# Mathematics Colloquium, Bar-Ilan University, November 2013

I gave a colloquium talk at Bar-Ilan University on November 10, 2013.

Abstract:

Paul Cohen celebrated solution to Hilbert’s first problem showed that the Continuum Hypothesis is independent of the usual axioms of set theory. His solution involved a new apparatus for constructing models of set theory – the method of forcing. As Cohen predicted, the method of forcing became very successful in establishing the independence of various statements from the usual axioms of set theory. What Cohen never imagined, is that forcing would be found useful in proving theorems (that is, implications).

In this talk, we shall motivate the forcing machinery, and then present a collection of results that were proved using the method of forcing.

The talk was intended for a general audience, and comments are very welcome.

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### 4 Responses to Mathematics Colloquium, Bar-Ilan University, November 2013

In A nonconstructible $\Delta^1_3$ set of integers, the following is proved by Robert Solovay using forcing:

Theorem. Assuming the existence of a Ramsey cardinal, there is a $\Delta^1_3$ set of sets of integers, X, which is not constructible from any set of integers A.

The following is stated about the proof of the above theorem:

It is amusing to note that the proof uses Cohen’s notion of a generic set of integers. This is probably the first application of Cohen’s method to set theory yielding an absolute result rather than a relative consistency result.

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• saf says:

Thanks for this finding, Mohammad! It is an amazing fact that already in 1967, Solovay proved a ZFC theorem using forcing.

p.s.
Here is an interesting quote of Shelah (from here):

At the end of my lecture, Paul was rather complimentary and said that he expected his method of forcing to be good in set theory but not for problems in other fields. He said that he felt like a father whose sons have taken things far further than he could have hoped and, moreover, that using forcing to prove theorems in ZFC thrilled him.

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2. Dear Assaf,

The only forcing free proof of the Gitik-Shelah theorem (forcing with a sigma ideal cannot be isomorphic to a product of Cohen and random forcings) that I used in the rational distance result is probably due to Burke and Fremlin as I read here:

https://www.essex.ac.uk/maths/people/fremlin/n96j01.ps

Kamburelis’ paper seems to cover the cases of random and Cohen forcings but not the product. Do you know another published version of the Cohen times random case?

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• saf says:

Hi Ashutosh,
I very much like this theorem of yours concerning irrational distances!

Thanks for your feedback – I corrected the slides. Next time I see Gitik, I will ask him about known forcing-free proofs of the Cohen times random case. I asked him: he is not aware of additional forcing-free proofs.

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