I Do you believe that continuum is Aleph-2, not Aleph-1?

[tzimie]

This negation of CH is based on Woodin's work: https://en.wikipedia.org/wiki/Ω-logic

Of course, you can only believe his result because you need to believe his axioms first. But for me it is really convincing for multiple reasons:

1. While it is, of course, a negation of CH, it does not really break everything because the sequence of Aleph numbers is preserved, just the names assigned to different alephs change
2. His conclusion is based on the quantification over possible forcings, and it looks really powerful - as forcing is used to prove independence of so many large cardinal axioms, so quantification over forcings must be extremely powerful. So it is like (in physics) expanding universe of sets into the multiverse!
3. \omega_1 is now less than continuum. And \omega_1, at least for me, looks intuitively "almost" countable, as the sequence of ordinals is explicitly well ordered. Of course, any set can be well ordered if we assume AC, but often no constructive example of such ordering can be provided.
4. Interestingly enough, Goedel himself had suspected that continuum = \aleph_2

I am Platonist, so for me it sounds more like a discovery. Not like a formal game (with this axiom we can do this, and with another we can do that). Do you feel the same?

 

[micromass]

Many set theorists believed that the continuum does not have cardinality $\omega_1$. This includes Gödel and Cohen. For example, in Herrlich's book on the axiom of choice, he calls the GCH something that is widely seen as something false.

You know, I don't believe in a single set theory. To me there are multiple ones, some which satisfy CH and some which don't. You can transfer between these set theories very easily using topos theory, which I think is the future of this kind of math.

 

[zinq]

[In what follows, we identify each cardinal with the least ordinal having that cardinality. Thus we may use cardinals as subscripts of alephs.]

Hugh Woodin told me about five years ago that he no longer believed his earlier work, that the continuum c satisfies

c = aleph₂;​

I don't know what he believes now.

Also, about 20 years ago I spoke with the late Paul Cohen, who said it was possible that c satisfies

c = aleph_c

(!). In fact, he pointed out that the ZFC axioms are consistent with the hypothetical axiom that

c = aleph_k

for any ordinal k whose cofinality cf(k) is not equal to aleph₀.

(Note: The cofinality cf(k) of any ordinal is the least ordinal having the order type of a cofinal subset of k. A cofinal subset X of an ordinal k is a subset such that for every element y ∈ k, there is an element x ∈ X such that y ≤ x.)

This observation, derived from Koenig's Theorem, implies for instance that it must be the case that

c ≠ aleph_(aleph0).

 

[pwsnafu]

You can transfer between these set theories very easily using topos theory, which I think is the future of this kind of math.

Yay, someone who actually likes topos theory. Everyone I talk to doesn't know it exists. Makes me happy.

 

[tzimie]

Yay, someone who actually likes topos theory. Everyone I talk to doesn't know it exists. Makes me happy.

I've seen links in Wiki to topos theory, but Wiki claims that this theory is strongly inaccessible to the idiots like me:
https://en.wikipedia.org/wiki/History_of_topos_theory

The level of abstraction involved cannot be reduced beyond a certain point

But is there some kind of simple explanation? )))
Is it some kind of Multivese?

 

[tzimie]

Hugh Woodin told me about five years ago that he no longer believed his earlier work, that the continuum c satisfies

​

Also, about 20 years ago I spoke with the late Paul Cohen, who said it was possible that c satisfies

c = aleph_c

(!). In fact, he pointed out that the ZFC axioms are consistent with the hypothetical axiom that

c = aleph_k

for any ordinal k whose cofinality cf(k) is not equal to aleph₀.

What a pity...
If there are so many cardinalities in between \aleph_0 and continuum (not just \omega_1), then these cardinalities create a fuzzy set like in Banach-Tarsky paradox, and no example of set of these cardinalities can be provided constructively.

 

[Stephen Tashi]

Many set theorists believed that the continuum does not have cardinality $\omega_1$.

I'm curious whether "the continuum" refers to a structure that satisfies a specific set of axioms - or whether it is a term of common speech (at least among set theorists) - similar to terminology like "the universe", which refers to a common notion, but not one that is defined by a unique set of axiom.

 

[pwsnafu]

I'm curious whether "the continuum" refers to a structure that satisfies a specific set of axioms - or whether it is a term of common speech (at least among set theorists) - similar to terminology like "the universe", which refers to a common notion, but not one that is defined by a unique set of axiom.

See linear continuum.

 

[Stephen Tashi]

See linear continuum.

I see.

That is an axiomatic definition of "a" linear continuum. Is there a theorem that any two linear continuua have the same cardinality? That would justify speaking of "the" continuum - at least as far as the property of cardinality goes.

 

[zinq]

"See linear continuum."

This is not correct. A "linear continuum" can be any of various totally ordered sets that do not all have the same cardinality. For example, the "long line" L in that Wikipedia article and the set R of real numbers satisfy

card(L) > card(R).​

But in the sense it is used above, "the continuum" — often denoted in math just by the letter c (but for obvious reasons this is a bad idea in physics) — refers to a specific cardinality. This is usually described as either the cardinality of the real numbers, or equivalently as the cardinality of the set of all subsets of the integers.

In terms of other cardinalities, the continuum is usually expressed as the cardinal power

2^aleph₀,​

where, as usual, 2 denotes the cardinality of the set {0, 1} and aleph₀ denotes the cardinality of the integers.

 

[tzimie]

I see.

That is an axiomatic definition of "a" linear continuum. Is there a theorem that any two linear continuua have the same cardinality? That would justify speaking of "the" continuum - at least as far as the property of cardinality goes.

I guess if is a consequence of AC.

 

[tzimie]

But intuitively, do you interpret \omega_1 (as well-ordered sequence of all countable ordinals) as continuum ?
For me it is weaker than continuum...

(Note: this question does not make sense to a formalist, but I am Platonist)

 

[Demystifier]

I've seen links in Wiki to topos theory, but Wiki claims that this theory is strongly inaccessible to the idiots like me:
https://en.wikipedia.org/wiki/History_of_topos_theory

But is there some kind of simple explanation? )))
I

Try http://math.ucr.edu/home/baez/topos.html

 

[tzimie]

Try http://math.ucr.edu/home/baez/topos.html

Thank you.

Suppose you're a finitist and you only want to work with finite sets and functions between them. Then you want to work in the topos FinSet.

Suppose you're a constructivist and you only want to work with "effectively constructible" sets and "effectively computable" functions. Then you want to work in the "effective topos" developed by Martin Hyland.

Suppose you like doing calculus with infinitesimals, the way physicists do all the time - but you want to do it rigorously. Then you want to work in the "smooth topos" developed by Lawvere and Anders Kock.

But how is it different from a collection of axiomatic systems? How is it different from saying "take any axiomatic system and do what you want"?

 

[tzimie]

P.S.
Demystifier, and am surprised and happy to meet you here, in math, not in physics subforum, so I can't resist asking you as physicist:
Do you believe that stuff (CH, Large cardinal axioms etc) has any (potential) relation to physics?

P.P.S
What interpretation of mathematics do you prefer - formalism or platonism?

 

[Demystifier]

Do you believe that stuff (CH, Large cardinal axioms etc) has any (potential) relation to physics?

At the moment I don't see any relevance for physics, but one day, who knows.

What interpretation of mathematics do you prefer - formalism or platonism?

When I want an intuitive understanding of abstract math concepts, I am a platonist. When I need to compute something or formally prove a theorem, I am a formalist. When I think philosophically about mathematical ontology, I am often a constructivist and finitist.

 

[Demystifier]

But how is it different from a collection of axiomatic systems? How is it different from saying "take any axiomatic system and do what you want"?

Category theory is not a replacement for logic and axiomatic systems. As a foundation for mathematics, category theory is a kind of replacement for set theory.

 

[micromass]

Category theory is not a replacement for logic

It can be.

 

[stevendaryl]

My favorite intuitive argument against the continuum hypothesis is Freiling's "Throwing Darts at the Number Line". It's a simple enough argument that I think I can reproduce it here.

Imagine that there is some process for randomly selecting a real number with a flat probability distribution in [0,1]. You can't actually select a random real, because a real number requires an infinite amount of precision, but for the sake of argument, suppose that you can. Freiling describes it as "throwing a dart at the number line", and wherever the dart sticks, that's your random real. Now, elementary measure theory tells you that for any countable set X you pick ahead of time, the chance that your randomly chosen real, x will be in X is zero. The measure of any countable set is zero.

Now, suppose that you have a function F(x) which takes a real number in [0,1] and returns a countable subset of [0,1]. Then two players, Alice and Bob, can use this function to play the following game:

1. Alice randomly picks a real a in [0,1].
2. She computes a countable set F(a)
3. Bob then randomly picks a different real, b.
4. He also computes a countable set F(b)
5. If Bob picks a real from Alice's set, he wins. If Alice picks a real from Bob's set, she wins.

The question is: What are the odds that Bob will win? After Alice has picked her real, she can reason as follows: "There are only countably many reals in F(a). So the odds that Bob will pick a real from that set is zero. So Bob's chance of winning is zero."

We can also ask what are the odds of Alice winning. If Bob went first, then he could have used the same argument as Alice to argue that Alice has a zero chance of winning. Intuitively, it shouldn't matter who went first, so the conclusion should be that with probability 1, neither Alice nor Bob is going to win. This means that it is very likely (probability 1) that a \notin F(b) and b \notin F(a). This should be true, no matter what the function F is (as long as it always returns a countable set).

This argument motivates the following conjecture:

Symmetry Axiom: For any function F that takes a real in [0,1] and returns a countable subset of [0,1], there are two numbers a and b such that a \notin F(b) and b \notin F(a)

(The argument actually suggests that most such pairs of numbers have this property, but for what follows, it's enough that at least one pair has this property.)

But Freiling gives a simple proof that the above Symmetry Axiom contradicts the Continuum Hypothesis.

Proof: Assume the continuum hypothesis. Then that means that it is possible to arrange the reals in [0,1] in a well-ordering of type \omega_1, which means that we can map each real to an ordinal less than \omega_1. Since \omega_1 is the first uncountable ordinal, that means that we can associate each real x with a countable ordinal ord(x). So assume we have such a mapping, then we define a function F(x) as follows: F(x) is the set of all reals y in [0,1] such that ord(x) > ord(y). Since every countable ordinal has only countably many smaller ordinals, that means that for every x, F(x) is countable. But clearly, for any two reals a and b, either ord(a) < ord(b), or ord(b) < ord(a). So for any two reals a and b, either a \in F(b) or b \in F(a). This contradicts the Symmetry Axiom above.

Assuming that you find that plausible, Freiling goes on to argue, along similar lines, that the continuum can't be \omega_2, \omega_3, ...

 

[Demystifier]

It can be.

Can you elaborate?

 

[tzimie]

Category theory is not a replacement for logic and axiomatic systems. As a foundation for mathematics, category theory is a kind of replacement for set theory.

But as there are options for those who work with finite set only, or constructuble sets, for those who accept/deny the existence of inaccesible cardinals isn't it a "multiverse" of Set Theory?

 

[tzimie]

Assuming that you find that plausible, Freiling goes on to argue, along similar lines, that the continuum can't be \omega_2, \omega_3, ...

And I've heard it can't be \aleph_\omega either
What is it then?

 

[stevendaryl]

But intuitively, do you interpret \omega_1 (as well-ordered sequence of all countable ordinals) as continuum ?
For me it is weaker than continuum...

(Note: this question does not make sense to a formalist, but I am Platonist)

\omega_1 is the smallest uncountable ordinal. It may or may not be equal to the size of the continuum, where the continuum means the set of all real numbers (or equivalently, the set of all subsets of the natural numbers).

 

[tzimie]

\omega_1 is the smallest uncountable ordinal. It may or may not be equal to the size of the continuum, where the continuum means the set of all real numbers (or equivalently, the set of all subsets of the natural numbers).

I mean if we agree with Freiling's argument, how many alephs are between \aleph_0 and continuum?
Based on Freiling it can't be 1,2,3... etc.
It also can't be \omega

 

[Demystifier]

But as there are options for those who work with finite set only, or constructuble sets, for those who accept/deny the existence of inaccesible cardinals isn't it a "multiverse" of Set Theory?

I don't have a problem with such a multiverse, as long as I think of mathematics as a human construct (see also my signature). It's a problem only when I think as a platonist.

 

[stevendaryl]

I mean if we agree with Freiling's argument, how many alephs are between \aleph_0 and continuum?
Based on Freiling it can't be 1,2,3... etc.
It also can't be \omega

Well, if you accept the very strongest of Freiling's conclusions, then the reals cannot be well-ordered, at all, so there is no cardinality of the continuum. His arguments imply that the axiom of choice is false.

 

[zinq]

"As a foundation for mathematics, category theory is a kind of replacement for set theory."

This is simply not the case. Category theory is a way to organize various fields of math in relation to one another inside of a larger framework. But category theory makes extensive use of set theory, and so is in no way a replacement for it.

It is true that there is a natural *analogy* between set theory and category theory: sets correspond to objects and functions correspond to morphisms. But that is not the same as saying category theory replaces set theory.

 

[stevendaryl]

"As a foundation for mathematics, category theory is a kind of replacement for set theory."

This is simply not the case. Category theory is a way to organize various fields of math in relation to one another inside of a larger framework. But category theory makes extensive use of set theory, and so is in no way a replacement for it.

I think some people would disagree. According to the Wikipedia article on Category Theory:

https://en.wikipedia.org/wiki/Category_theory#Historical_notes

Certain categories called topoi (singular topos) can even serve as an alternative to axiomatic set theory as a foundation of mathematics.

 

[micromass]

"As a foundation for mathematics, category theory is a kind of replacement for set theory."

This is simply not the case. Category theory is a way to organize various fields of math in relation to one another inside of a larger framework. But category theory makes extensive use of set theory, and so is in no way a replacement for it.

It is true that there is a natural *analogy* between set theory and category theory: sets correspond to objects and functions correspond to morphisms. But that is not the same as saying category theory replaces set theory.

Category theory can replace set theory entirely if you choose to. See the work by Lawvere.

 

[Demystifier]

Category theory can replace set theory entirely if you choose to. See the work by Lawvere.

You still didn't explain (or even made a hint) how category theory can replace logic.

 

[micromass]

You still didn't explain (or even made a hint) how category theory can replace logic.

Sorry forgot. First, I want to ask you what you think logic means. Do you think we need logic before set theory? Or are you talking about mathematical logic that is only developed once set theory is?

 

[Demystifier]

Sorry forgot. First, I want to ask you what you think logic means. Do you think we need logic before set theory? Or are you talking about mathematical logic that is only developed once set theory is?

I think of logic as something we need before set theory. (At least first order logic, admitting that second order logic can be thought of as "set theory in sheep's clothing".)

 

[micromass]

I think of logic as something we need before set theory. (At least first order logic, admitting that second order logic can be thought of as "set theory in sheep's clothing".)

The logic before set theory can't be modeled by category theory, but I wouldn't call that first order logic. In my opinion, first order logic requires set theory. First order logic can be done with category theory perfectly.

 

[Demystifier]

In my opinion, first order logic requires set theory.

Can you give an argument or a reference for that statement?

 

[stevendaryl]

In my opinion, first order logic requires set theory.

The semantics of first-order logic perhaps requires set theory, but first-order logic itself is just syntax plus rules of inference. It certainly doesn't require set theory. It would be circular if it did, because set theory is axiomatized using first-order logic.

 

[micromass]

Can you give an argument or a reference for that statement?

See any logic book, eg Hinman. It will work inside set theory already.

 

[micromass]

The semantics of first-order logic perhaps requires set theory, but first-order logic itself is just syntax plus rules of inference. It certainly doesn't require set theory. It would be circular if it did, because set theory is axiomatized using first-order logic.

What is your definition of first order logic?

 

[stevendaryl]

What is your definition of first order logic?

First-order logic is a language together with axioms and rules of inference for sentences in that language.

The language has:

• propositional operators: and, or, not, implies
• quantification operators: forall and exists
• function symbols
• relation symbols
• variables
• constants

The axioms (axiom schemas, actually) are things such as

Phi(t) implies exists x Phi(x)

The rules of inference typically are just modus-ponens and universal introduction.

 

[micromass]

How many variables do you typically have?

 

[Demystifier]

See any logic book, eg Hinman. It will work inside set theory already.

In that book, a formal discussion of set theory does not appear before Chapter 6. Of course, the concept of a set is used already in Chapter 1, but it is used only in the informal sense, not in the sense of set theory.

 

[micromass]

In that book, a formal discussion of set theory does not appear before Chapter 6. Of course, the concept of a set is used already in Chapter 1, but it is used only in the informal sense, not in the sense of set theory.

And uuh, what exactly IS a set in the informal sense? Note that he uses the axiom choice in the first two chapters too!

 

[Demystifier]

And uuh, what exactly IS a set in the informal sense?

A collection.

Note that he uses the axiom choice in the first two chapters too!

Uhh, perhaps it tells more about the book than about logic. I believe that most logic textbooks do not commit that crime.

 

[micromass]

A collection.

I have no problem with that. The problem is that from the outset, they start working with countable or otherwise infinite sets. It is my point of view that you can't do this without a formal set theory in place.

Uhh, perhaps it tells more about the book than about logic. I believe that most logic textbooks do not commit that crime.

Then your logic books do not treat the compactness theorem, completeness theorem and Löwenheim-Skolem theorems, all of which can be proven to need the axiom of choice!

 

[Demystifier]

Then your logic books do not treat the compactness theorem, completeness theorem and Löwenheim-Skolem theorems, all of which can be proven to need the axiom of choice!

OK, you convinced me, it's very hard to talk about logic without having some notions of sets. But the opposite, to talk about sets without having some notions of logic, is even harder. So where should we start? Should we completely abandon the idea that mathematics has a well defined foundation? (Or perhaps my avatar was right that we should found mathematics on type theory, no matter how much things become complicated then?)

 

[micromass]

OK, you convinced me, it's very hard to talk about logic without having some notions of sets. But the opposite, to talk about sets without having some notions of logic, is even harder. So where should we start? Should we completely abandon the idea that mathematics has a well defined foundation? (Or perhaps my avatar was right that we should found mathematics on type theory, no matter how much things become complicated then?)

See also this: http://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent

It's a chicken vs the egg problem. What comes first? Logic or set theory. In either case, we are forced to accept some non-formalized stuff. Either we accept some "god-given" logic, or we accept a "god-given naive set theory", or both. I don't see a way out of this. It is my personal believe that we can give an accurate but nonformal description (or intuition) of what logic is, what a proof is and what a set is. With these we can create formal logic and formal set theory. We can then prove a lot of cool theorems about logic and set theory. But these will not be theorems about our naive nonformal system. Rather, it will be theorems about the logic and set theory we mimicked inside our nonformal system.

 

[Demystifier]

See also this: http://mathoverflow.net/questions/22635/can-we-prove-set-theory-is-consistent

It's a chicken vs the egg problem. What comes first? Logic or set theory. In either case, we are forced to accept some non-formalized stuff. Either we accept some "god-given" logic, or we accept a "god-given naive set theory", or both. I don't see a way out of this. It is my personal believe that we can give an accurate but nonformal description (or intuition) of what logic is, what a proof is and what a set is. With these we can create formal logic and formal set theory. We can then prove a lot of cool theorems about logic and set theory. But these will not be theorems about our naive nonformal system. Rather, it will be theorems about the logic and set theory we mimicked inside our nonformal system.

That definitely makes sense!

But consider this. Let non-formal logic and non-formal set theory be called Log1 and Set1. Likewise, let Log2 and Set2 be their formal incarnations. And suppose that Log1 and Set1 are given. As the next step, what should we develop first, Log2 or Set2? So far I thought that Log2 should be formulated before Set2, but now it seems that it doesn't matter.

 

[micromass]

That definitely makes sense!

But consider this. Let non-formal logic and non-formal set theory be called Log1 and Set1. Likewise, let Log2 and Set2 be their formal incarnations. And suppose that Log1 and Set1 are given. As the next step, what should we develop first, Log2 or Set2? So far I thought that Log2 should be formulated before Set2, but now it seems that it doesn't matter.

Indeed, it doesn't matter so much. However, in my point of view, I reject any use of infinite sets in Set1 including the axiom of choice which is a statement about infinite sets. I am prepared to accept potential infinity. If we do this, then we cannot develop the completeness theorem or Löwenheim-Skolem theorem in Log2 unless you already developed Set2. So if I want to formalize Hinman's book in my pet system it goes as follows: non-formal logic and set theory first Log1 and Set1 then I create a formalized set theory (trying to avoid actual infinity) Set2 that satisfies ZFC or the finitely-axiomatizable NBG. Here is where Hinman begins where he develops Log2 then in a later chapter he develops axiomized set theory which is Set3

Of course, if you have no problems with infinite sets in your nonformal logic and stuff like the axiom of choice (I can imagine that you're a Platonist that accepts these universes as really existing), then you can work entirely inside Set2 and Log2 and there is no reason for a Set3

But whatever we do we always can go on: we can build inside Setn a logical system Logn+1 and a set theory Setn+1

 

[Demystifier]

I reject any use of infinite sets in Set1 including the axiom of choice

This is something I always thought but was afraid to say. Thanks for spelling it explicitly!

 

[Stephen Tashi]

What comes first? Logic or set theory.

In addition to that question, we can ask when the notion of "order" is to be introduced.

Before we can observe that an author did one thing before another, we must have the notion of things being done in some order.

 

[Demystifier]

Before ...

And before defining the word "before" we must first define some words before that.