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How Big Is Infinity?
- coldacid 4y agoCertainly much bigger than a walk to the chemist's.
- deleted 4y ago[deleted]
- yodon 4y agoInfinite is a word like forever. We have no problem understanding "ten minutes more than forever" makes no sense (other than as hyperbole), but a surprising number of people are unable to grasp that "one more than infinite" makes no sense. Next up: The word unique, and why it similarly doesn't make sense to qualify it with phrases like "the most unique."
- coldacid 4y agoStill, there are bigger and smaller infinities.
- function_seven 4y agoIt’s totally fine to put gradations on “unique”. The “most unique” item in a set will have multiple aspects that are rare or one-of-kind. It may be the only green one, and also the only textured one and the only top-heavy one. The more unique something is, the more it stands apart from other objects in its class.
- yodon 4y agoUnique means there's only one thing like it. You're advocating there's "more only one thing like this" of something than something else. The word that takes qualifiers is "unusual." People commonly say "unique" when they mean unusual. One thing can definitely be more unusual than another thing.
- mejutoco 4y agoUnique: being the only existing one of its type or, more generally, unusual, or special in some way https://dictionary.cambridge.org/dictionary/english/unique https://dictionary.cambridge.org/dictionary/english/unique I think it depends on the context, but it does not seem an incorrect usage.
- function_seven 4y agoI know, but still insist that the things that separate this unique item are themselves small or large. If an item separates itself from the others on mutliple dimensions, then it is more unique than an item that differs only in one dimension. Both of them are the only things like themselves. I have a collection of vases. Almost all of them are roughly the same size (15cm), shape (round; tapered), and color (clear or white). But there are two that are different: "A" is green instead of white, with a square base. B is made from petrified buffalo dung, stands 1 meter tall, must be carefully balanced because it's so heavy on top (and skinny at the base), and is covered in velvet. A is unique. B is definitely more unique.
- yodon 4y agoAll snowflakes are unique. Would a red snowflake be more unique? You might personally find the color axis more interesting, but it's just one of a huge number of axes along which the snowflakes are different. The red one is more interesting to you but it isn't actually "more unique". The same is true for your vases.
- Kranar 4y agoMost of the dictionaries I just Googled disagree with your assessment. It looks like unique has multiple definitions and while unique can definitely refer to something "being without a like or equal", other definitions include "unusual or special in some way" as well as something "rare and distinguished". Like most arguments about words, it mostly comes down to context.
- paxys 4y agoWhy does "the most unique" not make sense? Object A can be different from all others in some way (so it is unique), but object B can be different in a much more extreme way. So B is "more unique" than A.
- yodon 4y agoUnique means it's the only one like it. It can't be more only one than another thing. It can be more unusual or more interesting but it can't be more only one.
- mejutoco 4y agoNegative numbers also made no sense for centuries and now we teach them to kids as meaning owning some amount of things. The same could be said non-euclidean space (what is the use in that, right? or complex numbers) but both turned up to be useful in some contexts. Same could be the case with fuzzy logic. Cantor and the different infinite sizes is nonsense to some people but for some reason it is still there in the history of Mathematics. Maybe someone can explain better than me if it is useful, but there is a certain intuition to it that is interesting.
- philipswood 4y agoWith ordinal transfinite numbers infinity + 1 infinity + 2 Infinity × 2 Etc. Are all nicely defined.
- drdeca 4y agoyes, though, it is generally prefered to call it "omega" rather than "infinity".
- lupire 4y agoOmega + 1 is an original number that is larger than omega, which is infinite. This is well defined mathematics. There are an infinite number of binary strings 0.xyzw... that are less than 1.0, and 1.0 is the first number after all of them. Infinity + 1.
- willis936 4y agoNo mention of the fact that infinity is not a number. Numeric operators only work on numbers.
- zardo 4y agoNumeric operators only work on numbers. Operators can work on all sorts of mathematical objects.
- drewrv 4y agoI don't think that's a satisfactory answer for a layperson. Infinity is a size/quantity, for most people that usually means "number". The reason it's not a number is because it doesn't exist on the real number line, but again I don't think the helps people intuitively understand what's happening.
- willis936 4y agoI think it does. A layperson scratches their head when trying to think about how to circle the square of "infinity + 1" or "infinity / infinity". Framing infinity as something outside of arithmetic is a starting point for a more interesting discussion of countability and types of infinities. Infinity is bigger than any number, so then how could it be a number?
- moate 4y agoUm, duh, you tip an 8 over. If it's not a number, why is it made out of a number? /s
- lupire 4y agoThere are numbers that aren't on the real number line. i, for one. Omega, for another.
- WastingMyTime89 4y agoIt’s fairly easy to extend R with plus and minus infinity. It’s called the affinely extended real number system and you can extend the usual operations to somewhat work in it. It loses most of the interesting property of (R,+,x) but keeps enough for arithmetic to work mostly as expected.
- helen___keller 4y agoI'm used to blogs, magazines, and the internet butchering this topic, so it was refreshing to see a reasonable introduction. Only complaint is the attempt to build intuition came very close to making an incorrect statement: > But there’s something unsatisfying about declaring the size of the set of real numbers to be the same “infinity” used to describe the size of the natural numbers. To see why, pick any two numbers, like 3 and 7. Between those two numbers there will always be finitely many natural numbers: Here it’s the numbers 4, 5 and 6. But there will always be infinitely many real numbers between them, numbers like 3.001, 3.01, π, 4.01023, 5.666… and so on. > Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation. The alluded property (the set of real numbers is dense) is not related to cardinality, in the sense that the rational numbers are dense but also countable. But I appreciate that the author is trying to motivate the remaining explanation about diagonalization, which can be a tricky topic for beginners.
- WastingMyTime89 4y agoI was equally annoyed by this part. You are very generous with your very close to making an incorrect statement. I would straight away say it’s both misleading and wrong.
- Kranar 4y agoThe author specifically states, and I quote: >no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between ... this doesn’t mean that the sets of real numbers and natural numbers have different sizes What is misleading or wrong about that? What specific statement is incorrect?
- floxy 4y agoNo matter how close any two distinct rational numbers are to each other, there will always be infinitely many rational numbers in between. But the cardinality of rational numbers and natural numbers is the same.
- deleted 4y ago[deleted]
- WallyFunk 4y ago> this mysterious, complicated and important concept For me it's not mysterious. I believe it's a fundamental phenomenon of the Multiverse and that the Universe was two or more Universes colliding which we call the 'big bang', a Universe among infinite amounts of Universes where multiple ones happen to collide all the time, sprouting new ones. How else can you explain our Universe spontaneously sprouting out of 'nothing'?
- warent 4y ago> How else can you explain our Universe spontaneously sprouting out of 'nothing'? It's turtles all the way down. Where did this multiverse come from? Some hyperverses colliding together? Where did the hyperverses come from? etc. etc. At some point, something seems to have sprouted from nothing. Either that, or "nothing" and "something" are fundamentally the same thing. The distinction of the two is an illusion of the human mind.
- eatsyourtacos 4y ago>How else can you explain our Universe spontaneously sprouting out of 'nothing'? You explained nothing. Where did the multiverse come from then.. did it sprout out of nothing?
- WallyFunk 4y agoYes but Infinity explains that. It just goes on and on...there is no beginning.
- swamp40 4y agoI stopped believing in infinity. In the real, physical world. That makes much more sense to me and fits in much better with my understanding of the universe.
- mr_mitm 4y agoThat's quite uncontroversial. But this is about math.
- idiotsecant 4y agoWell if we're going to start deciding what things to believe in based on whether they're comfortable or not I am going to stop believing in traffic and the texture of coconuts.
- WastingMyTime89 4y agoTo be fair, mathematics are not really concerned with the real, physical world. What’s interesting is that we can built axiomatic rules allowing us to properly define infinite sets and these constructions display surprising properties. It’s all purely abstract but it’s a fun thought exercise if you enjoy that kind of things. Plus it has practical applications sometimes which is good I guess.
- Der_Einzige 4y agoDon't let the assholes here gaslight you into thinking that Finitism isn't a somewhat hetrodox but still respected field of math. A bunch of great mathematicians historically have felt this way, and a few great ones today still do. Even if you're willing to accept the idea of cardinalities of infinity (and I think you ought to), I find that the more broad acceptance of 1. The axiom of choice (vs the axiom of determinancy, it's opposite) and 2. The law of excluded middle to be highly suspicious. If you reject the axiom of choice, you're just in alternative but still correct math. If you reject the law of excluded middle, you've ended up with intuitionistic or "constructive" logic, where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality. Notably, in intuitonistic/constructive logic, infinity is rejected until it can be "constructed", which also means that Cantors diagonalisation argument is not so naively accepted. While that diagonalisation itself was constructive, Other related and stronger theories from Cantor are not.
- sleton38234234 4y agoMath 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 agoThis? https://www.youtube.com/watch?v=uD4izuDMUQA https://www.youtube.com/watch?v=uD4izuDMUQA
- mym1990 4y agoI 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 agoRelated: 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 agoI'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.
- aarondf 4y agoI just watched "A Trip to Infinity" [1] on Netflix last night and really enjoyed it. I'm not a mathematician by any means, but I was still able to follow it all. Highly recommended. [1] https://www.netflix.com/title/81273453 https://www.netflix.com/title/81273453
- passion__desire 4y agoI liked this "music video" [1] inspired by the concepts involving proof of infinity [1] https://www.youtube.com/watch?v=tNYfqklRehM https://www.youtube.com/watch?v=tNYfqklRehM
- birriel 4y agoI second this recommendation. Excellent documentary.
- Der_Einzige 4y agoFor those who want some slightly more "sane" arguments against the foundation of mathamatics that is normally accepted, I highly recommend that folks read these wikipedia articles. Basically, if your concerned with the epistemological justifications for the traditional way that infinite set theory is treated, it turns out you're far from alone! A lot of top mathematicians agree with you. 1. https://en.wikipedia.org/wiki/Intuitionism#Infinity https://en.wikipedia.org/wiki/Intuitionism#Infinity 2. https://en.wikipedia.org/wiki/Intuitionistic_logic https://en.wikipedia.org/wiki/Intuitionistic_logic 3. https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)#Cardinality https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_... 3. https://en.wikipedia.org/wiki/Constructive_set_theory https://en.wikipedia.org/wiki/Constructive_set_theory 4. https://en.wikipedia.org/wiki/Law_of_excluded_middle https://en.wikipedia.org/wiki/Law_of_excluded_middle <---- (all of the above reject this naughty child, and rightfully so)
- umutcnkus 4y agoOh, I think this is the ideal place to share one of my favorite math blogs. https://infinityplusonemath.wordpress.com/archive/ https://infinityplusonemath.wordpress.com/archive/ There is a section about 'infinity' and I think it is very fun to read.
- mabbo 4y ago> In 1940 the famous logician Kurt Gödel proved that, under the commonly accepted rules of set theory, it’s impossible to prove that an infinity exists between that of the natural numbers and that of the reals. That might seem like a big step toward proving that the continuum hypothesis is true, but two decades later the mathematician Paul Cohen proved that it’s impossible to prove that such an infinity doesn’t exist! It turns out the continuum hypothesis can’t be proved one way or the other. This, to me, is the exciting bit of the whole article. There are questions that cannot be answered definitively. There is no way to say that it's true or false. Not simply that we haven't figured it out yet, but that no one, no matter how clever, will ever be able to prove the answer one way or another. If we meet an extraterrestrial alien civilization that is a million years of math theory ahead of us, they will not have an answer.
- sorokod 4y agoI think that a question has been answered definitely. Continuum hypothesis is not provable within the theory along with the other axioms the the theory is derived from. Similar to parallel postulate really https://en.m.wikipedia.org/wiki/Parallel_postulate https://en.m.wikipedia.org/wiki/Parallel_postulate
- scapp 4y ago> along with the other axioms the the theory is derived from Axioms are trivially provable in any system. Unless you mean prove them without using them, in which case you're actually talking about a system where they aren't axioms.
- karmakurtisaani 4y agoIf I remember correctly, this means that CH is independent of the other axioms of set theory. So in theory you could come up with another set of meaningful axioms (I.e. ones that produce and capture interesting mathematics), and in which you could prove CH. So, an extraterrestrial civilization could in fact have an answer!
- BeetleB 4y agoIt's certainly cool, but don't make more of it than it is. It's merely pointing out that with a given set of axioms, there will be questions that cannot be answered. It's not a statement about the Real World Truth, but about the models we use.
- andrewla 4y agoThe Cantor conception of "infinity" is bad and should be retired. It's a pure naval-gazing exercise that has set back practical mathematics significantly. The notion of the "continuum" is fundamentally flawed. Intuitionalism and constructivism are a much better framework for reasoning about mathematics. The response to the idea that the natural numbers and the even numbers and the rational numbers and the computable numbers have the same "cardinality" should be "who cares". All it means is that Cantorian cardinality is a bad standard against which to judge the "size" of an infinite set. It ends in the same trap as the Axiom of Choice, where you can produce lots of obviously incorrect results and remark about how amazing they are; similarly you can make nonsensical statements about things being true "almost everywhere" that are clearly actually true nowhere.
- sorokod 4y agoDavid Hilbert was a fan. Cantor's paradise he called it. https://en.m.wikipedia.org/wiki/Cantor's_paradise https://en.m.wikipedia.org/wiki/Cantor's_paradise
- lupire 4y agoMathematics is fine. Uncountable infinity is really only a problem in physics.
- BeetleB 4y agoSay Hi to Doron Z for me!
- horseAMcharlie 4y agoI'm not really familiar with math beyond linear algebra and googling was not very helpful; could you please give an example of an obviously incorrect result enabled by the axiom of choice? No need for detail, just the name of an example is good enough for me.
- bawolff 4y agoAs far as axiom of choice goes, they were probably referring to https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox Whether or not it is "wrong" is in the eye of the beholder. Why should infinite things be intuitive? After all even "real" things in the universe are highly unintuitive - e.g. quantum mechanics sounds obviously "wrong" at first glance.
- unity1001 4y agoInfinity is infinitely big. Conversely, it also must be infinitely small, almost nonexistent. Because, in order to be infinite, it must contain everything, including all concepts that are contradictory. So it ends up in the eternal philosophizing of many esoteric schools and religions: Infinity is the ultimate balance, tranquility, that is totally inert and irrelevant to outside because it contains all the contradictions inside itself in a balancing, canceling-out fashion. But... Infinity must also be the total opposite of that in order to be infinite - it must be the total opposite of infinitely balanced too. Again canceling out any description and defying identification. So it all begins and ends in mystery. Then again, it also must not be mysterious in order to be truly infinite, so...
- drdeca 4y ago> Because, in order to be infinite, it must contain everything This is wrong (or at least, using a poor choice of definition of infinite). I don't get why people think this?
- unity1001 4y ago> I don't get why people think this? Because, otherwise at least one thing will be missing from infinity, and it wont be infinite in that direction. And that's not the mathematical sense of directions, physics vectors etc. All of them are always limited and they cannot describe infinity. Infinity must be infinite in every way. Its the opposite: A lot of people get stuck at a certain point because they try to describe infinity using mathematical concepts or thinking. Mathematics, which is a framework that is limited in specific ways in order for humans to be able to understand it and calculate through it.
- drdeca 4y agoOk, so you are just using an IMO mostly useless definition of the term. Fine. That mostly just means you aren’t talking about the same thing as everyone else. Though, what do you call something which isn’t finite then? Or perhaps you define “finite” as “not ‘infinite’” where ‘infinite’ is being used as you use it? In which case, that also makes “finite” mostly useless imo, and I question what you would use to describe something which not only has limits of some kind, but is specifically limited in terms of how much stuff it has, in addition to being limited in terms of what stuff, and specifically, where the “how much” is limited as being less than some natural number.
- thomastjeffery 4y agoI always disliked this question. It's explicitly an exercise in futility. The real answer to the question is to point out that the question is broken. It's pitting the prescriptive against the descriptive, then acting surprised they aren't the same thing. The question "how big" only works with the set of "quantifiable". It's just a type error. Yet it's unsatisfying to say infinity isn't quantifiable, because when we do, we aren't being descriptive. Infinity is by definition unquantifiable, which is a prescriptive statement. Prescriptive answers just aren't any fun. We aren't learning anything from them, because we knew before we asked. When we talk about "bigger and smaller" infinity, we are just using infinity as an abstraction in the very same way we use variables. It's just as straightforward as going from "x+1>x" to "+1>". We all know that "plus one is more". The first statement is using nouns, and the second is using functions. It's just a type difference, nothing more. The thing we are spending so much time blathering in awe about is just the relative difficulty in describing abstraction. Abstraction is amazing, impressive, useful, often surprising or elegant. It is not however, mythical. There's this thing we do where what we are talking about doesn't have any substance. It's called nonsense. That's it. There is no "deeper meaning" behind the explicit absence of meaning. It can be entertaining to talk in circles, but we know they aren't getting us anywhere new.
- bawolff 4y ago> Prescriptive answers just aren't any fun. We aren't learning anything from them, because we knew before we asked. All of mathematics is arguably a tautology. I think what you are saying applies to everything in math all the way back to 1+1
- thomastjeffery 4y agoEvery thing, yes, but the things aren't what make math interesting. It's the way those things relate to each other that is so interesting. The patterns. The connections. It's the same with this discussion about infinity. The real substance in most of this article isn't infinity: it's set theory. Infinity is just being used as the hook to grab your attention.
- maxjones1 4y ago
- photochemsyn 4y agoI came across this amusing question a while ago: "How many computable numbers could a computer compute if a computer could computer all computable numbers?" The correct answer (from HN comment section) appears to be 'countably many'. Interestingly computable numbers don't go to infinity: https://en.wikipedia.org/wiki/Computable_number#Not_computably_enumerable https://en.wikipedia.org/wiki/Computable_number#Not_computab... > "While the set of real numbers is uncountable, the set of computable numbers is classically countable and thus almost all real numbers are not computable." But they do get very very big, i.e. you can theoretically compute numbers so large that: ...even the number of digits in this digital representation of Graham's number would itself be a number so large that its digital representation cannot be represented in the observable universe. Nor even can the number of digits of that number—and so forth, for a number of times far exceeding the total number of Planck volumes in the observable universe. https://en.wikipedia.org/wiki/Graham%27s_number https://en.wikipedia.org/wiki/Graham%27s_number
- deleted 4y ago[deleted]
- andrewla 4y ago> Interestingly computable numbers don't go to infinity: Computable numbers are an unbounded set, so in that sense they do go to infinity, just like the natural numbers go to infinity. They just aren't uncountable. Others and myself have noted in other comments that the Cantor notion of cardinality is not all that interesting or useful compared to the concepts of intuitionalism and constructivism. In particular, as noted in Wikipedia, "almost all real numbers are not computable", yet it is impossible (by definition) to actually produce or approximate one of these numbers through any sort of computational process. This sort of obvious nonsense is why mathematics has very very slowly been edging back from "Cantor's Paradise".
- andrewgleave 4y ago“A Window on Infinity” chapter in the Beginning of Infinity is worth reading as an introduction.
- bawolff 4y ago> Remarkably enough, no matter how close any two distinct real numbers are to each other, there will always be infinitely many real numbers in between. By itself this doesn’t mean that the sets of real numbers and natural numbers have different sizes, but it does suggest that there is something fundamentally different about these two infinite sets that warrants further investigation. This seems a really odd example to start with, without coming back to, because the same is true of the rationals and they do have the same cardinality as the naturals.
- drdeca 4y agoYes, seems like a mistake to mention that without saying that the difference is that they have different order types (and that there is a sense of some order types being "much bigger" than others, even if they have the same cardinality)
- yarg 4y agoThe distinction between the size of {I} and {I}/{I} (and {I}^n) is not the size, but the orderability. (Sorry for the bullshit syntax.) You can map any rational onto an integer index; you cannot do it so that that r[i - 1] < r[i] < r[i + 1]. This has no impact on cardinality, but there's something here that doesn't sit quite right with me.
- lupire 4y agoWhat doesn't sit right? Your orderability idea is about "monotonic" mappings. Rational numbers are interesting even if they aren't ordered at all (but still can add and multiply, like other non-ordered objects like the integers mod N).
- yarg 4y agoI don't quite know. It feels as if there's some deep structural aspects to the rationals that get casually tossed to the side in order to shoehorn them into integer indices.
- LudwigNagasena 4y ago
- bawolff 4y agoSo i've definitely heard of cantor's proof before, but one thing that struck me just now which i didn't really connect the dots on before, is how similar the diagonilaztion proof is to the proof of the halting problem. Maybe its just coincidence, or maybe its not as similar as i think it is, but it does feel a bit surprising to me.
- oldgradstudent 4y agoYour observation is correct. This is pretty much the same argument. https://en.wikipedia.org/wiki/Diagonal_argument https://en.wikipedia.org/wiki/Diagonal_argument
- Rapzid 4y agoHow real is an imaginary number? Best to think of them as mathematical constructs and not get too hung up on them IMHO.
- marginalia_nu 4y agoWhat does it mean for a number to be real (in the lower case sense of the word)?
- lupire 4y agoImaginary numbers are more real than transcendental "real" numbers, and much more real than the unnameable reals, which are almost every real. It's easy to set up situation that points to any algebraic complex number.
- scapp 4y ago> the unnameable reals, which are almost every real Is there a definition you have in mind for "unnameable"? If you mean definable [0][1], then this is independent of ZFC. The "standard" argument for this is flawed, and indeed there are models where every real is definable. [2] [0] "x is definable if there exists a first order formula with one free variable P such that x is the unique real number with P(x) true" [1] https://en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC https://en.wikipedia.org/wiki/Definable_real_number#Definabi... [2] https://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numb/44129#44129 https://mathoverflow.net/questions/44102/is-the-analysis-as-...
- floxy 4y agoMaybe the GP is referring to something like Chapter 5 of Chaitin's Meta Math!. https://arxiv.org/pdf/math/0404335.pdf https://arxiv.org/pdf/math/0404335.pdf
- bawolff 4y agoWhat does imaginary numbers have to do with the topic at hand?
- shrubble 4y agoI was told when younger that infinity is "at least twice as big as the biggest number you can think of". Somehow that definition still works for me :)
- lupire 4y agoDoesn't work with ordinal and cardinal numbers.
- maggs 4y agoDecent introduction to the concept of different infinities to the layperson. However, this line: > ... and some recent work has changed the way people think about the issue. had me pretty excited I was about to read some brand new development. Alas, there is absolutely nothing new in this piece.
- lisper 4y agoCan't raise this topic without mentioning the Large Numbers Page: http://www.mrob.com/pub/math/largenum.html http://www.mrob.com/pub/math/largenum.html With this link being particularly relevant: http://www.mrob.com/pub/math/largenum-10.html http://www.mrob.com/pub/math/largenum-10.html
- ravenstine 4y agoDoes the universe even support infinity at absolute precision? If the data of an interim number used to calculate infinity, and that number could be represented by the using the smallest physical particle possible, said number would be limited to the size of the universe at a given time and the number of these hypothetical particles that could exist at said given time. In other words, the only things that can exist within the universe are things the universe is complex enough to support. There's nothing about the physical universe that suggests it can support anything remotely infinite. Infinity as a concept is handy for mathematics, but that's because we're not even close to being capable of calculating infinity to absolute precision (one reason why "Infinity" in programming languages is really just magic). If infinity was an actual number (having a size), it would need to be extrauniversal unless it was possible for us to calculate it to the extent required, which has yet to be seen. The only candidate we have for proof that infinity exists is that the amount of time academics can spend pondering on infinity seems infinite.
- simonh 4y agoThere’s also nothing about the physical universe to suggest that it is finite, or that it cannot contain infinities. It’s simply an open question. > If infinity was an actual number (having a size), it would need to be extrauniversal What does the universe or its physical extent have to do with the nature of numbers? A number is simply a concept, or idea, not necessarily a physical thing. We imagine things that aren’t physical all the time.
- ravenstine 4y ago> There’s also nothing about the physical universe to suggest that it is finite, or that it cannot contain infinities. It’s simply an open question. The only evidence we have is that it's finite. Just because the universe expands does not mean that at any given time it is not finite. > What does the universe or its physical extent have to do with the nature of numbers? The only way to have scientific confidence in an idea is to test it. We can't calculate infinity because, infinity being something that isn't finite, every unit of information in our universe would have to be used to describe infinity. This doesn't work because there is never infinite information space in our universe at any given slice of time. Think of it this way; you can't take a modern video game and get it to perform exactly the same on a home computer from 1996 because it simply lacks the computing capacity. The only way it can work is to reduce certain aspects of the software to make it work at a much lesser capacity. There are no examples of any system that can describe another system more complex than itself with total accuracy. Thus, it makes no sense that a universe in which we are currently only able to describe through finite numbers would be able to support calculating what infinity as well as support the rest of its contents, if it can even do that at all. This is why it's not at all accurate that a set of numbers can be "infinite." It only seems infinite because, for all intents and purposes, we don't have the capability to keep dividing a range of numbers forever. Even if we tried, the inevitable dissipation of heat energy would prevent us from doing so, that is if we don't simply run out of finite resources before then. If there is something that is indeed numerically infinite, we too would have to be infinite in order to make sense of it. We can't actually do that. That would be a contradiction. To illustrate this, go write some code that calculates every single number that exists in the "infinity" between two numbers. You won't be able to. Your software will fail because your computer doesn't support it. That is unless it has infinite bits. A set where there's "infinite" numbers would more accurately be described as being indeterminate. The seeming "infiniteness" of one of these sets breaks down when you realize there's no way to even demonstrate that part of a set is infinite. From a conceptual standpoint, it's similar to how it might seem that a ball will fall straight down when you drop it, and it's generally useful to think of gravity in such a way, but that doesn't mean that it's actually so, just as believing that a set can be "infinite" might be useful, but the use of the term "infinite" for something finite like a set is incorrect. So no, to others who think I'm being off topic. This is entirely on topic. A set being "infinite" gives you the wrong idea. At best, it defines a vector too large for humans or even human computers to find an end to. It's entirely virtual until proven otherwise.
- sohkamyung 4y agoI learned about Cantor and the concept of comparing infinities through George Gamow's excellent book, "One, Two, Three ... Infinity" many years ago. While most of the book's content on cosmology and atomic physics may be out of date (or now wrong), it's chapters on mathematics and Einstein's relativity are still a fun read.
- pentagrama 4y agoAs a kid I have some religious friends and I was somewhat envious of them for not be able to feel something magical trough religion, but when I started questioning the universe and infinity stuff, oh boy, it was MY THING, I was so happy for not understand it.