8/8/2019 Forcing Final

http://slidepdf.com/reader/full/forcing-final 1/2

WHAT IS Forcing?

Thomas Jech 

What is forcing ? Forcing is a remarkably pow-erful technique for the construction of models of set theory. It was invented in 1963 by Paul Co-hen who used it to prove the independence of the

Continuum Hypothesis. He constructed a modelof set theory in which the continuum hypothesis(CH) fails, thus showing that CH is not provablefrom the axioms of set theory.

What is the Continuum Hypothesis? In 1873Georg Cantor proved that the continuum is un-countable; that there exists no mapping of theset N of all integers onto the set R of all realnumbers. Since R containsN, we have 2ℵ0 > ℵ0,where 2ℵ0 and ℵ0 are the cardinalities of R andN, respectively. A question arises whether 2ℵ0

is equal to the cardinal ℵ1, the immediate suc-cessor of  ℵ0. Cantor’s conjecture that 2ℵ0 = ℵ1

is the celebrated continuum hypothesis made fa-mous by David Hilbert who put it on the top of his list of major open problems in the year 1900.Cohen’s solution of Cantor’s problem does notprove 2ℵ0 = ℵ1, nor does it prove 2ℵ0 = ℵ1. Theanswer is that CH is undecidable.

What does it mean that a conjecture is unde-

cidable? In mathematics, the accepted standardof establishing truth is to give a proof of a theo-rem. Thus one verifies a given conjecture by find-ing a proof of the conjecture, or one refutes it byfinding a proof of its negation. But it is not nec-essary that every conjecture can be so decided: it

may be the case that there exists no proof of theconjecture and no proof of its negation.

To make this vague discussion more precise wewill first elaborate on the concepts of theoremand proof.

What are theorems and proofs? It is a usefulfact that every mathematical statement can beexpressed in the language of set theory. All math-ematical objects can be regarded as sets, and rela-tions between them can be reduced to expressionsthat use only the relation ∈. It is not essentialhow it is done, but it can be done: For instance,integers are certain finite sets, rational numbers

are pairs of integers, real numbers are identifiedwith Dedekind cuts in the rationals, functions aresome sets of pairs etc etc. Moreover all “self-evident truths” used in mathematical proofs can

be formally derived from the axioms of set the-ory. The accepted system of axioms of set the-ory is ZFC, the Zermelo-Fraenkel axioms plus theaxiom of choice. As a consequence every math-ematical theorem can be formulated and provedfrom the axioms of ZFC.

When we consider a well formulated mathe-matical statement (say, the Riemann Hypothe-sis) there is a priori no guarantee that there ex-ists a proof of the statement or a proof of itsnegation. Does ZFC decide every statement? Inother words, is ZFC complete? It turns out thatnot only ZFC is not complete but it cannot be

replaced by a complete system of axioms. Thiswas Godel’s 1931 discovery known as the Incom-pleteness Theorem.

What is G¨ odel’s Incompleteness Theorem ? Inshort, every system of axioms that is (i) recur-sive and (ii) sufficiently expressive is incomplete.“Sufficiently expressive” means that it includesthe “self-evident truths” about integers, and “re-cursive” means roughly that a computer programcan decide whether a statement is an axiom ornot. (ZFC is one such system, Peano’s systemof axioms for arithmetic is another such systemetc.)

It is of course one thing to know that, byGodel’s theorem, undecidable statements exist,and another to show that a particular conjectureis undecidable. One way to show that some state-ment is unprovable from given axioms is to finda model.

What is a model ? A model of a theory inter-prets the language of the theory in such a waythat the axioms of the theory are true in themodel. Then all theorems of the theory are truein the model, and if a given statement is false in

1

8/8/2019 Forcing Final

http://slidepdf.com/reader/full/forcing-final 2/2

2

the model, then it cannot be proved from the ax-ioms of the theory. A well known example is amodel of non Euclidean geometry.

A model of set theory is a collection M  of setswith the property that the axioms of ZFC aresatisfied under the interpretation that “sets” areonly the sets belonging to M . We say that M satisfies ZFC. If, for instance, M  also satisfies thenegation of CH, then CH cannot be provable inZFC.

Here we mention another result of Godel, from1938: the consistency of CH. Godel constructeda model of ZFC, the constructible universe L,that satisfies CH. The model L is basically theminimal possible collection of sets that satisfies

the axioms of ZFC. Since CH is true in L, it fol-lows that CH cannot be refuted in ZFC. In otherwords, CH is consistent.

Cohen’s accomplishment was that he found amethod how to construct other models of ZFC.The idea is to start with a given model M  (theground model ) and extend it by adjoining an ob-

 ject G, a sort of imaginary set. The resultingmodel M [G] is more or less a minimal possiblecollection of sets that includes M , contains G,and most importantly, also satisfies ZFC. Cohenshowed how to find (or imagine) the set G sothat CH fails in M [G]. Thus CH is unprovable

in ZFC, and because CH is also consistent, it isindependent , or undecidable.

A consequence of Godel’s theorem about L isthat one cannot prove that there exists a set out-side the minimal model L, so we have to pull Gout of thin air. The genius of Cohen was to intro-duce so called forcing conditions that give partialinformation about G and then to assume that Gis a generic set. A generic set decides which forc-ing conditions are considered true. With Cohen’sdefinition of forcing and generic sets it is possibleto assume that for any ground model M  and anygiven set P  of forcing conditions in the model M 

a generic set exists. Moreover, if G is generic (forP  over M ), then M [G] is a model of set theory.

To illustrate the method of forcing, let us con-sider the simplest possible example, and let usstipulate that G should be a set of integers. Asforcing conditions we consider finite sets of ex-pressions a ∈ G and a /∈ G where a rangesover the set of all integers. (Thus {1 ∈ G, 2 /∈G, 3 ∈ G, 4 ∈ G} is a condition that forcesG ∩ {1, 2, 3, 4} = {1, 3, 4}.) The genericity of  Gguarantees that the conditions in G are mutually

compatible, and, more importantly, that everystatement of the forcing language is decided oneway or the other. We shall not define here what“generic” means precisely, but let me point outone important feature. Let us identify the abovegeneric set of integers G with the set G of forcingconditions that describe the initial segments of G. Genericity implies that for any statement Aof the forcing language, if every condition can beextended to a condition that forces A then somecondition p in G forces A (and so A is true inM [G]). For instance, if  S  is a set of integers andS  is in the ground model M  then every condition

 p of the kind described above can be extended toa condition that forces G = S : simply add the ex-

pression “a ∈ G” for some a /∈ S  (or “a /∈ G” forsome a ∈ S ) where a is an integer not mentionedin p. It follows that the resulting set of integersG is not in M , no matter what the generic set is.

Cohen showed how to construct the set of forc-ing conditions so that CH fails in the resultinggeneric extension M [G]. Soon after Cohen’s dis-covery his method was applied to other state-ments of set theory. The method is extremelyversatile: every partially ordered set P  can betaken as the set of forcing conditions, and whenG ⊂ P  is a generic set then the model M [G] is amodel of ZFC. Moreover, properties of the model

M [G] can be deduced from the structure of P . Inpractice, the forcing P  can be constructed withthe independence result in mind; forcing condi-tions usually “approximate” the desired genericobject G.

In the 45 years since Cohen, literally hundredsof applications of forcing have been discovered,giving a better picture of the universe of sets byproducing examples of statements that are unde-cidable from the conventional axioms of mathe-matics.

A word of caution: If after reading this youentertain the idea that perhaps the Riemann Hy-

pothesis could be solved by forcing, forget it.That conjecture belongs to a class of statementsthat, by virtue of their logical structure, are ab-

solute for forcing extension: such a statement istrue in the generic extension M [G] if and only if it is true in the ground model M . This is thecontent of Shoenfield’s Absoluteness Theorem.

And what is Shoenfield’s Absoluteness Theo-

rem ? Well, that is for someone else to explain,some other time.