Wonderfull video about proven unsolvable mathematical questions.

Wonderfull video about proven unsolvable mathematical questions.
Well it is infinite
I find this area of mathematics really interesting. I got hooked when I first saw the diagonal argument demonstrated on a numberphile video. Just how one goes about doing these transfinite proofs is beyond me!
Uh, first the question "how big is infinity" makes no sense in mathematics. What does "how big" mean, what sort of infinity are you talking about?Now where did did you trawl that up from? You are partially correct, but you need to specify what sort of infinity you are referring to.Which means it can be put in one to one correspondence with a proper subset of itself.
Example: consider the natural numbers and the proper subset . If you can find a onetoone correspondence with the natural numbers, let's say that has infinite cardinality. Otherwise it is finite.
Which is it?
But you haven't answered my question!
Then by your earlier definition has finite cardinality, right? Is that what you mean?
Actually NO, let's not shortcut things (this is rarely good hygiene in mathematics). What is AlephZero ()? You haven't told us.Lets shortcut things by saying that the Natural numbers have the Cardinality of AlephZero
Is this different from the cardinality of an arbitrary subset of always or sometimes? Is the cardinality of a set some sort of number? What sort of number is it?
Is this the "same" cardinality as that of the reals , or is it somehow different? Indeed, are they related in any way (even by identity)?
PS by edit: I am not asking for unsupported assertions (which anyone can find by internet trawling) but arguments based on mathematical logic
Last edited by Guitarist; August 12th, 2012 at 10:06 AM.
I learned that a set was countably infinite if there exists a bijection from it to the natural numbers? Was that wrong?
Of course it's not wrong  it is exactly right!
As an aside, I would prefer not to use the term "bijection" in this context, rather say a onetoone correspondence,but that is a matter of personal taste only, and not important.
What I was trying to draw out of SigurdW is that a set is countable iff it can be placed in onetoone correspondence with a subset of the natural numbers. And since any set is by definition a subset of itself, then is countable and yet infinite. Hence the existence, as you say of countably infinite sets.
The thm our friend quotes (actually a corollary to a larger thm) refers ONLY to uncountable sets, as I argued earlier. It is important to unerstand the difference.
Hey! Ive been busy but I WAS going to answer. Your honest and serious questions showed ability and how could I refuse to participate?
At the moment theres some silly objections I have to take care of in:The Poor Claim That God Does Not Exist
Then theres more serious matters in: Is the age of the universe the same in all frames? This place, im afraid comes third but,hmmm, looking at the participators this thread might
catch my attention.
Edit 1
1 everything has a cause
2 There is at least one thing
3 Therefore there are many causes
4 Consider the set "all causes" and the set "each other cause"
5 The two can be put into one into one correspondence
6 Any set that can correspond to either is (edit2: at least) countable
(This was done in a spur of a moment and should therefore be fallacious,
but I felt it to be impolite to ignore guitarist any longer.)
Last edited by sigurdW; August 12th, 2012 at 04:08 PM.
I'm a bit confused here, a set is infinite if it can be put in a onetoone correspondence with a proper subset of itself. So why the fight?
Ahem... then of course I resisted doing so, but its awkward: Using instead the concept of cause Im perhaps inviting the cardinality of C? There possibly are causes between any two causes ordered by the principle of "cause and effect". And there are more problems coming...(But Ill die fighting!)
This does not work.
We do not have uncountable systems of logic. The best for you is countable formulas that are infinite to describe your system of causality.
Remember, causality must be describable and the reals have transcendental numbers that are not describable by any language or algebraic formulas.
So, the set of all causes cannot be mapped to the reals.
Transcendental numbers can definitely be described mathematically, just not via algebraic terms. At least, the well studied ones.
Thanks river_rat. I'm aware of the context. I was just throwing in my two cents on that specific point concerning derivations for transcendental numbers.
That woke me up! Let me think: Suppose every statement can produce two statements such that none of the statements are identical then you can place them as corners of a triangle? If you can similarly fill in all statements you can see them as fitting a list of all reals in the interval between 0.1 and 0? If there is a way to give every statement a number ,say 1 or 0, then the sets are isomorphic having the same cardinality? And there are as many statements as there are real numbers?
Of course theres some mistake made when you construct something impulsively out of thin air just like that, but it would be nice to understand the idea behind the proof that statements are at most countable. You wont perhaps tell?
Good.I want full knowledge of any transcendental number and write in down in any language such that all on this planet will agree for all n and we are able to fully understand all n.
In other words, I want to know all possible decimal digits of this transcendental number. I do not want only a finite number of the decimal digits, I want all of them.
You cannot escape countability with formulas.
Formulas have rules. You can claim to associate a formula with any real and then claim you have proven they are mapped to the reals, but that is vacuous implication and proves nothing.
You must first prove you have an uncountable set of formulas, to map them to the reals.
However, formulas are defined based on recursion and all recursive output is countable.
So, that is impossible.
What is an uncountable system of logic?
An infinite argument? An argument must have a beginning and an end(=conclusion)
but that does not force it to not be uncountable.
Causality? Is that the theory of causes? I see causes as undefined objects.
Also: Im suspicious of the concept "describable"...
Some predicates has the property: IF (x=Px) THEN (Px=PPx)
Obviously the right side being false forces the left side being false as well.
Such predicates must be handled with utmost care,
you run the risk of producing a paradox otherwise.
Exercise: Check out (P = "not true")...(yes I made it up myself)
If the set of all causes is of the same cardinality as the set of points on a segment of a line then they can be so mapped: All thats needed (it seems to this layman), is for any two different causes to have at least one different cause between them.
Sorry I let you wait but I sort of forgot the existence of this thread containing interesting people.
What is an uncountable system of logic?
There is no such thing.
Logic is defined recursively and the best you can claim is that formulas are countable.
A collection of formulas define your logic system commonly called a theory.
We do not have any uncountable system of logic so we cannot completely describe elements of the reals.
Therefore, we cannot completely describe causality if it is indeed uncountable.
For example, a planet is on a path of causality. It's path is defined in terms of the reals.
But, we only have calculus to describe it's path based on small finite sections. We cannot completely describe its path in terms of the reals because we would need to be able to list all the reals in order to completely describe its path, which is a contradiction.
As a rough sketch consider this counting argument:
Let be the set of all symbols we could use to build words to represent ideas. This set is countable. This is implies that (the set of all finite words / sentences) you can make to represent ideas is countable. This is because the set of all finite subsets of a given set has the same cardinality as the original set.
Isn't infinity beyond finite definitions?
Why not watch the video?
Or, I think Guitarist had a thread a while ago where he provided the formal definition.
Informally, and very briefly, one infinity is the number of natural numbers (strictly speaking the cardinality of the set of natural numbers), which is the same as the number of odd numbers, integers and any other countable infinite set. Another infinity is the cardinality of the set of real numbers. There are an infinite number of infinities.
OK?
I'm not sure what means. How do define "infinite" and "endless"? Note, we are talking about "infinity", the mathematical concept not some abstract philosophical (and therefore meaningless) "infinite".
Correct. But the size (cardinality) of a set of numbers can equal infinity. Another way of putting this is that there is no largest integer (i.e. you always add 1 to any number). Infinity is not a member of that set of numbers.No set of numbers equal infinity.
Correct again. The set of odd integers is different from the set of squares of natural numbers. But they both have the same size (cardinality, represented as , aleph null). On the other hand, the set of reals is different again and has a different, infinite, size).There is no one infinite set of numbers.
Incorrect. What do you base that on?To be a set means that there is a capp, an end, a limit.
The set of integers is countably infinite, the set of reals is uncountably infinite.Infinite is uncountable.
End less meaning no end. infinite meaning no end.
I wasn't talking about it in a specific mathematical context or some abstract philosophical context... just in the context of the english word infinite.
Maybe I did misinterpret 'set', im not sure. It seemed to me that a set is something that is complete, which is why i said it's capped.
Tut Tut strange! a loop is a finite thing!
If your very informal explanaition of the definition of infinity is anything to go by then i might save looking at the video until after i have learnt all the stuff that interests me... Like in an almost infinite amount of lifetimes from now.
So you have never done any computer programming, then.
I really, really (*) recommend reading about Cantor's wok on infinity. It is fascinating and involves some of the simplest but most profound mathematical proofs since Euclid's proof that there are an infinite (oh, there it is again) number of primes.If your very informal explanaition of the definition of infinity is anything to go by then i might save looking at the video until after i have learnt all the stuff that interests me... Like in an almost infinite amount of lifetimes from now.
If that is what the video is about, it is probably a good starting point.
(*) No, really. I mean really, really, really ...
Hoyle has my symphaty
but CANTOR
is one of the greatest heroes of all times!
I only want to add that it can be proved that a set containing all points between two different points in a line is not countable, and the same for points inside a circle in a plane and so on adding real dimensions.
And a word to: question for you if you want someone to prove something for you its a good idea to produce a proof of your own to show him you understand the hard work needed to produce a proof. Lacking that I think you should ask where the proof can be found. Best wishes! Your Friendly: sigurdW
Cantor! Not math gave us the definition!
It began with Galileo...he said:
Compare the natural numbers to the even numbers:
X: 1 2 3 4
2X: 2 4 6 8
Shouldnt there be more natural numbers than there are even numbers?
But to each and every natural number there corresponds one and only one even number,
Shouldnt it rather then be so that there are as many even numbers as natural numbers?
This is a paradox,G, said. and therefore there cannot be infinitely many numbers!
There is no infinity.
Everybody said: Good proof! But we already knew and understand this.
CANTOR said: Excuse me but thats just the way things are!
No paradox at all!
The natural numbers constitute the simplest infinity there is.
What Galileo did
(without understanding that THAT was what he did)
was to define infinity!
So from what I've read, there exists an infinite number of infinite cardinalities. Does this mean there are an infinite number of differentlydefined sets?
Do you think I made it up?!
Well I didnt, and a suggest we make a bet!
I will admit to anything you want me to admit
if I cannot tell you where I read this.
(Im usually too lazy to look up what people say:
I say: let them say it to my face...I dont go anywhere.)
But if you loose you will never ever bother me again! OK?
No I don't think you made it up.
I'm not interested where you got it from. I'm not exactly frilled by the prospect of getting you to admit anything.
If you don't understand what you're quoting then it's a waste of both our time discussing it.
If you don't want me to 'bother you again' then I suggest you don't misquote me and give friendly advice that I have shown no need for.
The nerve!
You would be unwise to accept the bet, before I cited from memory now I quote verbatim Galileo says:
"we can only infer that the totality of all numberis infinite, and the numbers of"
now I show a weakness: since My memory isnt eidetic I remember Galileos "squares" as "even numbers"
You are intelligent enough to not claim I intended this arent you?
Lets go on quoting The Master Galileo himself:
", squares is infinite...;neither is the number of squares less than the totality of all numbers,nor the latter greater than the former;and finally, the attributes "equal," "greater," and "less," are not applicable to infinite,but only to finite quantities"
Galileo was clear about what he meant wasnt he?
Edit:
PS Perhaps I misunderstood, if so I apologize!
But just coming from an attempt to discredit me by misquoting me,
I shot instead of answering politely.
The following remark made me explode:thats not a paradox, did galileo really think it was?
I am calmer now and no longer believe you were intentionally insulting me:
Yes, as you can see for yourself IF you wish,
Galileo believed he had proved the concept of infinity to be inconsistent.
Historically it has been referred to as Galileos Paradox,
but whether G himself used the term I cant tell.
Galileo was clear about what he meant wasnt he?
And Cantor showed that Galileo actually did the opposite of what he thought he had done!
Take away Cantor from history and we would perhaps still not believe in infinity.
Last edited by sigurdW; August 17th, 2012 at 04:56 PM.
All this logic is derived from the rank function.
In the metalanguage Kunen claims one can union all these infinities and does so with some of his proofs.
However, there is no set of all ordinals and no union of all ordinals is permitted in the formal language since that invokes the BuraliForti contradiction.
Suffice it to say in the current mainstream, you can create larger and larger infinities but you cannot know all infinities.
I know this is not satisfying, but that is the game.
I am only telling you the mainstream in mathematics. What I posted are the facts of the belief of the mainstream.
You can find it at wiki under
The cumulative hierarchy
Implementation of mathematics in set theory  Wikipedia, the free encyclopedia
I also checked it. Its good but since my...ahem...virtual? ... expertise?... is on the non existing? ... general theory of foundations, I never bother to learn any particular notation (except the introduction part) Im not sure how far set theory in wiki reaches above the extendible cardinals. (I would like to know.)
Thank you,that was an unecessarily kind thing to say.
(FIRST:I take the opportunity to demonstrate the strengh of my intuition: Had I remembered the facts correctly using squares instead of even numbers as the second set in the comparisation then the following funny pythagorean scenario would not later occur to me.)
So far so good, there really is nothing wrong in trying to understand what a master in the subject thinks about it. but you should move from the galileo position because it was proven to be wrong by Georg Cantor... another master actually still today having the last word in the matter! I both love and hate him: To prove him wrong is something I never really tried because I see no place to start!( Thats why I hate him )
I say galileos intrepretation of his own words were wrong...do you understand what I mean?
He begins with equality. When are sets equal? Its like a dance floor,there are as many boys as girls if nobody is not dancing! (Supposing dancing is a boy paired with girl business)
Nothing wrong so far with galileos work.
Next (and keep in mind that no fact is true from the point of view of real history, we only need to understand concepts here, we are not historians) Galileo notices that if odd numbers are boys and even numbers are girls then everybody is dancing. There are as many boys as girls. This is the fact and cantor pats galileo encouraging on the shoulder and says And yet you think there is not an infinity why? Galileo says: But if I take the clothes off all them girls and put the clothes of each girl in neat pile beside her and then let the boys and the girls unite...(g looks at c: ure not thinking what im thinking are u?) THEN surely there cant be as many piles as there are girls AND boys!
Pats cantor on his head: I suspect you dont understand arithmethics very well my dear little cantor.
Lets stop cantor from answering, and try to figure it out ourselves! (BRB)
PS let me say what comes into mind without precaution as on a teraphy sofa:
copyright the master of trolls
Last edited by sigurdW; August 17th, 2012 at 09:23 PM.
A thanks to river_rat and chinglu for answering my question. Very surprising. I would quote, but it looks like this thread's getting pretty cluttered already!
Wow! You employ sarcasm at expert level!
If you hadnt shown your cards this early Id lost the game! Burrr!
Ok I admit only of a slight surprise. So you actually sort of have a split vision:Ive sort of defending galileos view so convincingly that one should be accepting his view, but as now "know" he must be wrong but how on Earth can that be possible?(sort of)
You trust me AND all them guys repeating this old story ... eh...Yes that is what I should mean, then I should point out that however and wherever you look, you wont see a copy of the story you inspired me to write for you as a sign of friendship
PS Thanks for not letting me down. How about stagediving together next time?
Yours truly Fred Hoyle.
PPS you should copy since your own copy of my text was in an early stage the word "god" is replaced with what my subconscious surely knew id put there. Else you run the risk of me editing out this post altogether soon.
Last edited by sigurdW; August 17th, 2012 at 11:30 PM.
Yes sarcasm is easy... but remember it is only the lowest form of wit.
Did I let you down somehow sigurd? I am sorry if this is the case... I'm not much of a skydiver, tell you what, you go sky diving and i'll pack your shoot ok partner?
The thing i found genuinely disturbing was the boys and girls getting undressed on the dancefloor analogy... you genuinely are bonkers! which is a very attractive quality my friend.
I WAS going tosay: youre hired,
but my ... eh ... "intuition?"... reminded me
I have promised myself to accept only a genuine MASTER as my lawyer...
Check this thread out to get my point:
Is the age of the universe the same in all frames?
You there explain to me why I may continue speaking after I solemny swear to shut up!
Do the following please:
Copy the the dancefloor analogy, visit me in above thread, post it and ask me if I deny making it up...
hide your true feelings at first, make it possible that are joining the warriors storming my castle!
Eh...first convince yourself youre not cheating!
Remember you are reading from a manuscript...you are NOT trying to cheat!
You may explain anything you wish at any time you wish.Ok??
If you need some more advice from a true friend of yours I will EDIT in here!
(I said that for the last time: I believe the truth of that statement cannot change at any point in existence.)
Oh! a last word
yes I make it up as I go. I dont PLAN in advance , I use a scenario trusting my ability to improvise.
Last edited by sigurdW; August 18th, 2012 at 08:48 AM.
I'm not sure what you're game is sigurd! but morbid curiosity dictates I will play along. See you in the other thread.
I have no idea what is going on in this thread anymore... let's please bring the sanity back. Okay, another question.
Well that's different than what I was expecting. It sounds like there's an infinity to infinity (I know that's not valid in the formality of standard math, but is that sort of the idea? I'm sure this subject of cardinality sets is welldefined with no loose ends, right?)Originally Posted by river_rat
Is that your attempt at bringing it back to sanity? really?
Infinite is infinite, there cannot be any larger or any smaller infinites, it's just simply infinite. I said earlier in the thread i didn't beleive a set could be infinite... I was told i was wrong repeatedly.
According to river rat i was right... not bad for somebody who knows nothing about maths... all i know is the 3 r's.
Theres a lot of over complicated ideas and concepts on here that are blatently nonesense disguised as intelletualism. Not this thread in particular.
Forgive me but it seems to me as a layman that the vein of nonesensical thoughts and pursuits run deep through the disciplines of science and math. That's not meant as a smear on science or maths, just some of the people who are into it.
And you are wrong. It is pretty simple to prove that, for example, the number of (infinite) integers is not the same as the number of (infinite) reals. And from here, that there are an infinite number of infinities.
Now if only someone would create a video with a simple explanation ....
question for you, I don't understand where you are coming from. Why did you respond to me? Are you against me for some reason?
It also seems you're against the concept of multiple infinities. Why is that? It is proven mathematically. This is Cantor's work, and today it is real, standard mathematics. I'm not aware of any mathematician that is against the idea.
So the standard and established mathematics of today, which is agreed on by every mathematician who has spent years professionally studying the subject, is nonsense.
I mean no offense to you, but you are being closedminded. You are making an argument from ignorance. You don't understand it, you cannot prove it, and you don't know how it is done... therefore, it must be nonsense?
Just watch the video. It's as simple a demonstration as it gets.
What is the point. You seem to be happy remaining ignorant.
OK. There are two possibilities. There are an infinite number of integers or a finite number. If there is a finite number, then you could tell me what the largest integer is. But one of the rules defining the integers is that I can add 1. Therefore you have not told me the largest integer and, in fact, you always add 1.you cannot count an infinite number of integers
Therefore the number of integers is infinite. You can do the same for real numbers.
Now the clever bit is "Cantor's diagonal argument" where he showed that if you do any 1 to 1 mapping between the integers and the real numbers, it is always possible to insert another real number. Therefore there are more reals than integers. I'm not going to copy out the whole proof, you can look it up if you are interested (which, apparently, you aren't).
You're simply stating the obvious epi!
Oh so strange was being sarcastic when he said 'if only somebody could make a video to explain it'? I see, I see! went over my head that!
Tell you what... I promise that if i see this thread next time i return then i will watch the video... after i have watched it and understood it, I will get back to this thread and tell you what I think.
I am very intrigued to see if this is some mind boggling BS or whether it actually has some truth in it...
I hope we'r working with the same definitiomn of infinite? which means... endless, beyond counting?
P.S the standard maths of every other 'day' in our history has turned out to be wrong, much the same with scientific theories.... whats the statistical probability that this accepted maths is absolutely correct?
See you back here soon
Let's go to an even simpler simpler example.
There an infinite number of integers. Half of them are odd, and half of them are even. There are an infinite number of odd integers and an infinite number of even integers, and both those infinities are smaller than the infinite number of integers.
No?
Exactly, there is no "number" equal to infinity.
Really? Got an example? Maths is the only field where things can be absolutely proven.P.S the standard maths of every other 'day' in our history has turned out to be wrong, much the same with scientific theories....
Watch out speedy (may I call you that?), I believe the integers can be put in a onetoone correspondence with both the odds and the evens, meaning all three sets have the same cardinality. It's when you compare any set of aleph_zero with the reals that you see it's impossible to make such a correspondence. This is where we really see that the two "infinites" are not of comparable size, in simplistic terms.
Well, I admit I don't really understand this stuff.
And you can call me what you like.
The subject itself is quite well defined but it has many lose ends, so to speak. The main problem is that the standard way of doing mathematics leaves many questions about the sizes of sets and the relationships between them unanswerable. In short, if you want to know more you have to assume more. So there are a whole host of extra axioms you can add to the standard set of axioms for axiomatic set theory  with the continuum hypothesis being the most famous.
You actually caught speedfreek in making an error...
so he turns to a poor excuse and an Ad Hominem.
Its encouraging to see that somebody checks arguments.
I only wish there were more of you ;(
PS Speedfreek is no fool, he perhaps meant:smaller than the infinite number of reals.
Thanks for clearing that up sig...
I often feel that comments i make are being argued with because they have'nt been read properly... and also that arguments i make are right, yet not excepted, meaning the person has made an incorrect statement... hence the confusion.
Im not sure if that sentence made any sense but i tried.
Ah, I see. And adding more assumptions only opens more doors which leads to more questions. I'm a layman for the realm of set theory axioms, but from I've read, it seems like some hypotheses are unprovable in the standard system of axioms but can possibly be proven in a different system, correct?
Yes, I did. And he did make an error, and there's nothing wrong with that. He also finely admitted to it.
What I fail to see is why you're saying he turned to a poor excuse and used an ad hominem. Where and when did he do such thing? I see no problem at all. Do you?Originally Posted by SpeedFreek
Infinity is a sum of what? how many variables, and how uesful?
when infinity is reached, what then?
infinity is greater than the actual addition?
we make "what" when we seek an infinite?
Happy to oblige
Youve changed your mind on the largeness of the reals havent you?
The problem has its origin in the existence of points I think.
Between any two points there is a point:
In contrast to natural numbers...
having no natural number between two consecutive numbers.
Compare to the rational numbers,
they also have a rational number between any two rational numbers,
just like points but there is a difference:
You can marry rationals to naturals, but neither can marry the points! WHY?
I notice that any rational need but two naturals to exist,
and a natural (except the first) needs a preceeding natural...
But WHAT DOES A POINT NEED TO EXIST?
Can a universe consist of nothing but an individual point?
Infinity really isnt seen as a sum... its seen as an ordinal.
But I guess it can be seen as the sum of all naturals the point being that the sum cant ITSELF be a natural!
This fact : That there are totalities that are of another KIND than any part of the totality is surprising:
(If theres no traditional name lets be hasty and call it the principle of Q.)
Maybe you need to get your eyes checked.
The first sounds like a perfectly reasonable excuse and an honest admission. Why would you assume he is lying? Are you judging by your own standards?
The second was a response to "may I call you speedy". Even if it weren't I cannot think of any possible interpretation that makes it an insult.
I don't understand why you say that. He admits he doesn't really understand this stuff. What is wrong with that?
You likely misunderstood the context.the second is an Ad Hominem.
> When I responded to SpeedFreek, I called him "Speedy".
> I asked, "May I call you that?"
> He said, "You can call me what you like"
Do you still think it's an ad hominem?
Well, I do. You accused SpeedFreek of wrongdoing when no such thing happened. Mind you that SpeedFreek is a kind and wellrespected member of the forum and is probably 10x smarter than me, especially in physics. If anything, he deserves even more respect for admitting to his error. You made an error also. But since it was just a misunderstanding of context, we can forget about this and move on... I hope.
On I side note, I'll say this sigurdW. You're statements and posting style on this forum are suspicious to me. Thanks to your questionable and unnecessary actions, you look like a blatant troll. And I do not want to be taken as a fool. Are you a troll?
In the definition of set theory, infinite is defined as not finite.
So, this leaves open a large collection of ideas for the infinite.
However, the foundation of mathematics postulates there is a set that contains all natural numbers.
This is called the axiom of infinity.
Axiom of infinity  Wikipedia, the free encyclopedia
This is the starting point of infinity for logic in mankind.
Now, I can stimulate your curiosity.
If there is no largest natural number, how can you create all natural numbers in any formal reasoning?
Sorry if I contradict but...
He might, perhaps, possibly mean something such as:
1 if x is a natural number then (x+1) is a natural number.
Then he "creates" them all by saying:
2 "1" is the name of an existing natural number.
In mathemathics theres is a concept of existence.
Its often kept separate from "physical" existence.
So maybe,maybe,maybe indeed: There are separated levels of existence!
If you say that something does not exist then you are saying that
its not to be found in YOUR frame of existence?
Existence is relative?
There is no absolute frame of existence?
PS:Now there is an alternative formulation
of my infamous definition of god?
:God is the absolute frame of existence.
Last edited by sigurdW; August 20th, 2012 at 03:09 AM.
Its the proverbial pandora's box actually. We have very little intuition to guide us here, which makes these problems very difficult. It is custom to state any axioms you are assuming beyond ZFC (and sometimes to state if you are using the axiom of choice as well) for any result, which is peculiar if you think about it but that is where mathematics as a disciple is now a days.
To address your question about proving statements in other systems we can typically assume a stronger axiom and thus derive the weaker one. For example, if we assume ZF and the axiom of choice we can show the axiom of countable choice is true. But ZF and the axiom of countable choice is not enough to show that the axiom of choice is true for example. Similarly, Martin's axiom (which basically says that any subset of the real line with cardinality less than the continuum behaves as if it was countable) is implied by the continuum hypothesis (trivially since you are either countable or the continuum in this case) but Martin's axiom can be true with the Continuum Hypothesis assumed false.
Or in layman's terms, that every subset of the reals is either countable or can be put into onetoone correspondence with the entire real line.
There are some basic limits you can put on (like it cannot be an inaccessible cardinal) and its cofinality must be uncountable but other than that I do not know any other limitations on the number of cardinals we can legitimately slot between the countable and the continuum.
Interest is in the eye of the beholder here I guess, if you are studying point set topology or the topology of these results are quite important.
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