# How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing.

Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain techniques invented beforehand could be seen to lead upto a new technique like Forcing.

What techniques could Cohen have been influenced by to invent Forcing?

• The following article may be of interest: Paul Cohen, The discovery of forcing, Rocky Mountain Journal of Mathematics 32 (2002), no.4, pp.1071--1100, Project Euclid link. Feb 17, 2014 at 10:14
• You can read by Akihiro Kanamori, COHEN AND SET THEORY (The Bulletin of Symbolic Logic, 2008), but also in Dov Gabbay & Akihiro Kanamori & John Woods (editors), Handbook of the History of Logic, vol 6: Sets and Extensions in the Twentieth Century (2012), ch.1: Set Theory form Cantor to Cohen, again by Kanamori. Feb 17, 2014 at 10:31
• I'd also add Dana Scott's Foreword to John Bell's Set Theory, Boolean-Valued Models and Independence Proofs, which explicitly address this issue. Feb 18, 2014 at 21:21

I found A Beginner’s Guide to Forcing by Timothy Chow very helpful in explaining how forcing was invented. It’s online and only 16 pages long.

An important insight of Cohen’s was that it’s useful to consider what he called standard models of $\mathsf{ZFC}$. A model $(M,R)$ of $\mathsf{ZFC}$ is standard if and only if the elements of $M$ are well-founded and the relation $R$ is ordinary set membership.

...

The somewhat counter-intuitive fact that $\mathsf{ZFC}$ has countable models with many missing subsets provides a hint as to how one might go about constructing a model of $\mathsf{ZFC}$ satisfying $\neg \mathsf{CH}$. Start with a countable standard transitive model $M$.

...

Consider a function $F$ from the Cartesian product $\aleph_{2}^{M} \times \aleph_{0}$ into $2 = \{ 0,1 \}$, where $\aleph_{2}^{M}$ denotes the element of $M$ that plays the role of the second uncountable cardinal. We may interpret $F$ as a sequence of functions from $\aleph_{0}$ into $2$. As $M$ is countable and transitive, so is $\aleph_{2}^{M}$; we can thus arrange for these functions to be pairwise distinct. Now, if $F$ already lies in $M$, then $M$ satisfies $\neg \mathsf{CH}$.

...

However, what if $F$ is missing from $M$? A natural idea is to add $F$ to $M$ to obtain a larger model of $\mathsf{ZFC}$, which we might denote by $M[F]$. The hope would be that $F$ can be added in a way that does not ‘disturb’ the structure of $M$ too much.

The careful creation of $M[F]$ is called forcing.

• This explains the idea behind the mechanics of forcing. But not the historical context from which Cohen could have come up with the technique itself. Jul 15, 2016 at 20:20
• True. I interpreted the question as how did Cohen conceive of Forcing? I believe the article gives a good context for that. Jul 15, 2016 at 22:00
• @kibble: thanks for the reference. Jul 26, 2016 at 12:07
• I like Yurii, and I appreciate him hosting the text, but using an arXiv version is better. Jul 24, 2017 at 7:49