Forcing (mathematics) Totally Explained

Everything about Forcing Mathematics totally explained

In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen, for proving consistency and independence results in set theory. It was first used, in 1962, to prove the independence of the continuum hypothesis and the axiom of choice from Zermelo-Fraenkel set theory. Forcing was considerably reworked and simplified in the sixties, and has proven to be an extremely powerful technique both within set theory and in other areas of mathematical logic such as recursion theory. Descriptive set theory utilizes both the notion of forcing from recursion theory as well as set theoretic forcing. Forcing has also been used in model theory but it's common in model theory to define genericity directly without mention of forcing.

Intuitions

   Forcing is equivalent to the method of Boolean-valued models, which some feel is conceptually more natural and intuitive, but usually much more difficult to apply.
   Intuitively, forcing consists of expanding the set theoretical universe V to a larger universe V*. In this bigger universe, for example, one might have lots of new subsets of ω = . Because of the mutual interdefinability between r and G, one generally writes V[r] for V[G].
   A different interpretation of reals in V[G] was provided by Dana Scott. Rational numbers in V[G] have names that correspond to countably many distinct rational values assigned to a maximal antichain of Borel sets, in other words, a certain rational-valued function on I = [0,1]. Real numbers in V[G] then correspond to Dedekind cuts of such functions, that is, measurable functions.

Boolean-valued models

ť Main article: Boolean-valued model

Perhaps more clearly, the method can be explained in terms of Boolean-valued models. In it, any statement is assigned a truth value from some infinite Boolean algebra, rather than just a true/false value. Then an ultrafilter is picked in this Boolean algebra, which assigns values true/false to statements of our theory. The point is that the resulting theory has a model which contains this ultrafilter, which can be understood as a model obtained by extending the old one with this ultrafilter. By picking a Boolean-valued model in appropriate way, we can get a model that has the desired property. In it, only statements which must be true (are "forced" to be true) will be true, in a sense (since it has this extension/minimality property).

Meta-mathematical explanation

In forcing we usually seek to show some sentence is consistent with ZFC (or optionally some extension of ZFC). One way to interpret the argument is that we assume ZFC is consistent and use it to prove ZFC combined with our new sentence is also consistent.
Each "condition" is a finite piece of information - the idea is that only finite pieces are relevant for consistency, since by the compactness theorem a theory is satisfiable if and only if every finite subset of its axioms is satisfiable. Then, we can pick an infinite set of consistent conditions to extend our model. Thus, assuming consistency of set theory, we prove consistency of the theory extended with this infinite set.

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