Welcome to SE:AI!

I have wondered about applications of Georg Cantor's work towards A.I. (en.wikipedia.org/wiki/Georg_Cantor) Is this something you are interested in OP?

Most of us humans do not understand infinity well enough. Including me.

@Tautological Revelations I am looking for a model/way that assigns infinity number of things to "information" in discrete domain.

@naive but you have a notion infinity. I think from functional perspective humans understand infinity.

I think the broader question is - do the (digital) computers understand anything (any concept)? The only "concepts" they depend on are < = and >. Everything else is composed off of that.

@Amrinder Arora according to strong AI, we can assume that understanding is just pretending. Therefore, the model that can differentiate different signals somehow understand the signals or notions (what you call it).

> this model must differentiate infinitely many numbers < Could you please elaborate what are you trying to convey by the word differentiate. It will help in understanding your thoughts well as this word has been used quite often in your post and I think it needs some justification or thoughts from your end.

@naive I mean the model must recognise infinitely many things in principle like a human. It must show the thing is different from others. Otherwise, we can not be sure that it really works.

@verdery I think a digit recognition model can be programmed to 'differentiate' infinitely many different digits on infinitely many different cards, with each card having a different set of digits or integers as you may. This, in the remotest sense, does not imply that the model understands infinity.

@naive I share that the strong ai seems a little strange but as far as I understand, strong ai propose pretending is the same with understanding and pretending is the most closest goal that the model can achieve in the future. Therefore, first of all, we must achieve to create the model that behaves like understanding infinity. I think that the "real understading" is an another topic.

Is this what you are looking for? (worldscientific.com/worldscibooks/10.1142/6769)

@TautologicalRevelations your suggestion gives analysis of infinity but my problem is different a little.

I can't find much information on your topic. Maybe it is a problem from my side. Best wishes.

I just recently had a long-ish discussion with some very intelligent people who simply did not understand how there could be equally many integers, positive integers, even integers, even positive integers, and prime numbers. So, I would challenge your statement that humans understand infinity. Also, please note that mathematically, there is no such thing as "infinity". There are many branches of mathematics, which may all have different notions of infinity, and any one branch of mathematics may have no, one, or multiple notions of infinity. Then, there are even different "sized" infinities!

@JörgWMittag I agree with you and the intelligent people you talked to. Therefore I deliberately point that I assume "understanding" in a weakness form. Althouhg we use "understanding" in a weakness form, the digital computers are not capable to "understand" infinity like us. I do not claim that infinity can be "really" understood by human. But we can not ignore that human being has a notion of "infinity". I want to add also an example from physics this "understanding" issue to clear the subject. Humans cannot understand dual nature of the light, but, as far as we know, there exist duality.

Please define "understand" precisely. Thanks.

@BobJarvis I define "understand" as a functionalist. Please read for detail: functionalism. Therefore, I assume a vision model "really" understands when it identifies a number written on a card.

I'm a little confused nobody has pointed out that basically every computer already handles infinity - specifically with IEEE 754

@OrangeDog sounds like an answer in the making!

@JörgWMittag is correct. Infinity is a concept that is defined in different ways depending on the field of mathematics. IEEE754 defines a fairly consistent set of rules to handle infinities that underpins most real arithmetic systems on most computers. But there are other rules. An AI can be taught such rules. Whether it can invent new and better ones is outside my pay grade: en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

"if we have enough resources (time etc.), we can count infinitely many things" [citation needed] I think the amount of resources you need to count infinitely many things is infinite, so it is impossible to have "enough"

@craq I mean that if we have enough life time. I does not mean "bigger brains or more neurons".

@OrangeDog this is not related to representation of infinity. This issue is related representing infinity things in the same model.

@verdery we have those too - streams and generators. It seems people on this site know more about philosophy than computer science.

@OrangeDog Asking whether a computer "understands" infinity is fundamentally different to asking whether it can "represent" infinity or "represent" an infinite sequence. If the OP intended the later, they should edit the question to reflect this, then someone can say "IEEE floating points" and "Haskell's cons lists + lazy evaluation" and then we can all go home.

@Pharap we can all go home anyway because computers don’t “understand” anything

@OrangeDog Not yet at least (if ever). Arguably humans are merely biological computers.

define "understand". In the sense of the word as I understand it, computers actually understand nothing.

@DukeZhou The rationals also have an infinite number of rationals between any two rationals, yet the rationals are countable. Reals are uncountable for more complex reasons.

@Pace You left out a very important one: the ability to not make any mistakes. Even if a human had infinite memory, if it were the same quality as most human's memory, eventually they'd get two digits mixed up or make a similar mistake. One of the biggest advantages a computer has over a human is that as long as a computer's hardware is functioning correctly, they won't make any mistakes that aren't already present in their programming. Humans are prone to mistakes even when doing a task they are well practiced at.

This question is extremely confusing to me because there are multiple ways to parse infinity. Infinity by definition cannot be fully understood, because infinity, by definition, is without end; it involves "infinite information." So a few small refinements might be a good idea. This could simply be my background and training. Best wishes and thanks for the interesting thoughts!

Is this what you are looking for, mate? (mally.stanford.edu/Papers/computation.pdf)

@TautologicalRevelations it seems nice paper. I will work on it. Thanks.

Please let me know if I can help you any further. If you accept this paper, I'll just simply add it to my answer.

@TautologicalRevelations I think it is very good starting point to study further.

Let me keep trying.

The information I got were very similar to the already supplied answers. So good job everyone. I got the best results with the search term: "computational cognition of mathematical infinity". I tried a lot of search terms. It could just be a limitation from my side. I'm not suggesting anything strong.