Did Paul Cohen settle the Continuum Hypothesis?

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Discussion Overview

The discussion centers around the status of the Continuum Hypothesis (CH) in relation to Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC), particularly following Paul Cohen's proof of its independence from ZFC. Participants explore whether the independence remains valid today and the implications of additional axioms on the decidability of CH.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that Cohen's result regarding the independence of CH from ZFC is permanent and unchanging.
  • Others express skepticism about certain sources, labeling one link as "crackpottery" while finding another to be a credible summary of the current state of the CH debate.
  • A participant mentions that Woodin believes the CH might be decidable using additional axioms like Large Cardinal Axioms and the Axiom of Projective Determinacy, although this does not imply that ZFC alone can decide CH.
  • There is a discussion about the relevance of additional axioms and whether they can lead to a consistent theory that might challenge the status of ZFC.
  • One participant questions the terminology used in earlier posts, specifically regarding the "truth" or "falseness" of CH, suggesting it should be framed in terms of provability instead.
  • Another participant raises a philosophical question about the origins of the ZFC axioms and the implications of adding new axioms to the existing framework.
  • Some participants highlight that while ZFC is the conventional standard, the introduction of new axioms does not necessarily undermine its relevance.

Areas of Agreement / Disagreement

Participants generally agree on the independence of CH from ZFC, but multiple competing views exist regarding the implications of additional axioms and the interpretation of Cohen's results. The discussion remains unresolved on the potential for new axiomatic systems to influence the status of CH.

Contextual Notes

There are limitations in the discussion regarding the assumptions behind the axioms of ZFC and the implications of adding new axioms, as well as the unresolved nature of the consistency of various proposed systems.

SW VandeCarr
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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.
 
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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.
 


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
 
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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:
 


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.
 
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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.
 


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.
 


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.
 


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...
 

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