"Forcing" and CH

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Dec 25, 1999, 3:00:00 AM12/25/99
Does anyone know where I can find a readable account of Paul Cohen's method of
"forcing" to prove that both CH and ~CH are consistent with ZFC? Thanks.


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Dec 26, 1999, 3:00:00 AM12/26/99
In article <19991225075916...@ng-fs1.aol.com>,

serap...@aol.com (Seraph-sama) wrote:
> Does anyone know where I can find a readable account of Paul Cohen's
> method of
> "forcing" to prove that both CH and ~CH are consistent with ZFC?

Here is Weemba's response to the question of finding a readable
book or article on forcing. When he mentions Kunen, he's
talking about Kenneth Kunen.


In article <350427FC...@netvision.net.il>, Misha <vulcao@netvision
> Does anyone know if there is a good online reference to Cohen's
>proof of the independence of CH and the introduction of the forcing

There isn't.

> If not online, is there any good book that covers advanced
>set theory in an tutorial way?

Are you trying to learn how forcing works, or just a guide to start
using it in applications?

For the former, Kunen's text is probably the best reference, although
I prefer (or I think I do, it's been awhile) Shoenfield's "Unramified
Forcing" article.

For the latter, try Just & Weese (two volumes) and Ciesielski. It is
possible to go quite some distance in applying forcing without having
any understanding of the proof that it works. Both books take pains
to explain what they are doing. (J&W leave most steps as exercises
for the reader. Ciesielski covers just a small amount of material,
and some of his unimportant comments about the role of choice are
wrong or misleading.)

> I know that Saharon Shelah is about
>to publish a book about forcing, but I am not sure that it is out

PROPER AND IMPROPER FORCING is listed on the Springer Web page. It is
most definitely not for beginners.


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#4: People who call others idiots but don't know how to spell "idiot"
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#5: Christina Ricci.

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Dave L. Renfro

Dec 26, 1999, 3:00:00 AM12/26/99
Seraph-sama <serap...@aol.com>
[sci.math 25 Dec 1999 12:59:16 GMT]


>Does anyone know where I can find a readable account of
>Paul Cohen's method of "forcing" to prove that both CH and

>~CH are consistent with ZFC? Thanks.

Probably the best thing would be to go a university library
and look through the set theory books. Almost all of them
will be in the QA 248 classification area.

You may want to begin by looking at the last few chapters of
some of the standard elementary introductory set theory books
[Devlin, Enderton, Henle, Hrbacek and Jech, Monk, Moschovakis,
Roitman, Vaught, etc.]. You'll sometimes find fairly elementary
(compared to what you'd find in a more advanced text)
discussions of models of set theory, typically the constructible
sets. This is the model that Godel used to show that CH is
consistent with ZFC. Here are a couple of elementary treatments
you will *NOT* find in the QA 248 section. [I know, because
I have a copy of each and on their title pages the library
of congress call number is given. (When I *don't* give a call
number, it's because it's not given in the book.)]

[QA 9 .J64 1987] [See chapter 9: "Consistency and Independence"
(pp. 97-107).]

John N. Crossley, et al., WHAT IS MATHEMATICAL LOGIC?, 1972.
[QA 9 .W47] [See chapter 6: "Set Theory" (pp. 59-77).]

There are much easier set-theoretic independence proofs
than the independence of AC and CH in ZFC, and by reading over
some of these simplier results you'll be in a better position
to at least have an idea of what it *means* to prove something is
independent in ZFC and how one might go about it. See, for
example, Chapter 10 "Independence Proofs" (pp. 113-119) of
Robert Vaught's SET THEORY: An Introduction, 2'nd edition, 1995
[QA 248 .V38 1994]. There's also Kenneth Kunen's SET THEORY: An
Introduction to Independence Proofs [QA 248 .K75], Chapter IV
"Easy Consistency Proofs" (pp. 110-151), but Kunen's book is
so well known that I'm guessing you already know about it and
you wouldn't be posting this question if Kunen fit your
definition of "readable". [Kunen isn't really an *advanced
treatise* in set theory, but it is aimed at the beginning
graduate level and it's written for someone who is fairly
serious about learning some set theory.]

You might also want to look at the following:

Alexander Abian and Samuel LaMacchia, "On the consistency
and independence of some set-theoretical axioms", Notre
Dame Journal of Formal Logic 19 (1976), 155-158.

[Yes, yes ... the "let's blow up the Moon" guy!]

This paper gives elementary models of sub-systems of ZFC
in order to show that each of the axioms of extensionality,
replacement, power-set, union, and choice are independent
from the other four. The underlying sets in the models
are either very trivial (i.e. a singleton set) or sets
obtained by easy constructions with the positive integers
with membership taking the form of various divisibility-type
conditions. These results had been known long before this paper,
but apparently the "paper has some expository merits" (phrase
quoted from the first paragraph).

There are several books at what I would call the intermediate
introductory level (Kunen being at the advanced introductory
level) that you might want to look at:

Frank Drake and Dasharathj Singh, INTERMEDIATE SET THEORY,
1996. [See chapter 8: "Constructible Sets and Forcing",
pp. 127-192.]

Yu. I. Manin, A COURSE IN MATHEMATICAL LOGIC, 1977. [QA 9 .M296]
[See chapter 3: "The Continuum Problem and Forcing"
(pp. 103-148) and chapter 4: "The Continuum Problem and
Constructible Sets" (pp. 149-174).]

Winfried Just and Martin Weese, DISCOVERING MODERN SET
1997. [I don't have this book (I only have their elementary
introductory level "volume I", but would imagine that volume II
has some things that you'd be interested in.]

1997. [QA 248 .C475] [See chapter 9: "Forcing" (pp. 164-210)
and Appendix B: "Comments on the Forcing Method" (pp. 215-219).]

Azriel Levy, BASIC SET THEORY, 1979. [I don't have a copy of this
book (it's out of print and Levy told me a couple of years ago
that neither he nor the publisher has plans for a reprinting),
but it's in most college libraries in the QA 248 section.]

[I don't have a copy of this book but I would put it at the
intermediate introductory level. Cohen reviews all the logic
you'll need and I've heard his treatment is pretty good.
Typically, the earliest treatment of a new field *is not*
where you want to begin learning it, but this may be an

Finally, although I'd put this book in the advanced introductory
level, it seems like it'd be very accessible if your background
in algebra and/or topology is much stronger than your background
in logic:

IN SET THEORY, 2'nd edition, 1985. [QA 248 .B44 1985]

Incidentally, Boolean-valued models have been around much
longer than fuzzy math [Zadeh (1965), not what the latest reform
math is called by its opponents], but one almost never sees
any recognition of this in fuzzy literature. [In fuzzy logic
truth values can be any number belonging to the closed interval
[0,1]. In Boolean-valued models one confronts "truth values" that
belong to an arbitrary Boolean algebra.]

Here are some internet sites that may be helpful:

Nancy McGough's "infinite ink: The Continuum Hypothesis"

"Is the Continuum Hypothesis True, False, or Neither?"
[Some really old (1991) sci.math posts.]

An *excellent* expository essay [Burton Dreben and Akihiro
Kanamori, "Hilbert and Set Theory", Synthese 110 (1997),
77-125] is on-line (.ps file) at

Another *excellent* expository essay [Akihiro Kanamori,
"Set Theory from Cantor to Cohen", Bull. Symbolic Logic
2 (1996), 1-71] is on-line (.ps file) at

Thomas Jech has a 9 page expository paper (.ps file) titled
"The Infinite" at:

Stephen Simpson's lecture notes for MATH 558,
Foundations of Mathematics I (Penn. State Univ.).
.dvi, .ps, and .pdf formats are available. I just checked
and the .ps file is 1937 K. Assuming that it hasn't changed
since I made a print copy a few months ago, it comes to
99 pages. Chapter 9: "Axiomatic Set Theory" has some material
on constructible sets and transitive models, concluding
with a proof that "ZF consistent ==> ZFC + GCH consistent".

Peter Suber has an excellent collection of on-line logic
related items at

And here are Peter Suber's lecture notes for a mathematical
logic course . . .


The story of how Paul Cohen came to prove that ~CH is consistent
with ZF (and ZFC, for that matter), assuming ZF is consistent of
course, is quite interesting. I forgot where I read about
it, but you might want to look into it. By the way, Cohen was a
1950 graduate of Stuyvesant HS in New York City.

Dave L. Renfro

Anders Goeransson

Dec 27, 1999, 3:00:00 AM12/27/99
On 26 Dec 1999 17:32:27 -0500, dlre...@gateway.net (Dave L. Renfro)

[About introductory material on forcing.]

Recently I saw a book that used modal logic (S4) to give the desired
model. The authors claimed in the preface that this was a natural
framework for the idea of forcing.

Perhaps someone has read this book and can give a commentary???

Set Theory and the Continuum Problem (Oxford Logic Guides, 34)
by Raymond M. Smullyan, Melvin Fitting Hardcover (July 1997)
Clarendon Pr; ISBN: 0198523955.


Allan Adler

Dec 27, 1999, 3:00:00 AM12/27/99

When I was an undergraduate, I read the paper of Robert Solovay and
Dana Scott, then circulating as a preprint, which explained Boolean
valued models and constructed models where CH and AC failed, among
other things. I thought it was a very nice paper and I've long been
disappointed that it never appeared in print, especially since I
no longer have the preprint.

Allan Adler

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