*boolean valued-models*. I want to give a brief sketch of Scott's proof, but one should definitely read the extremely well-writen paper [1] for all the details.

The first nice thing about Scott's paper is that it starts by giving a completely natural formulation of CH. CH is often stated using cardinals as the equation $$2^{\aleph_0} = \aleph_1$$ so it's easy to dismiss it as abstract non-sense. But it can also be rephrase in concrete terms as:

*Given any set of reals, either the set of integers can be mapped onto the set, or the set can be mapped onto the whole set of reals*. So, the question is: Is there a set of reals whose cardinality lies strictly between the cardinalities of natural numbers and the whole set of reals?

Formally this can be written as $$ \forall X ( \exists f^{\mathbb{N} \rightarrow X} (\mbox{Img}(f) = X) \vee \exists g^{X \to \mathbb{R}} (\mbox{Img}(g) = \mathbb{R}) ). $$ I'm writing \( x^X \) as an abbreviation for \( x \in X \).

Scott then builds a model of set theory (a version slightly different than the usual presentation of the Zermelo-Frankel set theory) in which CH is false.

The first novelty is that the construction of the model is parametrised by a probability space \( (\Omega, \cal{A}, P) \). The real numbers are interpreted as

*random variables*, i.e. in the constructed model the set of real number \( \cal{R} \) consists of all \( \eta \colon \Omega \to \mathbb{R} \) such that $$ \{\omega \in \Omega \; \colon \; \eta(\omega) \leq r \} \in \cal{A} $$ where \( \mathbb{R} \) is the "real" set of real numbers. The whole point is to make the set of reals in the model \( \cal{R} \) much much bigger than the actual set of real numbers \( \mathbb{R} \).

The second novelty is that formulas are no longer either true or false, but they have a real (in the sense of \( \mathbb{R} \)) value between zero and one (saying how true it is, with one meaning "very" true). So, equality between two "real numbers" \( \eta \colon \Omega \to \mathbb{R} \) and \( \zeta \colon \Omega \to \mathbb{R} \) is given as $$ val(\eta = \zeta) = \{\omega \in \Omega \; \colon \; \eta(\omega) = \zeta(\omega) \} / [P = 0] $$ where one has to quotient out the set of events which have \(P\)-measure zero. This valuation function can be extended to all formulas, and the main result is that if a formula \( A \) is provable in set theory then \( val(A) = 1 \), independent of the probability space we started with.

Hence, in order to show that CH is not provable, it's enough to find some probability space \( (\Omega, \cal{A}, P) \) such that in that space \(val(\)CH\() < 1\). Scott fixes \( I \) an arbitrary set of cardinality greater than \( 2^{\aleph_0} \), and then takes \( \Omega = [0,1]^I \). Because \( I \) is huge, the set of sample points \( \Omega \) will also be huge, which means the set of reals \( \cal{R} \) in the model has an immense cardinality.

So, now we need to build a set \( X \subseteq \cal{R}\) such that neigher \( \exists f^{\mathbb{N} \rightarrow X} (\mbox{Img}(f) = X) \) nor \( \exists g^{X \to \cal{R}} (\mbox{Img}(g) = {\cal R}) \). One can check that the projection \( \xi_i \colon \Omega \to \mathbb{R} \) are indeed random variables, and hence are real numbers in the model (i.e. belong to \(\cal{R}\)). Pick a subset \( J \subset I \) which is uncountable but still of cardinality strictly smaller than \( I \). Our set of reals \( X \subset \cal{R}\) will be (informally) $$ X = \{ \xi_j \; \colon \; j \in J \} $$ which one can show that is not countable but has (inside the model) cardinality smaller than the reals \( \cal{R} \). Neat!

Anyway, I hope Scott will forgive me for such a sketchy account of his paper, but I hope this will serve as an advertisement for the curious ones to go and have a closer look at all the details in [1].

[1] Dana Scott, A proof of the independence of the continuum hypothesis, Mathematical systems theory, 1(2):89-111, 1967.

## 2 comments:

There is now a new proof using classical realisability!

Thanks for the effort to make this accessible!

One thing that confuses me - you say that in the model where CH is false the reals have huge cardinality, but don't they have to have cardinality 2^{א_0} for result to be meaningful?

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