Did Paul Cohen settle the Continuum Hypothesis?

In summary: It's not clear what you're trying to say. The axioms of ZFC are a foundation for the theory, not a theory in and of themselves. Adding more axioms doesn't change what can or cannot be proven from ZFC.
  • #1
SW VandeCarr
2,199
81
Did Paul Cohen "settle" the Continuum Hypothesis?

Paul Cohen proved the Continuum Hypothesis is independent of ZFC and concluded that it's truth or falseness is undecidable (1963). Is this still the case today? These links suggest there is still interest in proving or disproving it.

http://www.gauge-institute.org/cantor/HilbertFirstProblem.pdf

http://www.ams.org/notices/200106/fea-woodin.pdf

EDIT: I'm suspicious of the the first link. I can't find out much about the Gauge Institute except that it's located in St Paul,MN and I don't think this paper proves anything, but I'm not a mathematician.
 
Last edited:
Physics news on Phys.org
  • #2


SW VandeCarr said:
Paul Cohen proved the Continuum Hypothesis is independent of ZFC and concluded that it's truth or falseness is undecidable (1963). Is this still the case today? These links suggest there is still interest in proving or disproving it.

The CH is independent of ZFC; this is forever true.

The first link is crackpottery. The second article seems a good summary of the state of affairs; it certainly doesn't contradict Cohen's result.
 
  • #3


CRGreathouse said:
The CH is independent of ZFC; this is forever true.

The first link is crackpottery. The second article seems a good summary of the state of affairs; it certainly doesn't contradict Cohen's result.

Here's Part II of Woodin's paper. He seems to believe that the problem is decidable without resolving all instances of the GHC, utilizing the Large Cardinal Axioms (as part of a well ordered hierarchy) and the Axiom of Projective Determinacy along with ZFC (although he seems to think Choice no longer would be necessary) From what I could find, Woodin seems the be major (credible) investigator re the decidability of CH.

http://www.ams.org/notices/200107/fea-woodin.pdf
 
Last edited:
  • #4


Whether or not ZFC+Large Cardinals+Axiom of Projective Determinacy can prove the CH has no bearing whatsoever on whether or not ZFC can prove CH.

Investigating what can be proven from additional hypotheses is an interesting thing that set theorists may do, but that doesn't change what can or cannot be proven from ZFC alone.



Oh, I just noticed:
Paul Cohen proved the Continuum Hypothesis is independent of ZFC and concluded that it's truth or falseness is undecidable (1963).
That sounds like a misstatement -- where did "truth" or "falseness" come from? :confused:
 
  • #5


Hurkyl said:
Whether or not ZFC+Large Cardinals+Axiom of Projective Determinacy can prove the CH has no bearing whatsoever on whether or not ZFC can prove CH.

I don't think I indicated it did. No one is questioning the independence of CH and ZFC, but Woodin is attempting to use ZF along with some new axioms as a basis for deciding the status of CH.

Investigating what can be proven from additional hypotheses is an interesting thing that set theorists may do, but that doesn't change what can or cannot be proven from ZFC alone.

The point was never argued. I couldn't find anything in the Wooden papers that said ZFC was sufficient to decide CH. The whole of the two papers, as far as I can surmise, try to make a case for the new axiomatic system he's working on. He seems quite aware of the difficulties but remains cautiously optimistic. Clearly such a new system must be consistent with ZFC. As I said ZF is included in the system he's developing but indicates at the end of his second paper that C may not be needed given the new axioms.

Oh, I just noticed:

That sounds like a misstatement -- where did "truth" or "falseness" come from? :confused:

Poor choice of words. Just should have said the provability of CH.
 
Last edited:
  • #6


Hurkyl said:
Whether or not ZFC+Large Cardinals+Axiom of Projective Determinacy can prove the CH has no bearing whatsoever on whether or not ZFC can prove CH.

Investigating what can be proven from additional hypotheses is an interesting thing that set theorists may do, but that doesn't change what can or cannot be proven from ZFC alone.



Oh, I just noticed:

That sounds like a misstatement -- where did "truth" or "falseness" come from? :confused:

Hurkyl said:
Whether or not ZFC+Large Cardinals+Axiom of Projective Determinacy can prove the CH has no bearing whatsoever on whether or not ZFC can prove CH.

Investigating what can be proven from additional hypotheses is an interesting thing that set theorists may do, but that doesn't change what can or cannot be proven from ZFC alone.



Oh, I just noticed:

That sounds like a misstatement -- where did "truth" or "falseness" come from? :confused:

I would like to ask from where did the axioms of ZFC come from in the first place ?

Why is it that you consider the axioms of ZFC relevant , but adding a few more axioms to ZFC , just an exercise for set theorists ?

I am not a mathematician , and have very limited knowledge of mathematics , so what I maybe saying maybe complete rubbish , but as far as by understanding goes , ZFC is just a set of axioms that lead to a 'consistent' theory . If by adding any number of axioms , I still am able to generate a 'consistent' theory , then what reason would be left for someone to favour ZFC from this other new theory ?

In the above paragraph , my usage of the word theory might be inappropriate , but I do not know what is the right word that should be used.
 
  • #7


srijithju said:
I would like to ask from where did the axioms of ZFC come from in the first place ?
They were brought up in the opening post.

Also, ZFC is the conventional "standard" -- unless someone says otherwise, we should assume if they refer to set theory, they mean at least ZF, and usually C as well.
 
  • #8


srijithju said:
as far as by understanding goes , ZFC is just a set of axioms that lead to a 'consistent' theory . If by adding any number of axioms , I still am able to generate a 'consistent' theory , then what reason would be left for someone to favour ZFC from this other new theory ?

ZFC + "CRGreathouse is emperor of the world" is consistent if and only if ZFC is consistent... but I haven't been able to get people to adopt that axiom system for some reason.
 
  • #9


ZFC is just a set of axioms that lead to a 'consistent' theory

Actually, ZFC is a set of axioms that lead to a theory that we believe to be consistent. Consistency can only be proved in a stronger theory, and so on...
 

Related to Did Paul Cohen settle the Continuum Hypothesis?

1. What is the Continuum Hypothesis?

The Continuum Hypothesis, also known as the Continuum Problem, is a mathematical problem proposed by Georg Cantor in 1878. It deals with the concept of infinity and asks whether there is a set that is larger than the set of natural numbers but smaller than the set of real numbers. In other words, it asks if there is a set with a cardinality (number of elements) between countable and uncountable infinity.

2. Who was Paul Cohen?

Paul Cohen (1934-2007) was an American mathematician who made significant contributions to the field of set theory. He is best known for his work on the Continuum Hypothesis and for proving its undecidability in 1963, for which he was awarded the Fields Medal in 1966.

3. Did Paul Cohen settle the Continuum Hypothesis?

No, Paul Cohen did not settle the Continuum Hypothesis. In 1963, he proved that the hypothesis is undecidable within the standard axioms of set theory, meaning that it cannot be proven or disproven using those axioms. This result is known as Cohen's Independence Theorem and it showed that the Continuum Hypothesis cannot be settled using the methods and techniques available at the time.

4. Why is the Continuum Hypothesis important?

The Continuum Hypothesis has been a central problem in set theory and has sparked much debate and research since it was proposed. Its resolution would have a significant impact on our understanding of infinity and the structure of the real numbers. Additionally, the methods and techniques developed in attempts to settle the hypothesis have had broader applications in other areas of mathematics.

5. Is the Continuum Hypothesis still an open problem?

Yes, the Continuum Hypothesis is still an open problem in mathematics. Despite the efforts of many mathematicians since its proposal, a definitive solution has not been found. However, the work of Paul Cohen and others has led to a better understanding of the problem and has opened up new avenues for research and exploration in set theory and related fields.

Similar threads

Undecidable statement made into axiom?
    • Set Theory, Logic, Probability, Statistics
Replies
9
Views
3K
What is the difference between cardinal and ordinal numbers?
    • General Math
Replies
1
Views
10K
Mathematica This Week's Finds in Mathematical Physics (Week 247)
    • MATLAB, Maple, Mathematica, LaTeX
Replies
1
Views
3K

Hot Threads

Recent Insights

Back
Top