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edited Feb 6 2011 at 17:09

Gil Kalai
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Background

The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^{\aleph_0}$. In other words, it asserts that every subset of the set of real numbers that contains the natural numbers has either the cardinality of the natural numbers or the cardinality of the real numbers. It was the first problem on the 1900 Hilbert's list of problems. The generalized continuum hypothesis asserts that there are no intermediate cardinals between every infinite set X and its power set.

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper entitled "The Generalized Continuum Hypothesis revisited "). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

The proposed asnwer $ 2^{\aleph_0}=\aleph_2$ goes back according to François to Goedel. It is (perhaps) mentioned in Foreman's presentation. (I heard also from Menachem Magidor that this answer might have some advantages.)

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

There are by now a couple of long detailed excellent answers (that I still have to digest) by Joel David Hamkins and by Anders Caicedo and several other useful answers. (Unfortunately, I can accept only one answer.)

Update (Fenruary 2011): A new detailed answer was contributed by Justin Moore.

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edited Dec 29 2010 at 10:46

Gil Kalai
10.7k●6●80●144

Background

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper entitled "The Generalized Continuum Hypothesis revisited "). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

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edited May 24 2010 at 5:47

Gil Kalai
10.7k●6●80●144

Background

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

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edited May 22 2010 at 21:15

Gil Kalai
10.7k●6●80●144

Background

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

There are by now a couple of long detailed excellent answers (that I still have to digest) and several other useful answers. (Unfortunately I cannot accept more than one answer.)

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edited May 14 2010 at 14:08

Gil Kalai
10.7k●6●80●144

Background

Cohen proved that the CH is independent from the axioms of set theory. (Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

The proposed asnwer $ 2^{\aleph_0}=\aleph_2$ goes back according to Francois François to Goedel. It is (perhaps) mentioned in Foreman's presentation. (I heard also from Menachem Magidor that this answer might have some advantages.)

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

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edited May 8 2010 at 17:30

Gil Kalai
10.7k●6●80●144

The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^{\aleph_0}$. In other words, it asserts that every subset of the set containing of real numbers containing that contains the natural numbers has either the cardinality of the natural numbers or the cardinality of the real numbers. It was the first problem on the 1900 Hilbert's list of problems. The generalized continuum hypothesis asserts that there are now no intermediate cardinals between every infinite set X and its power set.

Cohen proved that the CH is independent from the axioms of set theory. (Eralier Earlier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

The proposed asnwer $ 2^{\aleph_0}=\aleph_2$ goes back according to Francois to Goedel. It is (perhaps) mentioned in Foreman's presentation. (I heard also from Menachem Magidor that this answer might have some advantages.)

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hope most from an answer is some friendly elementary descriptions of the proposed solutions.

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edited May 8 2010 at 9:17

Gil Kalai
10.7k●6●80●144

The Continuum Hypothesis (CH) posed by Cantor in 1890 asserts that $ \aleph_1=2^{\aleph_0}$. In other words, it asserts that every set containing of real numbers containing the natural numbers has either the cardinality of the natural numbers or the cardinality of the real numbers. It was the first problem on the 1900 Hilbert's list of problems. The generalized continuum hypothesis asserts that there are now intermediate cardinals between every infinite set X and its power set.

Cohen proved that the CH is independent from the axioms of set theory. (Eralier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hop hope most dor from an answer is some friendly elementary descriptions of the proposed solutions.

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edited May 8 2010 at 9:10

Gil Kalai
10.7k●6●80●144

Cohen proved that the CH is independent from the axioms of set theory. (Eralier Goedel showed that a positive answer is consistent with the axioms).

Several mathematicians proposed definite answers or approaches towards such answers regarding what the answer for the CH (and GCH) should be.

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

I am aware of the existence of 2-3 approaches.

One is by Woodin described in two 2001 Notices of the AMS papers (part 1, part 2).

Another by Shelah (perhaps in this paper). See also the paper entitled "You can enter Cantor paradize" (Offered in Haim's answer.);

There is a very nice presentation by Matt Foreman discussing Woodin's approach and some other avenues. Another description of Woodin's asnwer is by Lucca Belloti (also suggested by Haim).

The proposed asnwer $ 2^{\aleph_0}=\aleph_2$ goes back according to Francois to Goedel. It is (perhaps) mentioned therein Foreman's presentation. (I heard also from Menachem Magidor that this answer might have some advantages.)

François G. Dorais mentioned an important paper by Todorcevic's entitled "Comparing the Continuuum with the First Two Uncountable Cardinals".

There is also a very rich theory (pcf theory) of cardinal arithmetic which deals with what can be proved in ZFC.

Remark:

I included some information and links from the comments and answer in the body of question. What I would hop most dor an answer is some friendly elementary descriptions of the proposed solutions.

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edited May 7 2010 at 14:55

Gil Kalai
10.7k●6●80●144

typo

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edited May 7 2010 at 7:46

Igor Pak
8,206●2●17●57

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asked May 7 2010 at 7:40

Gil Kalai
10.7k●6●80●144

Return to Question

Background

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

Remark:

Background

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

Remark:

Background

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

Remark:

Background

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

Remark:

There are by now a couple of long detailed excellent answers (that I still have to digest) and several other useful answers. (Unfortunately I cannot accept more than one answer.)

Background

The question

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

More background

Remark:

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

Remark:

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

Remark:

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

Remark:

My question asks for a description and explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

My question asks for a description and axplaination explaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.

Solutions to the Continuum Hypothesis

My question asks for a description and axplaination of the various approaches to the continuum hypothesis in a language which could be understood by non-professionals.