I promised my friend Zoltán a translation of an older post in Spanish, Modelos del Universo, so here it is. It is almost a literal version.

After some reflection on independence proofs in Set Theory using the method of forcing, one of my conclusions was that it should not be too counterintuitive the fact of adding a new set, i.e., a set not obtainable from previous existing ones using the already established operations of set construction.  For instance, we don’t feel any remorse for taking an arbitrary ring $k$ and adjoining an element transcendental over it, obtaining $k[x]$.

While writing this, it is impossible to avoid noticing that the universe of all sets $V$ is a Boolean ring with the operations of symmetric difference and intersection. The question now, is it legal to “evoke” some $x$ and construct $V[x]$? If $V$ is really all that is, this won’t have any sense from a foundational perspective. The key out of this dilemma is not forcing. In fact, in the context of the general problem, forcing was created by Paul J. Cohen to break the formidable barriers to obtain a “transcendental” $x$ (technically called generic —just like in Baire category), and moreover, in a coarse caricature of this development, to construct “formally” $V[x]$. In spite of this, the tools that allow to prove the “existence” of $V[x]$ predate forcing.

The class $V$ of all sets is not a set, but it can be constructed in partial stages that are sets. These stages are known as the cumulative hierarchy of sets, and they start at the empty set, successively applying the powerset operation:

$$V_0\defi \emptyset,$$

$$V_{\al+1} \defi \P(V_\al).$$

This processes doesn’t come to an end with the natural numbers; hence, when the device of taking powersets is exhausted, we take the union of all the sets already obtained and proceed all through the ordinals. In conclusion, when $\ga$ is a limit ordinal we stipulate:

$$V_{\ga} \defi \textstyle\bigcup\{V_\be : \be <\ga\}.$$

The Axiom de Foundation (o Regularity) ensures that $V = \bigcup_{\al\in\On} V_\al$. Now I want to refer a tool called the Reflection Principle, proved by Montague in 1961, that states how these $V_\al$ converge to the universe $V$.

Theorem. For each first order formula $\phi(x_1,\dots,x_n)$  and sets  $a_1,\dots,a_n$, there are unboundedly many ordinals $\al$ such that  $V_\al$ satisfies $\phi(a_1,\dots,a_n)$ if and only if it holds in the universe.

Recall that a first order formula may contain quantifiers $\forall$ and $\exists$, so we can express rather complex properties by using them. An abstract example of this (when there are no parameters $x_i$) is that of the very axioms of set theory. For any finite family of them, we can take their conjunction as $\phi$ above and we may conclude that there are numerous sets that verify these axioms. In a more spectacular form, there is a set that models all mathematics proved to this date.

Returning to our thread of reasoning, the Reflection Principle provides us with a set $V_\al$ that satisfies every axiom we might need for each particular situation. The second tool, that states another reflection phenomenon, is both older and more surprising.

To perform Cohen’s construction, it is necessary to apply a version of Baire Category Theorem. As you may recall, this theorem states (in one of its variants) that the intersection of a countable sequence of dense open sets in a complete metric space is nonempty. But this theorem does not hold if one takes an uncountable family of dense open sets. Analogously, forcing requires a countable set $M$ in order to find the generic element $x$ and then to construct the extension $M[x]$. This is exactly what the Downward Löwenheim, Skolem, and Tarski Theorem, that we state here for the case of structures with a countable language (i.e., countably many distinguished operations and relations):

Theorem. Every first order structure  $W$ contains a countable elementary substructure $M$. That is, $M$ is closed under $W$’s operations, it respects defined relations, and for each formula $\phi$ and parameters $a_1,\dots,a_n\in M$, $M$ satisfies $\phi(a_1,\dots,a_n)$ if and only if it holds in $W$.

In the case of models of set theory, the language is always  $\{\in\}$, so that the hypothesis holds.

If we apply this theorem to the $W$ above, we will obtain a countable set $M$ such that its elements are sets, and equipped with the $\in$ relation it satisfies all properties of sets that have been used throughout the history of mathematics.

Skolem’s Paradox states that in $M$ we may prove that the set of real numbers exists, and that it is not countable. Nevertheless, there can be at most countably many elements in $\R$, since $M$ is countable.

The last paragraph illustrates, to my opinion, some of the salient difficulties that a mathematician might face in understanding the first independence proofs.