin finding/calculating an infinite value for every 0.1 on the two ranges": the two processes start at the same moment and they will never end; so what? – Mauro ALLEGRANZA Jan 09 '23 at 09:04
  • And see [this post](https://philosophy.stackexchange.com/questions/6785/one-infinity-greater-than-another-infinity). – Mauro ALLEGRANZA Jan 09 '23 at 09:05
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    Correctly or not, I am reminded of [Paul du Bois-Reymond's "infinitary pantarchy"](https://en.wikipedia.org/wiki/Paul_du_Bois-Reymond), which Conway and Guy (*The Book of Numbers*) assimilate to Cantor's complicated infinite ordinals and other surreal numbers/functions. – Kristian Berry Jan 09 '23 at 13:25
  • This is why the Greeks, reports suggest, avoided *infinity* like the plague - paradox after paradox after paradox (it was an affront to the Grecian mind). My all-time-favorite infinity paradox is that *the whole is equal to its part* @MauroALLEGRANZA. – Agent Smith Jan 09 '23 at 18:20
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    Cool! If the *infinity* between 0 - 1 is the same as that between 0 - 2, *why* does my computer take *longer* to calculate the latter than the former? Do remember though that the calculation in *both cases* has *no end*, they *don't terminate* and that is *equality* in the universe of *infinities*. – Agent Smith Jan 09 '23 at 18:24
  • I realized that the title should have been related to pondering about infinity... – Xebiq Jan 10 '23 at 05:57

2 Answers2

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I would recommend taking a look at the SEP: https://plato.stanford.edu/entries/infinity/.

Frank
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How would you know if one set of quantities is bigger than the other set?

You compare, ofcourse. One to one correspondence. Now if you are left with one set still having any thing in it while the other set is exhausted you know the first set is bigger than the other.

If both sets have infinite quantities then you never exhaust any set. So you cannot say which set is larger or even both sets are equal. Your data tell you nothing about it so you cannot conclude.

Always, always and always keep data above and theory below. Your theory is wrong if it don't match with data no matter how symmetric, beautiful and logical it sounds.

People will tell you that there are more numbers between 1 and 2 if you go into decimals than there are natural numbers. But how can they know? Have they counted? Have they compared one by one elements of each set?

Never let go of Aristotlean Logic that a thing cannot both exist and not-exist at same time, in same place, in same sense etc. Because its just consistency. If you let go of consistency any answer will fit in as no answer will mean anything.

An infinite set cannot be counted. Comparison of sets to find which has more elements in it is an operation that requires counting (thats by definition of the said comparison). Give these two statements some thought in light of Aristotlean Logic. You will conclude that there is no way to tell which of any two infinite sets is larger.

Atif
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  • What I meant was that, if you start from a certain amount of "points/numbers", the ranges of parallel calculationcan be compared and valued to be of different amount of calculations. Of course, I'm not yet sure if it then "certainly" means that there are different amounts of infinities in those ranges of calculations of infinities... – Xebiq Jan 09 '23 at 09:32
  • You can start, you can compare, but you cannot finish the comparison. Its because what you are comparing is infinite (has infinite elements). Also, don't put the word:certainly in quotes, either you are certain or you aren't. – Atif Jan 09 '23 at 09:38