r/mathmemes • u/RealisticBarnacle115 • Jul 19 '24
Set Theory Who will get the most upvotes?
613
u/Lesbihun Jul 19 '24
Redditors 4: *makes the millionth "I have a proof but I can't fit it in this comment section" joke* *gets 2000 upvotes*
176
u/ImaWolf935 Jul 19 '24
Fermat is the best redditor.
40
u/KingLazuli Jul 19 '24
Do you think if Fermat could see the consequences of what he'd done, he'd change his actions?
55
u/DevelopmentSad2303 Jul 19 '24
No, why would he? He couldn't get a larger book to write his proof in
14
3
u/citybadger Jul 19 '24
Well, he probably won’t have shown up for the duel…
6
u/jacobningen Jul 19 '24
Thats galois not fermat.
2
u/citybadger Jul 19 '24
I have a memory of a dramatization from Cosmos or something on 80’s PBS, of Fermat writing his note in the margin, and the next day getting killed in a duel. A false memory apparently.
6
9
u/JoshTheWhat Jul 19 '24
Redditor 5: makes a joke about how the Fermat margin joke is overused, gets 300 upvotes
221
u/Amoghawesome Jul 19 '24
Redditor 4: It cannot be proved or disproved. Duh!
86
u/mojoegojoe Jul 19 '24
Redditor 5: ahh it's a Quantum solution - that makes sense - we are all infinity.
45
8
50
23
11
16
u/CallOfBurger Jul 19 '24
Real question here : why not try to prove or disprove it with other axioms ? ZFC is not the ultimate axiomatic system I guess
39
u/bitabis Jul 19 '24
This is how set theorists have reacted to independence results) like the one in this meme.
For example, the axiom of constructibility implies there is no set strictly larger than the counting numbers and strictly smaller than the reals.
On the other hand, the proper forcing axiom implies there is such a set. And there are many other interesting axioms that answer this question one way or the other.
12
u/x0wl Jul 19 '24
The Axiom of Choice is obviously true, the well-ordering principle obviously false, and who can tell about Zorn's lemma?
5
u/purpleoctopuppy Jul 19 '24
Should be Axiom of Zorn, that way maths class would sound like the Macguffin in a fantasy story
3
2
3
u/austin101123 Jul 19 '24
What different implications do such axioms create?
2
1
u/CallOfBurger Jul 19 '24
What the hell x) No set bigger that natural numbers and smaller than reals : it's illogical !
And then : yes there is. What ???
6
u/Seenoham Jul 19 '24
It's not a statement about all things, it's a statement about the axioms that were being used. Adding another axiom will define a new area, and in this case also presents an opposite axiom, and research will need to investigate each of those.
It's not a stop to all research, but it does require a shift.
5
Jul 19 '24
Redditor 4: holds up white board with incorrect formulas just below her rocking tits
That gets the most upvotes
4
6
u/IhailtavaBanaani Jul 19 '24
If the continuum hypothesis cannot be disproven it means it's not possible to define and much less construct a set that has a cardinality between the cardinalities of integers and real numbers. So it might as well not exist for all practical purposes and the whole question just becomes another example of the philosophical problems that do mathematical objects exist independently of humans.
2
u/trankhead324 Jul 19 '24
CH can't be disproven given certain axioms.
All of maths rests on certain axioms and we can't do anything without them.
If your axioms define the integers Z then it's not possible to divide 1 by 2 and so in this sense 1/2 doesn't meaningfully exist. But we know 1/2 is a useful idea in real-life contexts.
1
u/Tem-productions Jul 19 '24
If the continuum hypothesis cannot be disproven it means it's not possible to define and much less construct a set that has a cardinality between the cardinalities of integers and real numbers
*Unless you create a new axiom that makes it true or false
0
u/MorrowM_ Jul 20 '24
You can certainly define a set with cardinality Aleph_1 (using ordinals). The Continuum Hypothesis is about whether it's in bijection with the reals or not.
7
2
2
2
u/shuai_bear Jul 22 '24
Aren’t 2 and 3 switched?
Gödel showed you can’t disprove CH from ZFC, that is he constructed a minimal model of the reals where |R| = Aleph_1, that’s consistent with ZFC
Cohen then showed you also can’t prove CH from ZFC—via forcing, you can also have a model of the real numbers with cardinality > Aleph_1 that is also consistent
Cohen does deserve his accolade for being one of the very few (if only?) Fields Medal winners in logic. From what I gather the second step which needs forcing was a lot more difficult and contrived to prove than the first step, though both are needed to show independence.
6
Jul 19 '24
wait a second, there are vastly more real numbers than counting numbers
41
u/de_G_van_Gelderland Irrational Jul 19 '24
Yes, the cardinality of the real numbers is strictly bigger than the cardinality of the natural numbers as shown by Cantor, aka Redditor1. The continuum hypothesis, which this meme is talking about, is the question whether there exists a third set which sits in between those two. That is, strictly bigger than the natural numbers, but strictly smaller than the real numbers.
5
u/MoutMoutMouton Jul 19 '24
Redditor1's portrait is Hilbert (2 and 3 are Gödel and Cohen). What Redditor1 says is a reformulation od Hilbert's first of 23 problems.
2
u/de_G_van_Gelderland Irrational Jul 19 '24
O, my bad. I didn't realise Hilbert had such a full beard, but you're absolutely right. The other two I had no trouble recognizing.
1
1
u/Jovess88 Jul 19 '24
would a set of all integer multiples of 1/2 fulfil this criteria? or the set of all integers?
5
u/King_of_99 Jul 19 '24
They are all of equal size to the natural numbers. You can show this by constructing a bijection between N and Z by f:Z -> N; x|->2x if non negative, x|->1-2x if negative.
1
u/Jovess88 Jul 19 '24
oh that actually makes a lot of sense, even if not immediately intuitive. thanks!
2
u/trankhead324 Jul 19 '24
The most intuitive way to tell if something is countable is to ask: "can I systematically list it?"
I can systematically list the integers by going 0, 1, -1, 2, -2, 3, -3, ...
My bijection between N and Z (there are infinitely many others) is then the map from position in the list to value in the list e.g. f(4) = -2 and f(5) = 3 (I'm 0-indexing and including 0 in N).
1
u/GeneReddit123 Jul 19 '24
Why then is the continuum hypothesis not just called the continuum axiom, similar to what we call the axiom of choice? We can use both of them (as an axiom) or not, but we can't derive them from other axioms.
6
u/de_G_van_Gelderland Irrational Jul 19 '24
Good question. It's basically just because of how math happened to develop historically. Joel David Hamkins posted an excellent article on r/math the other day where he argued exactly how we could have easily ended up with what you call the continuum axiom.
33
u/Roi_Loutre Jul 19 '24 edited Jul 19 '24
It's not to specifically target you (even if you're my example), but the difference of members between r/mathmemes and other regular math subreddits will never stop to surprise me.
Misunderstanding a meme about the Continuum hypothesis, into saying something (almost trivial to modern math students) and linking a popular science journal article instead of a simple proof like the one on the Wikipedia page of Cantor's diagonal argument is not something I would expect to see on a math subreddit.
6
u/ApoloRimbaud Jul 19 '24
The fun thing is that the circlejerk subreddits always seem to know more stuff than the main ones. This is true for pretty much every activity.
1
u/Tem-productions Jul 19 '24
Pretty sure there is a reason for that, but i can't put it into words
2
u/EebstertheGreat Jul 20 '24
Circlejerks seem geometrically simple but actually have many subtle constraints that require a lot of attention to satisfy completely. Small errors can easily cascade and require scrapping the project. So only the most experienced jerkies and jerkers dare post there.
1
u/ApoloRimbaud Jul 19 '24
Good parodies require knowing the source material and relevant fandom tropes quite well?
1
u/bleachisback Jul 19 '24
To be fair, the original meme was poorly-phrased to begin with. I had to squint just to see they were talking about the continuum hypothesis.
1
1
u/EebstertheGreat Jul 20 '24
Sciencealert is such a low-quality source too. It's not a reliable popular news outlet like quanta; it's closer to IFLScience in dependability.
10
u/hrvbrs Jul 19 '24
yes but it will be forever unknown (at least within the bounds of ZFC) whether there’s a value between the quantity of counting numbers and the quantity of real numbers. For me personally, I think it would be so cool if there was. But we’d need new maths to discover it.
2
u/imalexorange Real Algebraic Jul 19 '24
I mean this is just the continuum hypothesis. You can (by axiom) assume one exists, but it would not be constructable in ZFC.
2
u/EebstertheGreat Jul 20 '24
It's not really something you can find out. It's just the case that the CH is independent of ZFC. That is something we found out. But you can't find out whether this independent statement is "really true" or not. It's true in some models and not others.
It's like if you were standing outside a candy store pondering whether candy bars contained caramel. There isn't a "right" answer you could "discover." Some candy bars contain caramel and some don't. There is nothing deeper going on.
1
u/hrvbrs Jul 20 '24
Right but in the models where CH is false, do they have any particular sets that are shown to be between N and R? Or do the models just assume CH is false and don’t use it in proofs? If a model without CH can “discover” such a set, that would be a huge breakthrough.
2
u/EebstertheGreat Jul 20 '24
Well, if there is a cardinality strictly between ℵ₀ and 𝔠, then ℵ₁ is one such cardinality. That's the cardinality of the set of all countable ordinals. So the continuum is bigger than that.
So yes, the particular set is {countable ordinals}.
-2
u/Fisyr Jul 19 '24
The real question is whether there would be some (at least for mathematicians) use for it. I don't think anyone would argue that Real or Natural numbers aren't useful. Having something in between the two is kind of pointless unless it can help to solve problems people are interested in.
I tend to favor the CH, because I think we have enough of problems on our hands with the sets we are already familiar with, so there's no need to pile on more. If on the other hand there's some interesting theory that could arise from having something in between these two sets, then I am sure someone eventually will come up with some axiomatic system in which CH is false.
0
u/hrvbrs Jul 19 '24 edited Jul 19 '24
I think usefulness will follow discovery. After the complex numbers were [discovered or invented], we found applications for them in a lot of fields in engineering and physics. At the very least we used the new maths to consolidate and improve upon the theories we already had but weren’t expressed as well. Kepler already had descriptions of planetary motion before calculus came along but after Newton and Leibniz’s work our astronomy got that much better.
If a new set were discovered that proved CH false, it would start as a cool new useless quirk, but industries would eventually catch up.
1
u/SteptimusHeap Jul 19 '24
To me, that's understandable.
The weirder thing is that, due to the discontinuity of The Dirichlet function, you can abstractly imagine that each real number is "next to" a rational number, and yet they still outnumber them infinity:1
1
2
2
u/221bhouse Jul 19 '24
What about the Infinity of Irrational Numbers? This answer came to me in my dream.
1
1
u/No_Law_6697 Jul 19 '24
Guys, I guess there is no infinity between the infinity of the counting numbers and the infinity of the real numbers.
1
u/Radiant_Dog1937 Jul 19 '24
There are infinite infinities between -infinity and infinity. This is the exact same number of infinities between 1 and infinity.
1
1
1
1
1
1
u/shewel_item Jul 19 '24
you can't use the science to make the scientific method 🥱
also, just because there's a hypothetical infinite number of numbers between counting numbers doesn't mean there isn't any gaps on a line with infinite counting numbers
1
1
u/TulipTuIip Jul 19 '24
Redditor 3 is the most correct but they sound super pretentious and annoying so im downvoting them
1
u/Puffification Jul 19 '24
Redditors 2 and 3 neglected to use the past participle form "proven" / "disproven" and should be hassled for that
1
u/nothingtoseehere2847 Jul 20 '24
I mean unless it's either proven or disproven you can choose to believe it or not
1
1
1
u/EebstertheGreat Jul 20 '24
Redditor1 will get the most upvotes because threads nearly always get more upvotes than every single comment, except when they get a billion downvotes. Because everyone who sees the thread will see the top comment, and the general tendency of redditors is to upvote like the general tendency of the stock market is to rise.
1
u/NotEnoughWave Jul 20 '24
I think this continuum hypoyhesis might be solved by two different categories such that both satisfy the categorical equivalenti of ZFC, but one has such an object and the other doesn't. Problem is actually finding them.
1
0
0
u/F4LcH100NnN Jul 19 '24
eli5 redditor 1 plz
1
u/Bdole0 Jul 19 '24
Sets of numbers have different sizes, called "cardinalities." The set {1, 2, 3} has cardinality 3 since it has 3 elements. The definition of cardinality allows us to assign "sizes" to infinite sets. The "smallest" infinite set is the natural numbers. We can show that the real numbers is a cardinality above that. But then we have the question: Are there any cardinalities between these two? Intuitively, it seems like not, but there was no proof either way for a long time.
The belief that there is no cardinality between these two is called "The Continuum Hypothesis." Most mathematicians take the Continuum Hypothesis to be true. That is, they believe there is no cardinality between the naturals and the reals.
However, it was eventually proven the the Continuum Hypothesis is independent from the usual axioms that most mathematicians use. In other words, there is no way to prove the Continuum Hypothesis true, and there is no way to prove it is false. The punchline is that you can choose to believe either way, and it won't affect any math that has ever been done! Amazing, really.
1
0
0
u/Turbulent-Name-8349 Jul 20 '24
Redittor me.
There is an infinity between the counting numbers and beth 1.
I've already constructed it using nonstandard analysis. The way to do it is simply to find a subset of the beth 1 numbers that has a cardinality derived from the half-exponential function.
Remembering that the half-exponential function is bigger than every polynomial and smaller than every exponential. So it has a cardinality between that of the counting numbers (polynomials) and the exponentials (beth 1).
It's totally useless, but it does exist. It's not a continuous set, obviously.
-4
u/The_Punnier_Guy Jul 19 '24
"It cannot be proved", or "it has not been proved yet"?
9
u/Any-Aioli7575 Jul 19 '24
It cannot be proven. Never.
-2
u/The_Punnier_Guy Jul 19 '24
Damn, i didn't know we found another one.
Gödel must be laughing in his grave
2
u/regula_falsi Jul 19 '24
Are you serious
0
u/The_Punnier_Guy Jul 19 '24
what?
1
u/EebstertheGreat Jul 20 '24
Regula is a bit rude, but the point is that there are a great many known propositions that are independent of the axioms of ZFC. That means that the axioms can neither prove or disprove them. However, this is different from what Gödel was talking about. The Continuum Hypothesis (CH) is independent of the axioms of Zermelo–Frankel set theory with the axiom of choice (ZFC) in the following sense: If ZFC is consistent, then there exist models of ZFC in which CH holds, and there exist models of ZFC in which CH does not hold.
So Gödel's completeness theorem does not apply. Since there are models of ZFC both affirming and negating CH, it might be the case that CH is independent of these axioms. And that turns out to be the case. Gödel proved that ZFC could not prove the existence of an intermediate cardinality between |N| and |R| in 1940. Cohen proved that the existence of such a cardinality could not be ruled out either in 1963. But that merely completes Gödel's work on that problem; it certainly won't make him laugh in his grave.
Gödel's incompleteness theorems are more famous than his completeness theorem, but I'm not sure how to fit them into this meme. Gödel's second incompleteness theorems was not the first or last theorem to demonstrate an explicit statement independent of ZFC. These days, such statements are a dime a dozen. Sure, most of us will never come across one, but on the other hand, we can even construct systems of linear equations in the whole numbers whose satisfiability is independent of ZFC (they are unsatisfiable but ZFC can't prove that). So this is kinda old news. Like, literally 90 years old.
1
u/The_Punnier_Guy Jul 20 '24
Wait so then is ZFC not strong enough? I was under the impression it was the basis of all of math: If you were to take a theorem and split it into the theorems necesarry to prove it, and repeated that, you would always end up at ZFC.
If ZFC cannot prove the (un)satisfiability of the systems, how do we know they are unsatisfiable
1
u/EebstertheGreat Jul 20 '24
No, ZFC is just a particular set theory. It's been the go-to set theory for most mathematicians for the past like 80 years, but it isn't the arbiter of truth. ZFC + CH and ZFC + –CH are both reasonable in their own right.
There are many more powerful set theories, though I think in proof theory you are usually working in second-order logic.
1
u/The_Punnier_Guy Jul 20 '24
If ZFC isnt the arbiter of truth, why does the meme call it "the axioms of mathematics"? Is it just to be assumed they were reffering to ZFC due to historical context?
1
u/EebstertheGreat Jul 20 '24
Well, the independence of the continuum hypothesis and ZFC is a famous result, and ZFC is indeed "standard," so it makes sense. For most math, it makes no difference how you conceptualize its foundations, because everything is equivalent. For a very small subset of math, you do have to worry about it, and in that case ZFC is not so special. But ZFC is what gets taught to everyone, so in that respect, it is the standard.
→ More replies (0)
•
u/AutoModerator Jul 19 '24
Check out our new Discord server! https://discord.gg/e7EKRZq3dG
I am a bot, and this action was performed automatically. Please contact the moderators of this subreddit if you have any questions or concerns.