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I am very interested in learning the full solution of the continuum hypothesis!

For some background, I have a degree in physics that I earn many years ago and the highest mathematics that I dealt with is abstract measure theory and some elementary functional analysis.

The only reason I want to learn a detailed solution of this hypothesis in for self satisfaction, I feel that I need such a challeng and that my life would be more completed with it.

I am ready to study it for as much time as it takes and give it my everything. From my elementary readings I understand that there are many approaches to the problem so I would prefer one that includes the concepts from functional analysis if there exist such a approach!

Clearly, by solution I mean showing that we cannot prove or disprove it with the current mathematical methods.

I hope someone can give me a step by step guide and that my question comes not as arrogant to the readers.

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    $\begingroup$ Perhaps mathoverflow.net/questions/5117/… (By the way, the independence statement is really two statements: if ZFC is consistent then ZFC+CH is consistent, and if ZFC is consistent then ZFC+negation of CH is consistent. The two were proved decades apart using quite different methodology.) $\endgroup$
    – Ian
    Apr 23 '17 at 13:02
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    $\begingroup$ Both proofs (see Ian's comment above) can be found in Yu. I. Manin: "A Course in Mathematical Logic", Springer, 1977, Chapter III and IV. $\endgroup$ Apr 23 '17 at 13:10
  • $\begingroup$ I just found a copy of "Set theory and the continuum problem" (Oxford Logic Guides, 34. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1996. xiv+288 pp. ISBN: 0-19-852395-5) mentioned in that post and it looks pretty decent! I will try to go through it now! At the beginning I thought the solution needs a lot more background, like learning many other stuff before tackling the problem itself... $\endgroup$ Apr 23 '17 at 13:15
  • $\begingroup$ You can try Nik Weaver's book about forcing for mathematicians. I don't know if he fully proves both consistency results, but I think it can at least give you some accessible introduction to logic and set theory. I haven't read the book, but I did look at a few pages here and there, and it seems promising. $\endgroup$
    – Asaf Karagila
    Apr 23 '17 at 15:32
  • $\begingroup$ Also, not to tout my horn too loudly, but I wrote something for this sort of occasion, karagila.org/2016/the-five-whs-of-set-theory $\endgroup$
    – Asaf Karagila
    Apr 23 '17 at 15:34
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You don't say how much background in set theory you have. You might find it useful then to take a look at the Teach Yourself Logic Study Guide. §4.3 is called "Beginning Set Theory" which mentions some (alternative) books which would get you to the point of being able to take treatments of the independent of the Continuum Hypothesis from standard ZFC.

Then §7.1 suggests some (alternative) treatments of the independence proofs (more recent and perhaps more accessible than Cohen). There is no getting away from it, though, independence-by-forcing is a tough topic -- Timothy Chow famously calls it "an open exposition problem" (i.e. an open problem how to give an ideally clear, naturally motivated, account of what is going on).

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  • $\begingroup$ My impression is the formulation of Cohen forcing as sheaves does give a fairly natural account of what is going on. $\endgroup$ Apr 24 '17 at 4:01

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