i apologize for my lack of math knowledge, but how come there's countable infinity for natural numbers, uncountable infinity for real numbers, but nothing inbetween? like what type of properties are there for a set that is less countable than natural numbers, but is more countable than reals? is it even possible?
>>7723606
We don't know if its possible. The contiumm hypothesis is still unsolved.
>>7723610
has there been any insight as to what type of criterion is needed for it to even be possible?
>>7723606
Well, it's either countable or it's not. Sure, there are some countable infinities that are larger than others-- For instance, there are countably infinite primes, but obviously the set of all natural numbers is more dense.
reals simply aren't countable.
Why aren't integers countable?
>>7723614
wait so there's no scale for countability? it's just a matter of density?
>>7723613
Idk maybe some other anons could tell you but from what I've read it really seems like allot of people just accept the contiumm hypothesis's independence from ZFC as proof that its just unsolvable and it doesn't seem to be a major area of research anymore.
So we've kind of given up.
>>7723615
why wouldn't they be? couldn't you just start at 0 then go 1,-1,2,-2,n,-n? you would keep getting closer to infinity rather than being trapped like with the reals, where you can't really go past 0 because, what's the closest real number to 0?
>>7723615
They are, the reals aren't. Unless you're wildberger then the reals just don't exist.
>>7723634
That's what I'm asking. I don't see how Q isn't countable if N is. OP said we don't know if they are, so Im asking why.
>>7723613
>>7723633
it's not a matter of accepting. it has been proven absolutely that CH is independent of ZF and choice. this means that you can have your naive set theory and it is both compatible with a new axiom asserting the existence of a set strictly between them, as well as a different axiom saying there isn't. and in fact same goes for the generalized version, which says that the power set of x infinite is always the smallest Cardinal above x.
>>7723644
N is countable, Q is countable by diagonal argument, and I suggest you find your own bijection to show that Z is countable, as it's good practice.
>>7723654
i'm not accustomed to mathematical logic, what do you mean by independent?
>>7723630
If they're countable, they're countable. If they aren't, they're not.
I don't quite understand your question. Countable is a qualitative label. Whether two countably infinite sets are different doesn't change that they're both countable. If you're looking for some third label between countable and uncountable, then that doesn't really make sense, since it's either one or the other.
>>7723662
It means that neither is derivable from the other. they also do not contradict each other. being a cat is independent from being brown or black; I can choose one and then work in a black or brown cat system. they will be different cat systems, but equivalent insofar as we make no use of its black or brown ness.
>>7723667
but of course differ as they make use of the new black or brown axiom
>>7723654
I know its not a a matter of "accepting" its independence that's not what I was saying. The inpendance is proven fact I know that.
>>7723662
Basically it means ZFC can be true and the continuum hypothesis can be true or ZFC can be true and the continuum hypothesis can be false. Either is possible and leads to no inconsistencies.
>>7723663
is that the same for cardinality?
>>7723662
It cannot be determined just from those axioms. I can construct two systems of set theory that both satisfy the ZFC axioms but one has a true continuum hypothesis and the other, a false one.
>>7723675
I just took issue with saying that it has been accepted as unsolvable. it's not, we have an answer. both the affirmative and negative are compatible with ZFC. you can choose to set up new axioms that have it as a consequence, but that wasn't really the original question.
>>7723681
Yes, even though it's counterintuitive.
Obviously integers are going to be more common than primes, but that's the thing about infinity. Since there are infinitely many of each, there exists a prime that can be mapped to every integer.
Map the first prime to the first integer, nth prime to the nth integer, and even though the integer you're mapping it to is smaller than the prime (for n>3), you'll never run out of primes. For every n, there's an nth prime, no matter how large.
That's what it means to be countably infinite. Reals are not countably infinite, because you can't map every real to an element in a countably infinite set.
i still don't understand. so it's independent from ZFC, meaning its possibility isn't reliant on whether or not you follow the axioms? does this mean it IS possible?
>>7723614
dont listen to this guy, countability is not qualitative.
The set of natural numbers and prime numbers are the same size
>>7723697
It means that in terms of our standard set theory, CH doesn't mean anything because you can't prove it, its a choice. If you use CH in a proof, then its not really a proof in ZFC. If you use the negation of CH in a proof, then its not really a proof in ZFC.
The only proofs in ZFC are things that use its axioms and theorems proven from those axioms (and theorems proven from using theorems)
>>7723708
so you can't have standard set theory and a proof or disproof of CH? what would have to be changed in ZFC for it to be dependent?
>>7723693
Well there could be some other system where it can be proven true or false. Gödel thought we needed a new system, he believed we could do better than "its independent, end of story".
But is trying to surpass ZFC with a new model a major area of mathematical research now? Idk. From what I read on here most anons says modern mathematicians don't give a shit about this stuff anymore.
>>7723712
When I say standard set theory I mean ZFC + AC.
If you added anything to ZFC+AC and were able to prove or disprove CH, then those new additions would actually just be equivalent to CH, so you wouldn't make new axioms, you'd just be adding in CH.
What I think you are saying is "what kind of axioms or requirements would we have to add to ZFC to make CH true or false" and the answer is "CH is true" or "CH is false" anything else can be shown to be directly equivalent.
>>7723734
>If you added anything to ZFC+AC and were able to prove or disprove CH, then those new additions would actually just be equivalent to CH, so you wouldn't make new axioms, you'd just be adding in CH.
well damn. that is pretty fascinating. so the hypothesis becomes self evident?
i don't understand how one to one matching infinite sets can lead to sets of different sizes. i can't contend with the proofs but it's so counter intuitive.
>>7723606
I took an intro to proofs class this semester, and countability of sets was defined as follows:
Let A be a set
A is countable if and only if there is a bijection from the naturals onto A
From there you can prove that the naturals are countable by creating a suitable bijection
You can also show that there is no bijection from the naturals onto the rationals. We used a proof by contradiction.
>>7723757
>let A be a set
what if it isn't a set you shitlord
>>7723606
>how come there's countable infinity for natural numbers, uncountable infinity for real numbers, but nothing inbetween
Because V=L
>>7723734
>then those new additions would actually just be equivalent to CH, so you wouldn't make new axioms, you'd just be adding in CH
Not at all. The general continuum hypothesis is strictly stronger than the CH and the axiom of constructibility (V=L) is strictly stronger than GCH.
>>7723734
> If you added anything to ZFC+AC and were able to prove or disprove CH, then those new additions would actually just be equivalent to CH, so you wouldn't make new axioms, you'd just be adding in CH
err... this is false, you can add A such that A implies B and such that B doesn't imply A...
what you prove isn't equivalent to the set of axioms you use, it's only a consequence of it.
>>7723757
>You can also show that there is no bijection from the naturals onto the rationals
i think you mistake the rationals for the reals. The rationals are countable
>>7723789
he start his post talking about countability of sets... then he does... how can you attack that?
>>7723614
Infinity is always the same size: infinite. There's as many even numbers as there are natural numbers.
>>7723936
Yup. But there are more reals. There's countable infinite and uncountable infinity and we can't prove if there's anything between them or not.
>>7723830
Well yes, but I assumed he wanted something that minimally proved CH, and not something like V=L or its negation or one of those other large cardinal axioms like 0#
>>7724856
can't some proper subset of the reals, say the closest real to an integer, be segmented into a cardinal that's not equal to the set of reals?
