How Big Is Infinity?
quantamagazine.org
quantamagazine.org
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.
>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?
>this doesn’t mean that the sets of real numbers and natural numbers have different sizes
I think the writer might have been meaning to imply that there's a spectrum of properties from countably infinite to dense to uncountable, where each property is stronger than the previous one. But this is actually not the case. The Cantor set is uncountable, but nowhere dense. So density is not just something that's in between countability and uncountability---it's an orthogonal thing.
The author is writing the article for someone who could very well be learning this for the first time and may think to themselves "Hmm... this is something unusual and counterintuitive.". Since you are not that reader the author's writing could come across as annoying, but it's not wrong or misleading, it's a way for the author to hint to the reader that they are empathizing with them and will address this unintuitive notion further (which the author does).
> 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.
> 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.
So no, as rightfully pointed by other commenters, the article implies that density somewhat indicates that two cardinalities which is straight wrong as the rational are dense in the natural yet of the same cardinality. They are even dense in R which entirely defeat the point of the analogy. Density and cardinality are not related at all.
There is an additional problem with the argument as written, as there are in fact "relations" on the "decimal expansions" of real numbers. for instance .099... = .1000....
Notice that none of the places in this expansion agree with each other so this might indeed be the element you construct from the following list:
`.1000... .0100... .0010... ...`
You might say, ok well pick a number different from a_i and also different from 0 or 9, which does indeed get you a number not in your list, but it really begs the question of "why doesn't this work in binary?", and is also far less intuitive than factoring the problem into two steps of |N|~=|P(N)|=|R|. This approach is more general anyway.
wait, so how does diagonalization account for this? The argument, as i recall, was that every real number in the "fake list" differs from the generated real number by at least 1 position in the decimal expansion.
But doesn't this argument depend on the assumption that every real number has a unique decimal expansion?
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.
(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.
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).
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.
The closest mapping of the reals to a countable set is the set of rational numbers.
Or am I an idiot?
E.g ratios of integers and square roots.
That's some sneaky code injection.
Basically you can add a countable infinity of new countable infinite dimensions to any countably infinite set and get something that's still countably infinite.
That shit still feels wrong to me.
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.
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
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.
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.
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.
"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_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.
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".
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.
Continuum hypothesis is not provable within the theory along with the other axioms the the theory is derived from.
Similar to parallel postulate really
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.
So, an extraterrestrial civilization could in fact have an answer!
As a simple example, if I make the following set of axioms -- "each natural number is either big or small", "10 is a small number", "20 is a big number", "it cannot be simultaneously true that X is big and X+1 is small", -- then I can prove that 5 is small, or that 30 is big, but I can neither prove nor disprove that 15 is big. Because this set of axioms allow multiple possibilities for what "big" and "small" mean.
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.
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.
But do you have something better? It’s not as if mathematicians immediately accepted this as the way forward. There was a long struggle accepting that lots of statements that are true about finite sets do not extend to infinite ones (examples: “adding an item to a set makes it larger”, “when summing a set of numbers, the result doesn’t depend on the order you do it”)
I think that you either have to accept this as the best way to treat infinite sets, or have to give up the notion of infinite sets, and that has its problems, too. For example, it would mean there’s a largest integer.
Similarly the natural numbers are a process that can produce more natural numbers, and can do so for however large a limit you choose.
The problem of rearranging the elements of a set to get a different sum is handled here because you are looking at the limit of the sum, not the sum of the abstract set. Just as in classical math you have to make sure that to compute { sum_i=0^inf (-1)^i / i } that you are actually computing { lim N->inf sum_i=0^N (-1)^i / i }, with intuitionist math you can't help but do the latter, because the former statement makes no sense except as the latter.
The "adding an item to a set makes it larger" is only problematic if you're trying to define "larger" for non-terminating sets, and in addition trying to treat them as "sets" divorced from their underlying structure. The rationals and the integers may look the same to a set theorist, but they are clearly very different objects and it's not at all clear what additional power you get from introducing morphisms that don't preserve any underlying structure.
Just for the sake of non-mathematicians, allow me to note here that despite intuitionism/constructivism being around for many decades, 99%+ of mathematics research done today is not, in fact, performed in these frameworks. These are just a curiosity that few working mathematicians actually care about. These approaches do have some certain philosophical benefits, but they have some extreme practical disadvantages that result in overwhelming majority of mathematicians rejecting the notion that these are “better frameworks”.
Your comment to me reads like saying that autogyro is a better framework for powered aviation than fixed wing or helicopters: regardless of your actual arguments in its favor (which may in fact be good), the fact that 99%+ of the industry disagrees is rather telling.
This means that you don't get any safety by rejecting the law of excluded middle, nor by rejecting the axiom of choice. For this reason, I think intuitionistic logic is trading away a lot of power for basically no gain.
[0] https://en.wikipedia.org/wiki/G%C3%B6del%E2%80%93Gentzen_neg...
[1] https://en.wikipedia.org/wiki/Axiom_of_choice#Independence
> It's a pure naval-gazing exercise
> The response to the idea [...] should be "who cares".
If anything sets back mathematics, it's when people have this kind of attitude towards the parts of math they find unintuitive.
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.
sqrt(2)
End of the bullshit.
To quote David Hilbert, "The infinite is nowhere to be found in reality. It neither exists in nature nor provides a legitimate basis for rational thought... The role that remain for the infinite to play is solely that of an idea."
That way the term works equally well at different scales. There are infinitely many real numbers between 1 and 2. There are infinitely many natural numbers.
Math? Which physical laws/equations need infinity? Or do they just need "as x approaches infinity"?
But regardless, my only point was that in the real world everything is countable. The number of atoms in the sun, the distance to anywhere, the age of the universe, etc. Everything real is countable.
Pretty much all of them are phrased in the language of differential equations.
Perhaps some can be salvaged through finitist approaches, but that seems far from certain or trivial.
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.
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
2. https://en.wikipedia.org/wiki/Intuitionistic_logic
3. https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_...
3. https://en.wikipedia.org/wiki/Constructive_set_theory
4. https://en.wikipedia.org/wiki/Law_of_excluded_middle <---- (all of the above reject this naughty child, and rightfully so)
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."
The more unique something is, the more it stands apart from other objects in its class.
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.
https://dictionary.cambridge.org/dictionary/english/unique
I think it depends on the context, but it does not seem an incorrect usage.
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.
Like most arguments about words, it mostly comes down to context.
Unique has a clear meaning. People frequently use the word unique in a sloppy manner to mean something different than unique, just as people frequently use the word infinite in a sloppy manner.
In all these cases, they are using the word infinite to simply mean very very large.
Which is to say, you can't have 2 apples, you can only have one very specific, singularly unique apple and another very specific, singularly unique apple. If you get pedantic enough, nobody could ever have anything because the word invented to describe the first thing wouldn't technically describe the second thing perfectly (as they're not the same) so as to render speech useless.
Either accept that "unique" is a word for the poets and not the mathematicians, or accept that it's entirely useless as a descriptor.
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.
infinity + 1
infinity + 2
Infinity × 2
Etc.
Are all nicely defined.
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.
http://www.mrob.com/pub/math/largenum.html
With this link being particularly relevant:
Maybe its just coincidence, or maybe its not as similar as i think it is, but it does feel a bit surprising to me.
https://infinityplusonemath.wordpress.com/archive/
There is a section about 'infinity' and I think it is very fun to read.
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#Definabi... [2] https://mathoverflow.net/questions/44102/is-the-analysis-as-...
(The absolute disrespect for Adams here.)
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.
> 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.
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.
Mathematicians perform calculations that include infinities routinely nowadays. They have sophisticated methods for it, and have discovered powerful tools and useful results from doing so.
Oh really? How about Professor Hawking?
https://scitechdaily.com/constraining-the-multiverse-stephen...
> “We predict that our universe, on the largest scales, is reasonably smooth and globally finite. So it is not a fractal structure,” said Hawking.
> “We are not down to a single, unique universe, but our findings imply a significant reduction of the multiverse, to a much smaller range of possible universes,” said Hawking.
> I have never been a fan of the multiverse. If the scale of different universes in the multiverse is large or infinite the theory can’t be tested. [– Hawking]
Not that it particularly matters.
Cosmologists believe all sorts of things. Lots of cosmologists treat String Theory as more than a hypothesis despite how it hasn't yet panned out after decades and decades. Same goes for ideas like Dark Matter and Dark Energy, which are actually placeholders for some things we can't actually explain. That doesn't mean either of those things live up to their names or exist in any capacity. Hell, a substantial number of cosmologists profess that there's "got to be" alien life out there, despite no credible evidence what so ever. It's fine to hypothesize about all these things, but that doesn't make them even remotely fact, no matter what a cosmologist thinks. Science and mathematics aren't a majority vote.
> some believe it’s likely it’s infinite
They can believe that all they want, but that doesn't mean they're correct or that they have evidence. It's highly dubious whether it's even possible for us to verify the presence of infinitude.
> none I know of believe it’s a settled question.
That's because there are no settled questions. Any scientist or mathematician who professes that a question is settled immediately loses credibility by doing so.
The difference between infinitude and finitude is one can be demonstrated and the other has yet to be demonstrated. Guess which one hasn't been demonstrated. Just because mathematics can include a construct that allows describing an indeterminate range of numbers doesn't mean calling it "infinite" is accurate. An actual value that can even remotely be referred to as infinity hasn't been calculated and it's possible that it never will if the known universe doesn't even support calculating it. To do math with "infinity" is to use an indeterminate vector in a way that proves useful. It's no different than when someone uses a phrase like "this is taking forever."
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'?
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.
You explained nothing. Where did the multiverse come from then.. did it sprout out of nothing?
Both theories equally non falsifiable and make use of infinity.
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...
No, this is a common misconception but it is false. Consider the set of all even integers. It is infinite, but no matter how long you search you will never find 3. There is no sense in which "infinity" entails "all inclusive".
You can define "the set that contains everything", but it is also not terribly interesting that it contains "contradictions". Clearly the set that contains everything contains the propositions "2 is even" and "2 is not even", but... so what? All that implies is that contradictory claims exist, which is not even slightly profound. Prove contradictory claims are somehow both true and now you're cooking with philosophical gas, but the mere fact they can be defined is uninteresting. That is literally nothing more than the observation that both false and true statements exist.
Nope, don't: Mathematical approaches to infinity do not work. Because all mathematical approaches are limited. And the infinite cannot be expressed by using the finite.
Infinity must contain everything that exists and their antithesis in it. If even one thing is missing, the infinity won't be infinite.
> but it is also not terribly interesting that it contains "contradictions"
It's terribly interesting. Because:
> the propositions "2 is even
The infinity must contain the antithesis of a proposition. The antithesis of 2. The antithesis of even. The antithesis of everything involved in making that proposition. If it doesn't, then its not infinite because it is missing something.
I see this sort of approach a lot. It may give you a lot of warm fuzzy mystical feelings, but it's sterile. It can't go anywhere beyond that. You've locked yourself in a tiny little box with your fuzzy feelings and mixed up definitions, and you think you've got a hold of something amazing, but you're just... locked in a tiny box, with nowhere to go, with nothing allowed to come in and affect you and no ability to put anything out into the world.
What's more, if you came out into the greater world with the rest of us, you'd find that your content-free mystical feelings are really just a counterfeit of the wonder that is available in the real world. More complicated, more substantial, more interesting, and as far as I can see, an infinite (heh) vista of things to discover and communicate, rather than a tiny little concept posing as a large one.
Throw away your attachment to your idiosyncratic definition and join the rest of us on the real voyage of discovery. It may hurt a bit at first, and it certainly requires more work, but the pay off is worth it.
This is wrong (or at least, using a poor choice of definition of infinite). 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.
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.
Its a philosophical approach. Not a mathematical. It does not have to be one.
> 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.
That's a given set of specific limitations. Quite a mathematical concept. Like 'the set of natural numbers up to a given, large number'. It only makes sense within the given limitations and definitions, a precise mathematical use case. Otherwise if you dive into its details, like how an infinite amount of fractional numbers inhabit even that given large set of natural numbers, things go out of hand there too.
If I have a box, I might ask, “is there some number n such that there are fewer than n protons in this box?”. Well, really, the question would be phrased “are there finitely many protons in the box”, but I’m avoiding the word “finite” because you use the word “infinite” differently.
The question of “are there ‘un-split-able ones’ ?” goes way back, and is a question about the world, not about math considered abstractly. Aiui, it was considered philosophically. A question as to whether there is a limit to how much things can be split into smaller parts.
The question of “Does this thing have a limit of ‘how much’/‘how many’ ?“ is not a question which concerns only mathematicians, but which concerns anyone who seeks to understand the world.
So what you really want is the infinity of infinite information complexity, but we haven't discovered such a thing yet.
They mathematically are. They are not truly infinite. Because, they are countable and define-able as a set. They are infinite in only one direction.
> what you really want is the infinity of infinite information complexity
Still falls short. The infinity must contain the antithesis of information. And complexity.
Which is fine. It is, after all, what the mathematicians did.
But then taking your definition and insisting other definitions are wrong - that's problematic.