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Technique for proving consistency and independence results
- Forcing (mathematics) In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object.
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In the mathematical discipline of set theory, forcing is a technique for proving consistency and independence results. Intuitively, forcing can be thought of as a technique to expand the set theoretical universe to a larger universe by introducing a new "generic" object .
- 2. Executive summary
- 3. Models of ZFC
- 3.1. Apparent circularity. One common confusion about models of ZFC stems from a tacit expectation that some people have, namely that we are supposed to suspend all our preconceptions about sets when beginning the study of ZFC. For example, it may have been a surprise to some readers to see that a universe is de ned to be a set together with: : : . Wait a minute|what is a set? Isn't it circular to de ne sets in terms of sets?
- 3.2. Existence of examples. A course in group theory typically begins with many examples of groups. One then veri es that the examples satisfy all the axioms of group theory. Here we encounter an awkward feature of models of ZFC, which is that exhibiting explicit models of ZFC is di cult. For example, there are no nite models of ZFC. Worse, by a result known as the completeness theorem, the statement that ZFC has any models at all is equivalent to the statement that ZFC is consistent, which is an assumption that is at least mildly controversial. So how can we even get o the ground?
- 3.3. \Standard" models. Even granting the consistency of ZFC, it is not easy to produce models. One can extract an example from the proof of the complete-ness theorem, but this example is unnatural and is not of much use for tackling CH. Instead of continuing the search for explicit examples, we shall turn our attention to important properties of models of ZFC.
- 6. Boolean-valued models
- 7. Generic ultra lters and the conclusion of the proof sketch
- 8. But wait|what about forcing?
- 9. Final remarks
The negation of CH says that there is a cardinal number, @1, between the cardinal numbers @0 and 2@0. One might therefore try to build a structure that satis es the negation of CH by starting with something that does satisfy CH (Godel had in fact constructed such structures) and \inserting" some sets that are missing. The fundamental theorem of for...
As mentioned above, Cohen proved the independence of CH from ZFC; more precisely, he proved that if ZFC is consistent, then CH is not a logical consequence of the ZFC axioms. Godel had already proved that if ZFC is consistent, then :CH, the negation of CH, is not a logical consequence of ZFC, using his concept of \constructible sets." (Note that th...
In fact, we are not de ning sets in terms of sets, but universes in terms of sets. Once we see that all we are doing is studying a subject called \universe theory" (rather than \set theory"), the apparent circularity disappears. The reader may still be bothered by the lingering feeling that the point of in-troducing ZFC is to \make set theory rigor...
Fortunately, these di culties are not as severe as they might seem at rst. For example, one entity that is almost a model of ZFC is V , the class of all sets. If we take M = V and we take R to mean \is a member of," then we see that the axiom of extensionality simply says that two sets are equal if and only if they contain the same elements|a manif...
One important insight of Cohen's was that it is useful to consider what he called standard models of ZFC. A model M of ZFC is standard if the elements of M are well-founded sets and if the relation R is ordinary set membership. Well-founded sets are sets that are built up inductively from the empty set, using operations such as taking unions, subse...
To recap, we have reached the point where we see that if we want to construct a model of :CH, it would be nice to have a method of starting with an arbitrary standard transitive model M of ZFC, and building a new structure by adjoining some subsets that are missing from M. We explain next how this can be done, but instead of giving the construction...
At this point we have a powerful theorem in hand. We can take any model M, any complete Boolean algebra B in M, and any ultra lter U of M, and form a new model MB=U of ZFC. We can now experiment with various choices of M, B, and U to construct all kinds of models of ZFC with various properties. So let us revisit our plan (in Section 5) of starting ...
The reader may be surprised|justi ably so|that we have come to the end of our proof sketch without ever precisely de ning forcing. Does \forcing" not have a precise technical meaning? Indeed, it does. In Cohen's original approach, he asked the following funda-mental question. Suppose that we want to adjoin a \generic" set U to M. What properties of...
We should mention that the Boolean-valued-model approach has some disad-vantages. For example, set theorists sometimes nd the need to work with models of axioms that do not include the powerset axiom, and then the Boolean-valued model approach does not work, because \complete" does not really make sense in such contexts. Also, Cohen's original appr...
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Aug 29, 2016 · Let me say a little bit about the proof. The key idea is the forcing relation: Definition. For $\mathbb{P}\in V$ a poset and $p\in\mathbb{P}$, we say $p$ forces $\varphi$ - and write "$p\Vdash\varphi$" - if for every generic (over $V$) filter $X$ containing $p$, $V[X]\models\varphi$. (Here $\varphi$ is a sentence that maybe also refers to ...
I S . . . What is forcing? Forcing is a remarkably powerful technique for the construction of models of set theory. It was invented in 1963 by Paul Cohen1, who used it to prove the independence of the Continuum Hypothesis.
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Aug 20, 2020 · Two key facts about forcing are (1) the definability of forcing; i.e., the existence of a notion $\Vdash^\star$ (to use Kunen's notation) such that $p\Vdash \phi$ if and only if $ (p \Vdash^\star \phi)^M$, and (2) the truth lemma; i.e., anything true in $M [G]$ is forced by some $p\in G$.
More recent set theory texts often incorporate the double negation translation directly into the definition of the forcing relation, or use a more semantic definition that automatically gives classical logic. This can obscure the underlying intuitionistic nature of set theoretic forcing.
Feb 6, 2021 · A special method for constructing models of axiomatic set theory. It was proposed by P.J. Cohen in 1963 to prove the compatibility of the negation of the continuum hypothesis, ¬CH, and other set-theoretic assumptions with the axioms of the Zermelo–Fraenkel system ZF ([1]). The forcing method was subsequently simplified and modernized ([2 ...
