[HN Gopher] How Big Is Infinity?
___________________________________________________________________
How Big Is Infinity?
Author : theafh
Score : 44 points
Date : 2022-09-27 15:49 UTC (7 hours ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| bawolff wrote:
| So 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 wrote:
| Your observation is correct. This is pretty much the same
| argument.
|
| https://en.wikipedia.org/wiki/Diagonal_argument
| willis936 wrote:
| No mention of the fact that infinity is not a number. Numeric
| operators only work on numbers.
| gspr wrote:
| Why would they need to mention this?
| zardo wrote:
| _Numeric_ operators only work on numbers. Operators can work on
| all sorts of mathematical objects.
| floxy wrote:
| https://en.wikipedia.org/wiki/Hyperreal_number
|
| https://en.wikipedia.org/wiki/Surreal_number
| WastingMyTime89 wrote:
| It'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.
| drewrv wrote:
| I 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.
| lupire wrote:
| There are numbers that aren't on the real number line. i, for
| one. Omega, for another.
| willis936 wrote:
| I 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 wrote:
| Um, duh, you tip an 8 over. If it's not a number, why is it
| made out of a number? /s
| aarondf wrote:
| I 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
| birriel wrote:
| I second this recommendation. Excellent documentary.
| passion__desire wrote:
| I liked this "music video" [1] inspired by the concepts
| involving proof of infinity
|
| [1] https://www.youtube.com/watch?v=tNYfqklRehM
| ravenstine wrote:
| Does 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.
| bawolff wrote:
| This is misunderstanding what the article is about. It does not
| make sense to talk about calculating a cardinal to infinite
| precision.
| simonh wrote:
| 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.
|
| > 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 wrote:
| > 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.
| mabbo wrote:
| > In 1940 the famous logician Kurt Godel 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 wrote:
| I 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
| scapp wrote:
| > 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 wrote:
| If 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 wrote:
| It'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.
| andrewgleave wrote:
| "A Window on Infinity" chapter in the Beginning of Infinity is
| worth reading as an introduction.
| Der_Einzige wrote:
| For 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
|
| 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)
| [deleted]
| andrewla wrote:
| The 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.
| librexpr wrote:
| Cantor's work on infinity and the diagonal argument was hugely
| important in mathematics, paving the way for important results
| like Godel's incompleteness theorems, the halting problem, the
| creation of modern set theory which allowed unifying
| effectively all of known mathematics into one theory, etc.
|
| > 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.
| Someone wrote:
| > All it means is that Cantorian cardinality is a bad standard
| against which to judge the "size" of an infinite set.
|
| 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.
| xyzzyz wrote:
| > [Intuitionism] and constructivism are a much better framework
| for reasoning about mathematics.
|
| 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.
| librexpr wrote:
| In addition to this, I'd like to add that intuitionistic
| logic is consistent if and only if classical logic is. This
| follows from the Godel-Gentzen negative translation[0], which
| implies that for any contradiction in classical logic, you
| can get the same contradiction in intuitionistic logic more
| or less by adding "not not" before both sides of the
| contradiction. The same applies to the axiom of choice: set
| theory with choice is consistent if and only if set theory
| without choice is consistent[1].
|
| 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
| BeetleB wrote:
| Say Hi to Doron Z for me!
| lupire wrote:
| Mathematics is fine. Uncountable infinity is really only a
| problem in physics.
| horseAMcharlie wrote:
| I'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 wrote:
| As far as axiom of choice goes, they were probably referring
| to
| 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.
| sorokod wrote:
| David Hilbert was a fan. Cantor's paradise he called it.
|
| https://en.m.wikipedia.org/wiki/Cantor's_paradise
| maxjones1 wrote:
| photochemsyn wrote:
| I 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_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
| andrewla wrote:
| > 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".
| [deleted]
| bawolff wrote:
| > 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 wrote:
| Yes, 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 wrote:
| The 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.
| bawolff wrote:
| Note, you also can't do this type of monotonic mapping with
| the integers mapped to the natural numbers. So its not like
| the rationals are a dividing point here.
| yarg wrote:
| Isn't it really the same thing?
|
| The closest mapping of the reals to a countable set is the
| set of rational numbers.
|
| Or am I an idiot?
| zardo wrote:
| There's room for an infinity of closer mappings.
|
| E.g ratios of integers and square roots.
| lupire wrote:
| What 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).
| LudwigNagasena wrote:
| I guess the point is that you can't map them in a way that
| preserves "structure" in some sense.
| yarg wrote:
| I 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.
| helen___keller wrote:
| I'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, p, 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 wrote:
| I 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 wrote:
| The 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 wrote:
| No 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.
| Kranar wrote:
| That's correct and it's precisely what the author states,
| and I quote:
|
| >this doesn't mean that the sets of real numbers and
| natural numbers have different sizes
| housecarpenter wrote:
| They state that only as a disclaimer, after a bunch of
| other statements that hint at the opposite. Just adding a
| disclaimer to misleading writing merely makes it
| confusing and unclear for those who are paying attention,
| and still misleading for those who don't notice the
| disclaimer.
|
| 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.
| Kranar wrote:
| I would say your criticism is akin to the curse of
| knowledge [1], wherein you know all of the relevant
| background information and so you have difficulty putting
| yourself in the position of someone who does not.
|
| 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).
|
| [1] https://en.wikipedia.org/wiki/Curse_of_knowledge
| ouid wrote:
| Diagonalization might be a tricky topic for beginners, but that
| is also _who we teach it to_. It is usually in your
| introduction to mathematical proofs class, and even there it is
| taught in the first couple of weeks.
|
| 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.
| unity1001 wrote:
| Infinity 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...
| jerf wrote:
| "Because, in order to be infinite, it must contain everything,
| including all concepts that are contradictory."
|
| 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.
| unity1001 wrote:
| > Consider the set of all even integers
|
| 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.
| drdeca wrote:
| > 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 wrote:
| > 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.
| otabdeveloper4 wrote:
| Countable numbers (1, 2, 3, ...) are infinite, in the sense
| that they never end. However, the information complexity of a
| program to generate them is finite and only a few bytes long.
| Information complexity of generating digits of p is
| considerably more complex but still very finite.
|
| So what you really want is the infinity of infinite information
| complexity, but we haven't discovered such a thing yet.
| unity1001 wrote:
| > Countable numbers (1, 2, 3, ...) are infinite
|
| 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.
| BeetleB wrote:
| Looking at your comments, it seems you are merely defining
| infinity to what you want it to be.
|
| 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.
| Rapzid wrote:
| How real is an imaginary number? Best to think of them as
| mathematical constructs and not get too hung up on them IMHO.
| marginalia_nu wrote:
| What does it mean for a number to be real (in the lower case
| sense of the word)?
| lupire wrote:
| Imaginary 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 wrote:
| > 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/Defi
| nable_real_number#Definabi... [2]
| https://mathoverflow.net/questions/44102/is-the-analysis-
| as-...
| floxy wrote:
| Maybe the GP is referring to something like Chapter 5 of
| Chaitin's _Meta Math!_.
|
| https://arxiv.org/pdf/math/0404335.pdf
| yodon wrote:
| Infinite 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."
| philipswood wrote:
| With ordinal transfinite numbers
|
| infinity + 1
|
| infinity + 2
|
| Infinity x 2
|
| Etc.
|
| Are all nicely defined.
| drdeca wrote:
| yes, though, it is generally prefered to call it "omega"
| rather than "infinity".
| paxys wrote:
| Why 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 wrote:
| Unique 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.
| function_seven wrote:
| It'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 wrote:
| Unique 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 wrote:
| Unique: being the only existing one of its type or, more
| generally, unusual, or special in some way
|
| https://dictionary.cambridge.org/dictionary/english/unique
|
| I think it depends on the context, but it does not seem an
| incorrect usage.
| function_seven wrote:
| I 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 wrote:
| All 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.
| moate wrote:
| By their nature, all things are unique, depending on their
| framing. We make concessions because we don't all have
| infinite time.
|
| 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.
| Kranar wrote:
| Most 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.
| yodon wrote:
| There are also dictionaries that say the definitions of
| infinite include "great or very great." I don't think
| most people at HN would accept those sloppy definitions
| of infinite any more than we should accept those sloppy
| definitions of unique.
|
| 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.
| Kranar wrote:
| Yes people, including those on HN, use the word infinite
| to mean very very large as opposed to the mathematical
| definition of infinite as a set that contains a proper
| subset with equal cardinality. Doing a Google search for
| uses of infinite on HN will reveal just as much, for
| example people complaining about websites with "infinite
| scrolling", or that the Fed has printed an infinite
| amount of money recently, or that software can be copied
| infinitely many times, or someone claiming that their
| friend has an infinite amount of memory.
|
| In all these cases, they are using the word infinite to
| simply mean very very large.
| mejutoco wrote:
| Negative 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.
| coldacid wrote:
| Still, there are bigger and smaller infinities.
| lupire wrote:
| Omega + 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.
| thomastjeffery wrote:
| I 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 wrote:
| > 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 wrote:
| Every _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.
| umutcnkus wrote:
| Oh, I think this is the ideal place to share one of my favorite
| math blogs.
|
| https://infinityplusonemath.wordpress.com/archive/
|
| There is a section about 'infinity' and I think it is very fun to
| read.
| [deleted]
| maggs wrote:
| Decent 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 wrote:
| Can't raise this topic without mentioning the Large Numbers Page:
|
| http://www.mrob.com/pub/math/largenum.html
|
| With this link being particularly relevant:
|
| http://www.mrob.com/pub/math/largenum-10.html
| shrubble wrote:
| I 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 wrote:
| Doesn't work with ordinal and cardinal numbers.
| WallyFunk wrote:
| > 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 wrote:
| > 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.
| WallyFunk wrote:
| Yes but Infinity explains that. It just goes on and
| on...there is no beginning.
| lupire wrote:
| A name is not an explanation.
| eatsyourtacos wrote:
| >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 wrote:
| Yes but Infinity explains that. It just goes on and
| on...there is no beginning.
| comboy wrote:
| If that's what you need multiverse for, then it's perfectly
| compatible with our current understanding of the big bang
| to claim that it is without any beginning and that it was
| expanding and expanding from smaller and smaller thing
| forever.
|
| Both theories equally non falsifiable and make use of
| infinity.
| coldacid wrote:
| Certainly much bigger than a walk to the chemist's.
| moate wrote:
| That's just peanuts to [infinity].
|
| (The absolute disrespect for Adams here.)
| swamp40 wrote:
| I 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 wrote:
| That's quite uncontroversial. But this is about math.
| idiotsecant wrote:
| Well 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 wrote:
| To 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.
| SoftTalker wrote:
| For me, I stopped thinking of "infinity" meaning "bigger number
| than you can imagine" to just simply meaning "unlimited" or
| "never ending."
|
| 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.
| Der_Einzige wrote:
| Don'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.
| anthk wrote:
| Ahem. Let me show you a simple example. Basic geometry. The
| diagonal of a right triangle. sqrt(2)
|
| Show me the end of a diagonal making the angle infinitely
| sharp no matter how much you "zoom" it.
|
| End of the bullshit.
| peppertree wrote:
| Do you believe in 0?
| gspr wrote:
| How do you reconcile the fact that the best tool we have, by
| far, to describe the "real, physical world" depends on certain
| notions of infinity?
| sleton38234234 wrote:
| 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 wrote:
| This? https://www.youtube.com/watch?v=uD4izuDMUQA
| daveslash wrote:
| " _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
|
| [1] https://en.wikipedia.org/wiki/God_Created_the_Integers
| bmacho wrote:
| 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.
| yamrzou wrote:
| Related: Is Infinity Real? --
| https://www.quantamagazine.org/the-infinity-puzzle-solution-...
| mym1990 wrote:
| 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!)
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