>The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers)
>proven to be impossible to prove or disprove within the Zermelo–Fraenkel set theory with or without the Axiom of Choice (provided the Zermelo–Fraenkel set theory with or without the Axiom of Choice is consistent, i.e., contains no two theorems such that one is a negation of the other).
Can you explain this plz for someone who doesnt know wtf this means? More specific - what is the Axiom of choice, I dont get it how it has anything to do with mathematics.
>>7747248
It's a really easy concept dude just read it more. Basically there is no set with a cardinality, k , such that aleph0<k<aleph1, this is abuse of notation though.
who wants to know
pay attention in class don't come to /sci/ to do your homework
The axiom of choice states that one can "choose" one element from each set in an arbitrary family of nonempty sets (that is, there exists a surjective function from the family to a collection of singletons, where each singleton is a subset of some set in the family of sets, such that each set is sent to a singleton contained by the set). Sounds obviously true, but gives rather strange results.
>>7747248
The sets of Naturals, Integers and Rationals are all the same kind of infinite.
The set of Reals are a different (larger) kind of infinite.
Continuum hypothesis: There is no set that is larger in size than the first group, and smaller than the second.
Axiom of Choice: You can always choose an element from a nonempty set even if you have no way of knowing how to chose it. Basically like a black box, but when you reach in you can always fish something out.
>>7748235
*also holds for infinite black boxes with no defined way of "reaching in"
>>7747248
>The continuum hypothesis (that is, there is no set whose cardinality is strictly between that of the integers and that of the real numbers)
Usually we think of the cardinality of a set as the "number of elements" it contains. This doesn't quite work for infinite sets so we give a more general definition. We say set |A| <= |B| if there is a 1-1 function from A to B.
(Think about two bags of lollies and putting all the lollies in the first bag in correspondence with the lollies in the second bag. If you exhaust all the lollies in the first bag then the first bag contains the same amount, or less, than the second bag.)
Then we can say that set A and set B are "the same size" if there are injections from A to B and from B to A.
From this basic definition, it eventually turns out that the set of Naturals and the set of Real numbers are not the same size (see: Cantor's diagonal proof, a proof by contradiction). So the Reals are strictly bigger than the Naturals, by this definition.
The Continuum hypothesis is that there is no "size" or "quantity" between the Reals and Naturals. Just like there's no whole number between 1 and 2.
>>7748239
(cont.)
You have to choose what truths to take as axiomatic before you can begin doing proofs within some mathematical model. Axioms are things assumed to be true "a priori". They're just kind of self-evident truths that form the backbone of the system.
You want any axiomatic system to be consistent, meaning that any statement in the model (formalised in first-order logic or whatever) can only be proven true or false, and not both or neither. If you can prove something is both true and false with your axioms then you're in a bit of trouble.
ZF is a set of axioms. ZFC is the set ZF with the axiom of choice, which basically says that for any nonempty set (finite or infinite) you are always able to choose an element from it. It is not known whether ZFC is a consistent set of axioms (meaning it's uncertain if you're able to prove things both true and false).
Now it turns out that, whether you're working in ZF, or you're working in ZFC and we assume ZFC is consistent (i.e. can't prove things are true and false as t there is no formal proof from the axioms which shows the Continuum hypothesis to be true or false.
>>7747248
Wouldn't the set of even reals be between reals and integers?
>>7748235
>You can always choose an element from a nonempty set even if you have no way of knowing how to chose it.
No. This is part of ZF actually. If A is nonempty, there exists x in A. This is not choice.
Choice is that for every collection of sets, you can make a new set which contains exactly one item from each of them.
>>7748537
No, it's just as big as the reals. What you need to do is (assuming axiom of choice):
Exhibit a set S such that N < S and S < R where < denotes that there exists an injection but not a bijection
>>7748549
I'm struggling to understand how there is a 1-to-1 mapping of even-reals to reals but not with rationals to reals.
Obviously there are more reals than rationals, but I would say too there are 'clearly' more reals than just the even reals.
>tfw forever math pleb
>>7748596
It's not something you should try to just come up with on the spot. Go and read a book with an introduction to set theory. You want to learn about what "countable" means, cantor's diagonal argument, etc.
>>7748597
Yeah nah fuck math and its autistic made-up rules masquerading as objective truth.
>>7748597
I'm starting Calc 2 next semester, I plan on taking Analysis after that... is it possible to get into set theory without already being competent in those? I'm not naturally good at math but set theory looks interesting
>>7748607
>look mom i'm trolling
>>7748629
It's true though. Every other field has the intellectual honesty to admit that their work is a human product and nothing more.
You won't see a programmer saying that the C programming language is an inherent fact of the universe.
You won't hear a physicist denying that we invented relativity theory and quantum mechanics.
But the mathematician is a special kind of retard, for he thinks he is -discovering- rather than inventing. And he will proudly tell this to everyone else also, even though it is blatantly false.
>>7748655
>All mathematicians are realists
Most mathematicians are platonists, and the realist position's not far fetched at all t.b.h, Things are because they are, and you're finding out which things are based on unquestionable simple facts. Shitpost somewhere else.
>>7748660
>Most mathematicians are platonists
This is what I was ragging on.
>>7748660
>realists
And on second thoughts I would rag on these as well as platonists. Numbers don't real.
The "set of even reals" would just be the set of even integers, in which case the set of even integers and the set of integers is the "same size" .
>>7748596
>I'm struggling to understand how there is a 1-to-1 mapping of even-reals to reals
There isn't one, like I've already said the "even reals" are just the even integers, if there was then since [math] \mathbb{ R } \sim \mathcal{ P } ( \mathbb{ Z}^{+} ) [/math] and [math] \mathbb{ Z }^{E} \sim \mathbb{ Z }^{+} [math] then [math] \mathbb{ Z }^{E} \sim \mathbb{ Z }^{+} \sim \mathbb{ R } \sim \mathcal{ P } ( \mathbb{ Z}^{+} ) [/math] but then [math] \mathbb{ Z }^{+} \sim \mathcal{ P } ( \mathbb{ Z}^{+} ) [/math]. But we know from the most general statement of Cantor's theorem That is no set, A, for which it is true that [math] A \sim \mathcal{ P } (A) [/math], so we have a contradiction. Thus we conclude that there is no bijection between the even integers and the reals.
>>7748709
There are more interpretations.
Formalism/fictionalism would be more in line with my thinking than either realism or platonism.
>>7748726
>>He believes in factionalism
>Hahahahahahahahahaha!
You know those religious people who argue about what their made up deity thinks on some matter?
That's basically how I see mathematicians.
>>7748733
Further proof, as if it was needed, that you aren't that bright.
>>7748749
Look man it's cool, I like fiction too, just could you please stop pretending it's real?
>>7748758
Actually I'd say that embodied mind theories is probably a closer approximation to my views on math than either formalism or fictionalism
>>7747325
>aleph0<k<aleph1
No no no. There is no cardinality k such that "aleph0 < k < aleph1" by definition. Aleph1 is not the cardinality of the real numbers if the CH fails.
The continuum hypothesis states that there is no cardinality k such that "aleph0 < k< 2^aleph0" .
>>7748534
Consistent doesn't mean you necessarily can either prove a statement A true or false. That would be completeness.
>>7747248
A set is smaller than another one, if you can embed the first into the second but not the other way around. This is more formally defined using injective functions.
If [math] {\mathbb N} [/math] denotes the natural numbers (let’s rather write [math] \omega_0 [/math] for it here) and [math] {\mathbb R} [/math] denotes the real numbers, then you can find an injection from the former set into the latter. This means there is a function [math] i : \omega_0 \to {\mathbb R} [/math] so that for any distinct two [math] n,m \in \omega_0 [/math] you have [math] i(n) \neq i(m) [/math]. For example [math] i(n) := e^n [/math] does the job.
However, Cantor proves that you can’t find such an injective function in [math] {\mathbb R} \to \omega_0 [/math]. So the set of reals is bigger than the set of natural numbers.
Btw., note that the space of functions [math] \omega_0 \to {\mathbb R} [/math] can be represented internally to a category of sets. I.e. there are representations of this collection of functions from [math]\omega_0[/math] to [math] {\mathbb R} [/math] as a set, and this set (up to bijection) is written [math] {\mathbb R}^{\omega_0} [/math].
If [math] \{0,1\} [/math] is some set with only two elements, Cantor also shows that even [math]\{0,1\}^{\omega_0}[/math] is bigger than [math]\omega_0[/math]
Given that we have a notion of bigger now, call [math]\omega_1 [/math] the next big infinite set after the naturals and [math]\omega_2[/math] the set after that one.
(cont. >>7748952)
Let’s now consider a large category of sets (or classes, or model, or however you want to call those collections) which obeys the laws of our standard theory of sets (ZFC). There is an object in this category which you denote by [math]\{0,1\}^{\omega_0}[/math], but what this thing actually contains contains depends on how you stuff it. The only rule is that it’s content doesn’t violate the laws of ZFC, but this still leaves some freedom.
It’s now possible to find a category where the sets called [math]\{0,1\}^{\omega_0}[/math] and [math]\omega_1[/math] are in bijection.
(By the standard construction of the real numbers from [math]\{0,1\}^{\omega_0}[/math], this also means the reals are the next big things after the naturals.)
BUT in the 60’s the mathematician Cohen showed you can define a category of sets that simultaneously fulfills the laws of ZFC, but where the sets are such that it’s possible to find an injective function from [math]\omega_2[/math] to [math]\{0,1\}^{\omega_0}[/math]! This means taking [math]\omega_0[/math] to [math]\{0,1\}^{\omega_0}[/math] jumps in size over a whole kind of infinity, namely [math]\omega_1[/math].
Since the category obeyed ZFC, then if ZFC is a consistent theory, this implies that you can’t use the formal laws of the theory ZFC to prove [math]\{0,1\}^{\omega_0}[/math] and [math]\omega_1[/math] are in bijection. We say that the statement is independent of this theory of sets.
(cont. >>7748955)
Cohen invented a new technique to construct the above injection [math]\omega_2 \to \{0,1\}^{\omega_0}[/math]. It’s called „forcing“ and a since then big deal in logic. Basically, he first notes that
[math]\omega_2 \to \{0,1\}^{\omega_0}[/math]
is in bijection to a characteristic function
[math]\omega_2 \times {\omega_0} \to \{0,1\} [/math]
He uses ZFC to construct the big set of functions
[math] \{ f : X \to \{0,1\} \} [/math]
where X’s are finite subsets of [math]\omega_2 \times {\omega_0} [/math].
Next comes some order theory, and „filters“ where patches the f’s together and ends up with one huge function [math] F [/math], so that for different [math] \alpha, \beta \in omega_2 [/math], the functions
[math] n \mapsto F(\alpha, n) [/math]
and
[math] n \mapsto F(\beta, n) [/math]
are pairwise distinct. This then does the trick.
The continuum hypothesis being independ is somewhat ugly, I’d say. It implies, amongst other things, that from ZFC you can’t prove that given a big and a small set B and S, you cannot even prove that the number of subsets of B are more than the number of subsets of S.
So even if your axioms of a theory of sets are in-your-face strong, like ZFC is, it’s hard to capture what you’d like to be true for sets. [spoiler]>2015, using sets
>>7748701>>7748660
>platonists
>>7748995
Nevermind the shitposting in the −1/12 thread.
I'm having a lazy Christmas week.
Semirelated:
A friend just points out to me that there is a lot of free stuff here:
http://link.springer.com/search?facet-series=%22136%22&facet-content-type=%22Book%22&showAll=false
>>7748545
>Choice is that for every collection of sets, you can make a new set which contains exactly one item from each of them.
This sounds obviously true as well to my noob ears.
If we take a valid mathematical question that has (as of yet) no answer, and I think that even though has found the answer it does exist in some way, does that make me a platonist? I.e. is this the crux of the matter or is it something else?
>>7749027
*even though nobody has found the answer
>>7749027
No.
>>7749020
i dont understand, - lets say i have the empty set A and a set B with the number 3 and 4 in it, now i cant pick from A because there is nothing in it, I could pick A itself but A wasnt defined to be an element in itself ( A = {} != A={A} )
>>7749039
The set's are required to be non-empty.
>For any set X of nonempty sets, there exists a choice function f defined on X.
>>7749020
I have a bookshelf with 7 rows and all are full of books. Come up with a way of choosing a book from each row.
Okay, you might say "standing in front of the shelf, choose the left-most book of each shelf."
That's a good choice function, breh.
Also, for Christmas I got 7 gift boxes, each full of books. They send to me by mail and the boxes are totally made up. Come up with a way of choosing a book from each box.
Okay, you might say "Unbox each box, look at the first page and find out the day the books were printed. For each box, take the oldest book."
That's a good choice function, breh.
I've bot a closet with 7 drawers and each contains a pair of socks. They are the same. Come up with a way of choosing a sock from each drawers. Okay, if I'm not hard on you I'll let you pass with "Randomly grab into each drawer and take the sock of the two you feel first."
Okay, now I've got a bunch of sets and their elements all don't have any distinguishable property or at least relation to each other I can formulate you right now. Come up with a way of choosing a element from each set.
You can't. How would you, a priori. The axioms of a set theory define what a "set" is supposed to be in the first place. The axioms of choice, "there exists a choice function", is a requirement of sets ought to be. But opposed to the paring axiom, say, i.e. "If you're given a set X and a set Y, then {X,Y} is a set", the axiom of choice is not constructive. The pairing axiom has a programmatic equivalence, it's an axiom. The axiom of choice/well ordering/Zorns lemma is an existence claim. If you adopt it, you narrow down what a "set" should be - same is true if you adopt it's negation. If you are ambivalent to weather it's true or not, you can prove less. Many more categories of sets are models of that theory now. And e.g. you can't prove Banarch Tarski.
>>7749027
>>7749028
This question is too vague.
Firstly, there are
>valid mathematical questions
that provably have no answer. E.g.
>The mortal matrix problem: determining, given a finite set of n × n matrices with integer entries, whether they can be multiplied in some order, possibly with repetition, to yield the zero matrix. This is known to be undecidable for a set of six or more 3 × 3 matrices, or a set of two 15 × 15 matrices.[2]
https://en.wikipedia.org/wiki/List_of_undecidable_problems
Secondly, Platonism is much about if you believe that there are primal objects, which are the things theories try to talk about. If you just think that provable (in principle and from a completely formal theory, given axioms etc.) theorems do have a truth value assigned to them even thought no human has done that yes, this alone doesn’t make you a Platonist. I’d say almost all people who believe in a capital T truth believe this.
[spoiler]>2015, capital T truth
>>7749041
if i have the set of all natural numbers and i have countably many copies of that in another set S (S=N0,N1,N2...) and i take 1 element out of each set in S to generate new sets looking like 1,1,1,... and 2,2,2,... and now i add those sets to my previous set S, and S gets bigger everytime i add a new element - is S then still countable?
>>7749052
I'm not entirely sure what you're asking, but I think it's something like:
>If I have a collection, [math] \mathcal{ C } [/math] of countably many sets of the natural numbers (I'm not entirely possible if that's true in axiomatic set theory, but we'll continue any way) and then construct another set using elements of [math] \mathcal{ C } [/math], then add that set to [math] \mathcal{ C } [/math]. Is the resulting set also countable?
Honestly this sounds to me like "a union of countably many sets", in which case yes, it is countable. But someone else might want to give it a better analysis, I'm going out in a bit.
>>7749065
>If I have a collection, C of countably many sets of the natural numbers (I'm not entirely possible if that's true in axiomatic set theory, but we'll continue any way) and then construct another set using elements of C, then add that set to C. Is the resulting set also countable?
yes thats what im asking, thanks for the answer, I would also like to hear another opinion
>>7748627
Check out an introductory text on discrete math senpai
>>7749065
>>If I have a collection, C of countably many sets of the natural numbers (I'm not entirely possible if that's true in axiomatic set theory, but we'll continue any way) and then construct another set using elements of C, then add that set to C. Is the resulting set also countable?
oh sry its not exactly what im asking but its also a good question.
Here is what I meant:
If we have a collection C of countably many sets of the natural numbers and then construct another set C' using elements of C, then add that set to C' so that C'= C+1 and then we repeat this with C' ->
...We have a collection C' of countably many sets C +1 and then construct another set C'' using elements of C', then add that set to C'' so that C''= C'+1 , then we do this again...
...We have a collection C'' of countably many sets (C' +1) and then construct another set C''' using elements of C'', then add that set to C''' so that C'''= C''+1 , then we do this again...
repeat forever
Is C''''''... countable? I would say yes because I can number the recursive depth
>>7748534
>the axiom of choice, which basically says that for any nonempty set (finite or infinite) you are always able to choose an element from it
No, that's the definition of being nonempty. The axiom of choice states that for any family of nonempty sets, you can simultaneously choose an element from each set of the family.
>>7749441
That is, the Cartesian product of nonempty sets is nonempty.
