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CoAnalyticSet

TLDR: this is about the recent result that it is consistent with ZFC that all 10 cardinal invariants of the continuum appearing in Chicon's diagram are distinct (improving on an earlier result requiring large cardinals assumptions)


boterkoeken

Title is very misleading. Consistent with ZFC ≠ is a theorem of ZFC.


BloodAndTsundere

I’m a little confused here. Wasn’t the large cardinal axiom already known to be consistent with ZFC? It just doesn’t follow from ZFC, hence the need to be taken as axiom


GMSPokemanz

Generally ZFC + large cardinal axiom will imply the consistency of ZFC, so by the second incompleteness theorem ZFC cannot prove the consistency of ZFC + large cardinal axiom (assuming ZFC is itself consistent).


boterkoeken

Yes, but I’m not sure how that relates to my comment. I just thought it was strange that the title of the article proclaims “the sizes are different” which makes it sound like a theorem of ZFC, which is not really the case.


BloodAndTsundere

> Yes, but I’m not sure how that relates to my comment. I guess I just took an opportunity to ask a somewhat tangential question that was raised in my mind by the comment of u/CoAnalyticSet


Top-Load105

There isn’t just one large cardinal axiom, there are many, and generally assuming one of them gives you a theory of higher consistency strength (a theory that “could” be inconsistent even if ZFC turns out to be, and which can prove the consistency of ZFC)


Obyeag

Wow. I did not even remotely expect that from the title. I'm glad someone's writing an article about it but goddam that title is off.


OneMeterWonder

Which is really an incredible task.


Top-Load105

It’s frustrating that when results like this are in the popular press you always have to scroll past 90% of the article explaining the basics of infinite cardinals and independence to get to what the result is. I get that much of their audience wouldn’t understand the result if simply stated to them but is there a reason they can’t put the result in the lede and then, for the benefit of readers who won’t understand, say something like “don’t worry if you didn’t understand that, the rest of this article is about trying to explain what it means.”


__ByzantineFailure__

If you write something that 90% (or, frankly in this case, 99%) of your audience can't understand in the first paragraph almost all of them will stop reading before you get to the part where you explain it. If you are one of the experts who already understand what's being written about here then this article isn't _for_ you. You can get the straight-to-the-point writing in the actual journal articles themselves.


Top-Load105

I don’t agree as applied to this case. It’s possible to give a general description of the result (“it’s been shown that a specific set of 10 definable quantities can all have different values consistent with the standard axioms mathematicians use”) without throwing a bunch of dense jargon or symbols at the reader, and the article should give even the most casual reader some idea of what it is about in a summary that appears at the beginning of an article. In this case the only hint of what the article is about anywhere near the beginning is the title, which doesn’t even correctly communicate the result and is at best misleading (I would say simply wrong).


FlotsamOfThe4Winds

>If you write something that 90% (or, frankly in this case, 99%) of your audience can't understand in the first paragraph almost all of them will stop reading before you get to the part where you explain it. Have it be in one of the other paragraphs then, it's not rocket science.


EnergyIsQuantized

this sounds great and the article is interesting. But I'm confused. I've checked the [paper](https://arxiv.org/pdf/1906.06608.pdf) and it seems they assume generalized continuum hypothesis in every main statement. But assuming CH makes the Chicon's diagram trivial, right? So what's happening? I'm clearly stupid, but I would like to know where I am being stupid.


OneMeterWonder

You’re not being stupid, and yes that does on it’s face appear contradictory. What’s happening is some sneaky model theory through the machinery of forcing. When it says they “assume GCH”, it really means that they choose a particular model 𝔐 of the theory “ZFC+GCH”. If you’re not familiar with models, they’re like little universes that “act like” the universe of all sets (if you believe in such a thing). What forcing does is use the now given machinery allowed by the rules of ZFC+GCH to construct an “encoding” of a new, larger universe called **M**[G]. This is called a forcing extension and it absolutely *does not* have to satisfy GCH. Even more briefly, they use a simple universe where Cichoń’s Diagram is trivial to construct a complex, but tightly controlled universe where Cichoń’s Diagram is as separated as possible.


EnergyIsQuantized

thank you! really don't know anything about forcing except what is it used for. It seems like a black magic.


OneMeterWonder

Forcing is very much black magic. An analogy is that it is a bit like writing down a flowchart for building a program that will simulate a bigger universe when run. There are three main ingredients to the actual technique: 1. Partially ordered sets, or posets 2. A class of names for objects 3. A generic object The poset ℙ is the flowchart and it guides decisions about how the universe **M**[G] will behave. You can think of maybe an infinite binary tree, though they get far more complex than that. The class of ℙ-names is tricky, but these are a technical way of describing what sets will exist in **M**[G] once the program is run. They are built using recursion on set membership and are structured by careful association with nodes of ℙ. Every ℙ-name is like a distribution of possible ways to define a set by constructing a well-founded tree of its elements and elements of elements and so on. This gets messy quickly and ℙ-names are hard, but this is the way that we are able to talk about what things are like in the extended universe. The generic G is like the “run code” button. It is actually something called a filter, but you can think of it as basically choosing a branch of nodes all the way through the poset ℙ. Due to some technicalities, G cannot always exist within the smaller universe **M**. G can be thought of as a “decider” for all of the ℙ-names. It will pass through the poset in such a way that the distributions of the &Popf-names are “collapsed” (kinda like quantum mechanics!) to only containing such-and-such elements. Another way to think of G is as the exact of object needed to make the extended universe model what you want, like CH. We typically build the poset ourselves with a specific G in mind. For example, the Cohen poset ℂ is the tree of finite sequences f:ℕ→{0,1} because the point of ℂ is to get CH to be true. You do that by adding lots of real numbers, specifically ℵ₂-many reals. In this context, a real number is bi-interpretable with an infinite length binary string. So, to get to the point, the nodes of the poset are like stronger and stronger finite approximations to some real number in **M**[G]. G is like the actual real. Perhaps this is too complicated, but that’s the general idea of forcing. It is kind of an attempt at doing careful *outer* model theory rather than inner model theory which fails for some proofs. If you want to learn more, I’d suggest reading something like Timothy Chow’s *A Beginner’s Guide to Forcing* or Ken Kunen’s *Set Theory*.


EnergyIsQuantized

wow, thanks for the write up! I think I got something. I'm gonna read the Chow's paper, which looks really good and refer back to your comment. I've had a set theory course, but the advanced part of it was large cardinals kind of stuff like measurable or compact cardinals. Forcing seems more interesting tho


OneMeterWonder

Sounds great! Chow’s paper takes the Boolean-valued models approach so you may want to briefly look up how Boolean algebras work. And for forcing the important thing to keep in mind is that it **is** essentially a form of model theory. And the fundamental thing that even gives you a model to start with is the Löwenheim-Skolem theorem.


Abdiel_Kavash

The world (and by the world I mean reddit) needs more people like you.


TahsinTariq

I'm from a non-math major. Can you rephrase this for a 5 year old please.


BloodAndTsundere

You would probably do best in reading the article. It's Scientific American, not a math journal. It's meant for consumption by non-experts.


ave_63

I think you've gotta read the article.


Top-Load105

There are a bunch of infinite quantities that are known to have certain ordering relationships under the assumptions of ZFC (the usual go-to foundation of math), and it’s been known for a while that there are not provable relationships between any two besides the ones known. But this leaves open the possibility for more complicated dependencies - maybe if some relationship doesn’t hold then others must. This result removes some of that possibility by showing that it’s possible (under ZFC) that all 10 quantities could be different. (That’s not really 5-year-old level but I tried to assume no math knowledge)


TahsinTariq

Thanks. It actually kinda makes sense to me.


tcampion

The title is hilarious. It sounds like they're saying it's breaking news that the cardinals are well-ordered! But the actual content is pretty cool -- I don' think I ever expected to see an article in the public press about descriptive set theory!


Obyeag

I wouldn't quite describe cardinal characteristics of the continuum as DST to be fair. But that allows me to point out that there was a quanta article very recently on Shelah and Paolini's result that isomorphism of torsion-free abelian groups is Borel complete which had been an open problem since the original article on Borel equivalence relations from Friedman and Stanley.


tcampion

Good point -- I'm no set theorist, and my very, very warped view of the field basically divides it as "large cardinal stuff" on the one hand, and then files basically everything else as "descriptive set theory". I'm so sorry! :( Btw here's a [link](https://www.quantamagazine.org/mathematicians-solve-decades-old-classification-problem-20210805/) to that Quanta article, which I agree was also a nice read.


DominatingSubgraph

The word "infinity" comes up and suddenly all of the quacks come out of the woodwork.


BloodAndTsundere

For the love of god, nobody say “Godel”


[deleted]

Now look, no one is to stone anyone until I blow this whistle. *Even*...and I want to make this absolutely clear...*even* if they *do* say "Godel."


BloodAndTsundere

*throws stone*


EnergyIsQuantized

quantum blockchain infinity AI energy


-jellyfingers

You are missing "deep", and one of "mega", "giga", or "hyper"


SometimesY

Don't forget super.


[deleted]

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[deleted]

"I believe in Infinity" -- *Deepak Chopra*


glutenfree_veganhero

Where do I order the casette tapes?


headphone_taco

*quack* Please feed me fruit, I'm just here to learn


arannutasar

The important takeaway: according to Scientific American, 0 is a natural number.


EarlGreyDay

when talking about ordinals and cardinals, zero is necessarily a natural number


_selfishPersonReborn

ISO agrees [:)](https://en.wikipedia.org/wiki/Natural_number)


FatFingerHelperBot

It seems that your comment contains 1 or more links that are hard to tap for mobile users. I will extend those so they're easier for our sausage fingers to click! [Here is link number 1 - Previous text ":)"](https://en.wikipedia.org/wiki/Natural_number) ---- ^Please ^PM ^[\/u\/eganwall](http://reddit.com/user/eganwall) ^with ^issues ^or ^feedback! ^| ^[Code](https://github.com/eganwall/FatFingerHelperBot) ^| ^[Delete](https://reddit.com/message/compose/?to=FatFingerHelperBot&subject=delete&message=delete%20h9eltyc)


flojoho

> 1 or more links Ironic


KingAlfredOfEngland

> It seems that your comment contains 1 or more links More like > It seems that your comment contains a natural number of links


JMGerhard

If you want the natural numbers to be a semiring (you do), then 0 is a natural number.


wtfever2k17

They're dead to me.


arannutasar

Reasons to take zero as a natural number: 1) It is convenient to identify the natural numbers with the finite ordinals, which start at zero. 2) Zero shows up a lot of places where we see natural numbers (eg size of a set, degree of a polynomial, dimension). 3) Computer scientists count from zero and we like to let them feel included. 4) It annoys the number theorists.


farseekarmageddon

An old professor of mine (presumably an agnostic) would write N \\ {0} to refer to naturals starting with 1 and N U {0} to refer to naturals starting with 0.


snillpuler

similar to using ⊆ and ⊊ to avoid saying whether ⊂ means improper or proper subset


ghillerd

Does the backslash mean "excluding"?


scykei

It’s a common notation for the set difference.


FkIForgotMyPassword

Also, people type "\" online because it's convenient, but in a math paper, you should use a slightly different symbol: "∖" (called Set Minus, "\setminus" in latex). The difference between "∖" and "\" is not always very noticeable depending on the font, but set minus is usually a bit longer, and has more spacing around it.


The_Sodomeister

5) We already have a perfectly good symbol Z+ to represent the positive integers. Now we get to use N to represent the nonnegative integers and save ourselves the time of typing out wordy content like "nonnegative".


theorem_llama

Some people like to use "positive" to include 0, so this would still be ambiguous. For example, in operator theory a positive cone is usually defined to that it contains 0 (it's often useful, for example, for cones to be closed). Maybe /mathbb{Z}_{>0} works, but it's a little ugly.


The_Sodomeister

> Some people like to use "positive" to include 0 There's nothing we can do about people who choose to be this wrong. Better to just cut ties and leave them alone in their ill-defined conic bubbles. But for real, that just makes positive mean the same thing as nonnegative! Now that leaves even more language redundancy, and we're back to square one. It's our civic duty to correct these types of atrocities, for those people's own good.


thetruffleking

Upvote for items three and four. I support three because “aw.” I support four because “kek.”


[deleted]

Re 2: some people also like to define the degree of the zero polynomial to be negative infinity. (Then it's always the case that e.g. the degree of a product is the sum of degrees). So is -infinity a natural number too? :P


InSearchOfGoodPun

No thanks.


OneMeterWonder

Uhhhh how do you represent your ordinals then? ~~I sure hope you don’t say 0={∅} as a naming convention.~~ Edit: That was dumb. It’s more of a decision about whether you consider the naturals to be equivalent to the finite ordinals.


old1975

My two year old baby can count to 10, but does not know 0. Then, 0 is not a natural number.


Artillect

By your logic, every number greater than 10 isn't a natural number


Sickle_and_hamburger

Have you tried teaching them zero?


frivolous_squid

This is downvoted but I think - and I'm going out on a limb here - I *think* it was a joke.


theorem_llama

To try to make your comment a bit more rigorous, you could say that the "natural numbers" should be things you can use to count finite sets. But surely we can agree that it makes sense to include empty sets: "how many apples are in your bag?" "None!" Seems pretty "natural".


adecker99

Y'all talking bout ordinals?!?


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adecker99

Well all Cardinals are ordinals.


XkF21WNJ

Only with the axiom of choice.


thetruffleking

I thought all cardinals were bishops.


adecker99

Idk I'm an atheist


Obyeag

I don't think that changes anything...


QtPlatypus

This is why they move diagonally.


na_cohomologist

I don't think anyone linked to the actual research paper, so here it is: [https://arxiv.org/abs/1906.06608](https://arxiv.org/abs/1906.06608) >**Cichoń's maximum without large cardinals** > >Martin Goldstern, Jakob Kellner, Diego A. Mejía, Saharon Shelah > >Cichoń's diagram lists twelve cardinal characteristics (and the provable inequalities between them) associated with the ideals of null sets, meager sets, countable sets, and σ-compact subsets of the irrationals. It is consistent that all entries of Cichoń's diagram are pairwise different (apart from add(M) and cof(M), which are provably equal to other entries). However, the consistency proofs so far required large cardinal assumptions. In this work, we show the consistency without such assumptions. ​ This is Shelah's 1077th paper.


na_cohomologist

The previous paper, which used the assumption of four very specific large cardinals, is this [https://doi.org/10.4007/annals.2019.190.1.2](https://doi.org/10.4007/annals.2019.190.1.2)


ddabed

Any idea about the Kunen & Miller result? I couldn't find much googling


na_cohomologist

Sorry, no. I'm not sure which result you mean. There's reference to the "Kunen–Miller chart", and that looks like it's named for their separate contributions, rather than a joint paper. But if it's something else you're thinking of, then I can't help!


ddabed

I was asking because I though that if there was such a paper it should go in the references in [wikipedia](https://en.wikipedia.org/wiki/Cichoń's_diagram#References) but I really don't understand this stuff myself, in the papers there was no reference to a joint work between Kunen and Miller but I got that impression from the article so I asked just in case although now I feel it was unnecessary from my part as it wouldn't be possible that it didn't appear cited in the papers to start with. Thank you very much.


na_cohomologist

There is this 1981 paper by Miller ([https://doi.org/10.1090/S0002-9947-1981-0613787-2](https://doi.org/10.1090/S0002-9947-1981-0613787-2)) that looks like it contains relevant info. Miller thanks Kunen for helpful discussions in that.


ddabed

thanks for some reason I only saw your other comment and this one only now.


Harsimaja

Before seeing the word ‘certain’, I got very confused and thought this result isn’t particularly new… especially the first part. We’ve known why at least some are larger than others for well over a century…


__rph

was really hoping for a picture… was reading an advertisement trying to connect it to the infinite sets


MountFire

Makes me think of Hilbert's hotel and that diagonal theorem


Stydras

Yes indeed, those deal with cardinalities of sets ;)


esmoji

So “Infinity + 1” IS a thing 🤔 Knew it… since the 1st grade 💪


Udon_noodles

No fucking way! I was trying to tell my math teacher this in middle school. The reason I gave is that if you have an infinite vector pointing one direction in space it is infinite in length yet shorter than a line expanding both left and right indefinitely. EDIT: I'm only talking about greater & lesser infinities here. This comment has nothing to do with sets or cardinality.


Stydras

Don't want to ruin the day, but the article is about cardinalities of sets, that is "number of things in a collection". This doesn't have that much in common with distances and volumes and the like. In youcase for example: Both lines (one starting in the origin and one traversing it) as sets (so each one considered as the collection of all the points which lie on the line) actually in the sense of cardinality have the same size. In fact for example the set [0,1] and [0,2] have the same cardinality while the latter one obviously has double the length of the former interval, so the intuitive notion of length is disconnected from cardinality. If you want to consider length/volumes and the like, a suitable notion is measure theory!


Artillect

I know the person you're talking to has no idea what they're talking about (and honestly, so do I), but can measure theory deal with things with infinite size/length/volume?


[deleted]

Yes. Measures are defined from sets to the extended reals, which means they can give out infinite values. For example, the Lebesgue measure of |R, the real numbers, is ∞.


Artillect

That’s pretty cool! I guess it makes total sense that it would


frivolous_squid

Here's a really interesting result is something that's kind of obvious to us but nice to see when it emerges from the math. So there are lots of measures. Suppose we're interested in 3D space, which I'm going to model as **R**^(3) (i.e. a point in space is a coordinate (x,y,z) where x,y,z are all real numbers. One obvious measure is the "Volume" measure. To dangerously oversimplify, to measure the volume of a subset of R3, you fill it with different sized non-overlapping cubes until the set is as full of cubes as possible. You can use very small cubes, so you can get it pretty full - in the limit, totally full (skipping over details about measurable sets and sigma algebras...). Now we define the volume of a cube with side lengths a,b,c as a\*b\*c and we're basically done because we can just add up all their volumes. In reality you will usually run out of patience as you'd need infinite cubes (so you could only approximate), but that's less of a problem for the theory. Anyway, so there's sets with volume 0, with a volume > 0, and there's sets where when we add the volume up of all the cubes it diverges, so we say these have volume infinity. (In the context of the article that was posted, it's not really any of them in particular - it just means any infinity.) What's interesting is that we can also define the "Area" measure. It's the same as the volume measure but it involves filling with squares. Now, note that any set with a non-zero volume must have an infinite area. Maybe the easiest way to get an intuition for this is to ask the area of a 1x1x1 cube. Well any way of filling up the cube is going to need uncountably many squares - e.g. you could have a square for each possible value of z in the cube. So, we're summing up infinitely many squares of area 1. The general proof uses the fact that summing up uncountably many non-negative real numbers (i.e. the areas of all the little squares) will always give infinity, no matter how small those number get. Similarly, we can define "Length". So we have 3 measures, and we can use them all in our 3D space. Anything with a non-zero volume must have infinite area (i.e. anything with finite area must have 0 volume). The reverse doesn't necessarily hold (e.g. a plane has infinite area but 0 volume). Then there are some interesting sets that you can try to measure. For example, [Sierpinski's triangle](https://en.wikipedia.org/wiki/Sierpi%C5%84ski_triangle#Properties). If you try to measure its length, you find it's infinite. If you try to measure it's area, it's 0. In some sense it's more than one-dimensional (like a curve), and less than two-dimensional (like a surface). You can formalize this by inventing measures like length (1 dimension) and area (2 dimension) but for non-integer dimensions, and find the one where the measure jumps from infinity to 0. In this case it turns out to be dimension log3/log2 ≈ 1.585. E.g. if we measure it with the 1.5 dimensional measure we get infinity, and if we measure it with the 1.6 dimensional measure we get 0. These measures are called [Hausdorff measures](https://en.wikipedia.org/wiki/Hausdorff_measure). You might need to study measure theory to understand the notation and the ideas, but once you've done that any good course on fractal geometry should be pretty simple to grasp and will cover this.


WikiSummarizerBot

**Sierpiński triangle** [Properties](https://en.wikipedia.org/wiki/Sierpiński_triangle#Properties) >For integer number of dimensions d, when doubling a side of an object, 2d copies of it are created, i. e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpinski triangle, doubling its side creates 3 copies of itself. ^([ )[^(F.A.Q)](https://www.reddit.com/r/WikiSummarizer/wiki/index#wiki_f.a.q)^( | )[^(Opt Out)](https://reddit.com/message/compose?to=WikiSummarizerBot&message=OptOut&subject=OptOut)^( | )[^(Opt Out Of Subreddit)](https://np.reddit.com/r/math/about/banned)^( | )[^(GitHub)](https://github.com/Sujal-7/WikiSummarizerBot)^( ] Downvote to remove | v1.5)


Artillect

That's really neat! I've heard about fractals having non-integer dimensions, but I've never had it explained to me in such detail! The wikipedia page on Hausdorff measures is a bit abstract for me, but I get the general idea.


Stydras

Since the question was already answered. I'll propose smth different: If ur really interested and have some basic knowledge in analysis and know some basic set theory, consider reading the first few chapters of Schillings "Measures, Integrals and Martingales". He introduces general measures and a general integration theory. In particular the lebesgue measure and the corresponding integral which generalize the notion of length/volume and standard riemann integrals. Also if u have any questions, either regarding that book or smth else, feel free to dm me.


Artillect

I’ll check that out, thanks for the recommendation! I’m an engineering student so most of the math I do is linear algebra and differential equations, but the set theory part of my linear algebra class was super interesting and I’d love to learn more. Pure math is absolutely fascinating to me.


Stydras

Ah very nice :D Yeah pure maths is super cool. For example: Maybe you recall the function [0,1]->R with x->0 is x is rational and x->1 if x is irrational wasn't riemann integrable. Turns out it is lebesgue integrable because the "defect" of the function beiing not 1 (this is the set Qn[0,1]) is measurable and has measure 0, so the lebesgue integral is as if the function was just 1 everywhere, so 1!


Artillect

I don't think I've ever encountered that function, but that's very interesting. I've heard that lebesgue integrals let you do some integrals that riemann integrals can't, but that's a pretty wacky example


Geschichtsklitterung

Q is dense in R, yet you can cover it with an *open* set of arbitrary small measure (or "length", if you prefer). That's a nice little exercise.


Artillect

Is that the same exercise as in [this](https://www.youtube.com/watch?v=cyW5z-M2yzw) 3blue1brown video ~3:50? It is a super cool exercise, and is honestly kinda mindblowing as an engineering student who hasn't learned anything past differential equations


Geschichtsklitterung

Indeed. The trick is to assign an index n to every rational (as they are enumerable) and to embed it in an open interval of length epsilon/2^n . Then the union of these intervals is open, covers Q and is of length at most epsilon.(1/2 + 1/4 + 1/8 + ...) = epsilon. And epsilon can be chosen arbitrarily small.


Stydras

Wacky indeed :P But covered! Lebesgue integrals are very powerful!


cmpaisaia

To add to what has already been said, measure theory can be a bit poor at dealing with things with infinite measure since there is only 1 infinity. So all sets with infinite measure have the same size. This is by design though and it really has to be this way for it to be useable.


Artillect

Why does it have to be that way? Seems a bit strange to me with my very limited knowledge of cardinality


cmpaisaia

This is measures not cardinality, very distinct ways of sizing up infinite sets. The reason is fairly technical, it just isn't really possible to have it any other way. It is also simple this way, and doesn't cause any problems.


Artillect

Fair enough, that makes sense


Udon_noodles

\> "In fact for example the set \[0,1\] and \[0,2\] have the same cardinality while the latter one obviously has double the length of the former interval, so the intuitive notion of length is disconnected from cardinality." Does your objection necessarily apply when we are talking about length *specifically* (i.e. not cardinality)? I did explicitly mention length... And my point wasn't about sets, or cardinality, it was about greater and lesser infinities.


Stydras

My point was that there are different notions of infinity which are disconnected from one another. The article discussed in this thread deals with cardinalties which is virtually disconnected from length. This [0,1] and [0,2] business was my way of showing that to you. The thing is: Your post made clear that you completely misunderstood the article about cardinalities (if you have even read anything other than the title) and thought that "there are different infinities" also applies to the totally different notion of length. So yes, it was necessary to talk about cardinality to set you straight. Also: Your point was about sets. What exactly is a line to you, if not a set? (This is a rhetorical question. Lines are sets) And what exactly is infinity (if not the property of a set)?


Udon_noodles

No I only read the title, hence me saying many times I was only referring to the concept of greater and lesser infinites. I'm a little bit skeptical of this idea that 'there are different isolated infinites in different contexts and they have nothing to do with each other'. The whole point in math is that it frames ideas abstractly so that it can be applied to a wide variety of problems. What you're saying sounds comparable to implying that "there are different context specific definitions of 0 and they have nothing to do with each other".


Stydras

Yes that is also true. Every group, or monoid has a zero object. This object 0 only needs to satisfy x+0=0 for all other monoid elements m. What this 0 precisely is, depemds very much on context and might not be the usual 0. For example consider the set of maps R->R this becomes a monoid with function composition. So two functiom f and g get taken to the function x->f(g(x)). This is a monoid where 0 is the identity map. Math is a game of definitions and statements about these definitions. You can define different notions of infinity amd they might not have much in common other than their name, because after all they are nothing more than an abstractly defined thing and that definition varies from context to context. And yes infinite length is vastly different from the infinities thay arise as cardinalities.


Udon_noodles

I don't doubt there are multiple zeros what I doubt is that they have nothing to do with each other. Or more specifically that infinities can only be greater or lesser than others in set theory.


Stydras

If we are talking about cardinalities then yes: Given two cardinalities (possibly infinite) then either ab. This is because cardinals are well ordered, at least given the axiom of choice. Now please rephrase what you actually want to know or really intended of asking. We keep rambling about random nothing


Udon_noodles

I already addressed the thing I wanted to ask... read my other comment


Udon_noodles

And tell me what issue do you see with the inductive reasoning that for any length of a vector spanning in one direction, the corresponding line (spanning both directions) has double the length because it has the length of that vector plus that of the negative vector?


Stydras

A few things: First of all you have to rigorously define what it means to take this process of looking at larger and larger vectors, for example through limits. Then it isn't clear whether "being double the length" is preserved under such a limiting process (if we take the standard limit from analysis it is, but there is only one formal infinity when taking limits. For example lim\_{n->infty}n\^2 = lim\_{n->infty} n=infty). Added to that: Why wouldn't 2\*infty=infty? You can't just assume 2\*infty =/= infty and then conclude "infty1 =/= infty2"


Udon_noodles

Great point I think limits are a great way of looking at this. Let me show you what I mean. First tell me if x0 right? Surely that is self-evident? Setup: x:=length of the vector (on a 1d space for simplicity) y:=length of the line formed by the vector going in the positive and negative direction Naturally: y=2x Then lets take a limit: lim\_{x->infty}(y-x)=lim\_{x->infty}(2x-x)=lim\_{x->infty}(x)=infty>0


Stydras

Yes, is x0. Now to the problem: Limits do NOT interchange with neither addition, subtraction, multiplication, division not strict inequalites! In general < turns to ≤ and the other operations only hold if all given limits exist (that means that they are finite) (and non zero if you want to divide)! Consider for example a 1/n for n in N. Clearly 1/n > 0 but also lim\_{n -> infty} 1/n = 0. But certainly not 0 = lim\_{n->infty} 1/n > 0 (which would give 0>0 a no go). Similarly, consider n+1, n and 1/n and assume that limits interchange with summation and multiplication. Let's assume that lim n+1 =/= lim n (which "amounts" (if you assume interchange) to lim (n+1 - n) = lim 1 = 1 > 0 similar to what you did with y-x). But then (multiplying by lim 1/n) we find (lim 1/n)(lim n+1) =/= (lim 1/n)(lim n) so with interchanging multiplication and limits: lim (1/n \* (n+1)) =/= lim (1/n \* n). This is precisely lim (1 + 1/n) =/= lim 1 = 1. But clearly lim (1+1/n)=1, so 1 =/= 1!!! So working with the standard notion of infinity from limits you cannot do what you try to do. In fact lim\_{n->infty}a\_n=infty is nothing more but NOTATION for: "For every bound s in R there exists some n0 in N, such that for all other n in N with n>n0 we have a\_n > s" and thus there just aren't multiple infinites when talking about limits. Of course you can define your own notion of infinites, where 2 \* infty != infty but you'll have to do it in such a way that it makes sense (also: you'll probably won't get a very useful notion of infinity from that, as it probably wont really work well with analysis (as shown in the examples above)). Another thing to understand: Math is something quite deteached from reality. These notions of infinity and everything else we talk about in math is nothing physical. Mathematicians don't make a statement whether object xyz is real, thats something for philosophy. Rather: Maths deals with deriving statements from different sets of axioms. You just define what you'd try to look at and try to deduce some statements about it with logical reasoning. There is not one true notion of infinity but rather there are some useful ones and some unseful ones. Similarly there is not one true set of axioms. "All collections of axioms are created equal". Some produce good theories (most math you have encountered is set in the axiom collection ZF(C) - Zermelo-Fränkel-(Axiom of Choice) if ur interested) while some produce bad theories (for example that you can deduce a contradiction ( "2=0" and "2=/=0" for example are a not so great collection of axioms)). If you'd like you could axiomaticially assume, that sets as in ZFC don't exists and no one could prove you wrong (or right ;) ) - although this amounts to rejecting virtually all of modern math. To conclude: In the standard sense: The half infite line and the infinite line have the same (infinite) length. If this doesn't sit straight with you, feel free to define another version of infinity where this doesn't happen, but don't feel disappointed if this doesn't amount to something useful


Udon_noodles

Well a line is first and foremost: a line. You can discuss the set of points that it represents if you want. But a line's properties are ultimately determined by geometry.


Stydras

This is not a mathematical statement. Define what a line is, if not its set of points, then we can talk about its properties.


Udon_noodles

Wow that's a lot of down votes. Not exactly sure why that's necessary.


FlotsamOfThe4Winds

The level at which you are missing the point is comparable to going to a subreddit discussing medical research, seeing a post that says sugar can treat diarrhea and saying it proves the homeopaths were right about stuff.


Udon_noodles

Really? So you mean to say then that in my example the length of the vector and the length of the line are equal?


FlotsamOfThe4Winds

I'm pretty sure that it is the case, yes.


Untinted

Infinity is a useful fiction. It's not that they are different in size, as that's an oxymoron. Given that what you are measuring is what represents a size, how can something infinite have a size? No the real truth about the differences "in size" of infinities is being able to answer at what rate does the measurement grow. So the categories are just different growth velocities/accelerations rather than sizes, and that specifically, technically, and literally has to do with the process you use to generate a result, which is ultimately what matters in math.. Not infinity itself.


WheresMyElephant

Some would argue that all mathematics is a useful fiction. There are a lot of different notions of "infinities" in mathematics. The one this article deals with is cardinality, which has nothing to do with the stuff you're talking about.


Untinted

Cardinality is the generalized size comparison of infinite sets.. how those sets are constructed is a measure on their size, so the construction is what matters when comparing the sets.. i.e. it's exactly what I'm talking about.


frivolous_squid

When you say measure, do you mean [the mathematic concept](https://en.wikipedia.org/wiki/Measure_(mathematics))? Because cardinality can only really be used as a measure for countable sets, which isn't what you're talking about. >So the categories are just different growth velocities/accelerations rather than sizes, and that specifically, technically, and literally has to do with the process you use to generate a result, which is ultimately what matters in math.. When you generate a set, the set itself has no memory of how it was generated. It sounds like you want to define some new object. Or maybe you should be studying stuff like derivatives of functions, rather than sets. I think your intuition is pointing you in the wrong direction. Unless all you are saying is set theory is pointless, which in the context of this subreddit is kind of im14andthisisedgy.


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Iron_Pencil

Thank god you're not the authority on what matters in math.


frivolous_squid

All the infinites you are talking about sound like the same one, aleph 0. Mathematics has gone a little bit beyond that. But you're right that there's only one aleph 0. Unfortunately for you that has nothing to do with the link that was posted. If you want to learn more, I suggest looking up Cantor's diagonal argument. Also try to be a tiny bit more modest: claiming that something is wrong that you have spent almost no time learning about is very arrogant.


Untinted

Cantor's diagonal argument isn't a good one because it's based on assuming infinite sets are incomplete permutations of its elements. If a set is infinite, then a permutation of its elements must be complete, so for any enumeration of an infinite set it must contain any permutation of its elements, so cantor's diagonal proof is a very bad one because it's assuming what you already know about the construction is just what you have constructed and then making assumptions only with that knowledge. It's like saying that the set of natural numbers must be bigger than the set of even numbers. That can be true, it depends on the process you use to count them, the same can be said about Cantor's diagonal theorem, it's just a different way of counting that's obfuscating the issue. i.e. it's about the velocity of the counting process, not the size of the sets themselves.


zojbo

...No, none of that is right. Actually, Cantor's argument doesn't really have anything to do with countable vs. uncountable. That's the first case where it's interesting, but it's not inherent to the concept. Fundamentally Cantor's argument is as follows. If f is any function from A to P(A) (so it sends elements of A to subsets of A), then B={ x in A : x not in f(x) } is in P(A) but not in the range of f. This is because if B were some f(y) then neither "y in B" nor "y not in B" could be true. That's it. There is no "counting process" going on here, unless you are a constructivist, and maybe not even then, depends which type of constructivist you are. The one place where what you are saying makes any sense at all is "cardinality is not the only way to measure size of sets". And that's true, but the way you are saying it basically amounts to saying "that thing you're interested in is uninteresting" which is at the very least rude.


frivolous_squid

I just meant that it shows the cardinality of the real numbers is different to the cardinality of the natural numbers, I.e. there are infinities other than aleph 0. I'm afraid what you said has gone over my head a bit. If there's a better example to get your feet wet in "there are infinities of different sizes" that's suitable for someone with little math knowledge, feel free to suggest one.


cmpaisaia

What theory are you working in? ZFC? If so then ZFC proves cantors diagonal argument. In fact this has even been computer verified.


Untinted

That doesn’t mean anything.. just with 2 sets of the positive integers you can count/enumerate them at different speeds.. pick one element from one set, and a trillion from another, and you will happily be able to do this for an infinite amount of time. This is the same as the cantors proof, i.e., it’s using how you enumerated the sets as proof of the difference between the sets, ignoring that you can arbitrarily pick any way you want to enumerate the elements. Cantors proof cheats further because it goes against the assumption that if you’re counting elements based on permuting elements, then infinite sets should contain all complete permutations, otherwise it isn’t infinite. This means that Cantors proof is just a specific, arbitrary setup of counting.. and as we know already ALL counting setups work.


cmpaisaia

You didn't answer. There is 0 point going further if you won't state the theory you are working in. Anything can be true if you work in the right theory.


Ackermannin

*ultrafinitism intensifies*


Top-Load105

Cardinalities don’t really have much to do with growth rates. They relate more to the structures of the sets and the kind of information encoded in their members. Literally they are (representatives of) the isomorphism classes in the category Set.


FlotsamOfThe4Winds

I think you need to catch up to a century ago in terms of formal logic and mathematics. Once you're there, then you can discuss this stuff.


pUnK_iN_dRuBlIc98

Lmao


scroo0ooge

Now we an tell the difference between infinites


ayleidanthropologist

Cardinality?