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Martin's Axiom Chapter 14 Martin’s Axiom In this chapter, we shall introduce a set-theoretic axiom, known as Martin’s Axiom, which is independent of ZFC. In the previous chapter we have compared forcing extensions with group extensions. Similarly, we could also compare forcing exten- sions with field extensions. Now, if we start, for example, with the field of rational numbers Q and extend Q step by step with algebraic extensions, we finally obtain an algebraic closure F of Q. Since F is algebraically closed, we cannot extend F with an algebraic extension. With respect to forcing extensions, we have a somewhat similar situation: If we start, for example, with Gödel’s model L, which is a model of ZFC + CH, and extend L step by step with forcing notions of a certain type, we finally obtain a model of ZFC which cannot be extended by a forcing notion of that type. The model we obtain in this way is a model in which Martin’s Axiom holds. In other words, models in which Martin’s Axiom holds are closed under certain forcing extensions, like algebraically closed fields are closed under algebraic extensions. As a matter of fact we would like to mention that in the presence of the Continuum Hypothesis, Martin’s Axiom is vacuously true. However, if the Continuum Hypothesis fails, then Martin’s Axiom becomes an interesting combinatorial statement as well as an important tool in Combinatorics which has many applications in Topology, but also in areas like Analysis and Algebra (see RELATED RESULTS 83 & 84). Filters on Partially Ordered Sets Below, we introduce the standard terminology of partially ordered sets, as it is used in forcing constructions. In the context of forcing, a binary relation “ ” on a set P is a partial ordering of P if it is transitive (i.e., p q and q ≤r implies p r) and reflexive (i.e., p p for every p P ). Notice≤ that we do≤ not require that≤ “ ” is anti-symmetric (i.e.≤, p q and q ∈ p implies p = q), as we have done in Chapter≤ 6. If“ ”is a ≤ ≤ ≤ 341 342 14 Martin’s Axiom partial ordering on P , then (P, ) is called a partially ordered set. If P = (P, ) is a partially ordered set, then≤ the elements of P are usually called conditions≤ , since in the context of forcing, elements of partially ordered sets are conditions for sentences to be true in generic extensions. Two conditions p1 and p2 of P are called compatible, denoted p1 p2, if there exists a q P such that p1 q p2; otherwise they are called incompatible| , denoted p ∈ p . ≤ ≥ 1 ⊥ 2 A typical example of a partially ordered set is the set of finite partial functions with inclusion as their partial ordering: Let I and J be arbitrary sets. Then Fn(I,J) is the set of all functions p such that • dom(p) fin(I), i.e., dom(p) is a finite subset of I, and ∈ • ran(p) J. ⊆ For p, q Fn(I,J) define ∈ p q dom(p) dom(q) q = p. ≤ ⇐⇒ ⊆ ∧ |dom(p) If we consider functions as sets of ordered pairs, as we usually do, then p q is just p q. We leave it as an exercise to the reader to verify that (Fn(I,J), )≤is indeed a partially⊆ ordered set. ⊆ Let P = (P, ) be a partially ordered set, and for the moment let C P . ≤ ⊆ • C is called directed if for any p ,p C there is a q C such that p q p . 1 2 ∈ ∈ 1 ≤ ≥ 2 • C is called open (or upwards closed), if p C and q p implies q C. For example with respect to (Fn(I,J), ), for every∈ x I≥the set p Fn(∈ I,J): x dom(p) is open. ⊆ ∈ { ∈ ∈ } • C is called downwards closed if p C and q p implies q C. ∈ ≤ ∈ • C is called dense if for every condition p P there is a q C such that q p. For example with respect to (Fn(I,J)∈, ), for every x ∈I the set p Fn(≥I,J): x dom(p) is dense. ⊆ ∈ { ∈ ∈ } • Anon-emptyset F P is a filter (on P ) if it is directed and downwards closed. Notice that this definition⊆ of “filter” reverses the ordering from the definition given in Chapter 6. • Let D P(P ) be a set of open dense subsets of P . A filter G P is a D- generic⊆ filter on P if G D = for every set D D. As an example,⊆ consider again (Fn(I,J), ).If F∩is a filter6 ∅ on Fn(I,J), then∈ F : X J is a function, where X is some⊆ (possibly infinite) subset of I. → S PROPOSITION 14.1. If (P, ) is a partially ordered set and D is a countable set of open dense subsets of P , then≤ there exists a D-generic filter on P . Moreover, for every p P there exists a D-generic filter G on P which contains p. ∈ Filters on Partially Ordered Sets 343 Proof. For D = Dn : n ω and p := p, choose for each n ω a pn Dn { ∈ } −1 ∈ ∈ such that pn pn , which is possible since Dn is dense. Then the set ≥ −1 G = q P : n ω(q pn) ∈ ∃ ∈ ≤ is a D-generic filter on P and p G. ∈ ⊣ It is natural to ask whether the restriction on the size of D can be weakened. In particular, one can ask whether D-generic filters also exist in the case when D is uncountable, or whether it is at least consistent that such fiters exist. In order to get a positive answer to these questions, we have to put a restriction on the partial ordering: Let (P, ) be a partially ordered set. A subset A P is an anti-chain in P if any two distinct≤ elements of A are incompatible. As mentioned⊆ in Chapter 6, this definition of “anti-chain” is different from the one used in Order Theory. A partially ordered set P = (P, ) satisfies the countable chain condition, denoted ccc, if every anti-chain in P is≤ at most countable (i.e., finite or countably infinite). Now we are ready to formulate Martin’s Axiom in its general form. Martin’s Axiom (MA). If P = (P, ) is a partially ordered set which satisfies ccc, and D is a set of less than c open dense≤ subsets of P , then there exists a D-generic filter on P . In other words, MA(κ) holds for each cardinal κ< c. If we assume CH, then κ < c is the same as saying κ ω, thus, by PROPO- ≤ SITION 14.1, CH implies MA. On the other hand, MA can replace the Contin- uum Hypothesis in many proofs that use CH. Furthermore, MA is consistent with ZFC + CH, as we shall see in Chapter 19. ¬ Instead of requiring that D < c, we can require that D κ for some cardinal κ. | | | |≤ MA(κ). If P = (P, ) is a partially ordered set which satisfies ccc, and D is a set of at most κ open dense≤ subsets of P , then there exists a D-generic filter on P . On the one hand, MA(ω) is just PROPOSITION 14.1, and therefore, MA(ω) is prov- able in ZFC. On the other hand, MA(c) is just false as we will see in FACT 14.5. So, we cannot generalise MA by omitting D < c. Another attempt to generalise MA would be to omit ccc. However, this attempt| | also fails. FACT 14.2. There exists a (non ccc) partially ordered set P = (P, ) and a set D ≤ of cardinality ω1 of open dense subsets of P such that no filter on P is D-generic. Proof. Consider the partially ordered set (Fn(ω,ω ), ). Notice first that, for ex- 1 ⊆ ample, 0, α : α ω1 is an uncountable anti-chain, and hence, (Fn(ω,ω1), ) does noth satisfyi ccc.∈ Now, for each α ω , the set ⊆ ∈ 1 Dα = p Fn(ω,ω ): α ran(p) ∈ 1 ∈ 344 14 Martin’s Axiom is an open dense subset of Fn(ω,ω1): Obviously, Dα is open. To see that Dα is also dense, take any p Fn(ω,ω1). If α ran(p), then p Dα and we are done. Otherwise, let n ω be∈ such that n / dom(∈ p) (notice that∈ such an n exists since ∈ ∈ dom(p) is finite). Now, let q := p n, α ; then q Dα and q p. Similarly, ∪ {h i} ∈ ≥ for each n ω, the set En = p Fn(ω,ω ): n dom(p) is open dense. ∈ { ∈ 1 ∈ } Let D = Dα : α ω En : n ω ; then D = ω . Assume that G { ∈ 1}∪{ ∈ } | | 1 ⊆ Fn(ω,ω ) is a D-generic filter on Fn(ω,ω ). Since for each n ω, G En = , 1 1 ∈ ∩ 6 ∅ fG = G is a function from ω to ω . Now, since for each α ω , G Dα = , the 1 ∈ 1 ∩ 6 ∅ function fG : ω ω1 is even surjective, which contradicts the definition of ω1. S → ⊣ In order to show that ccc cannot be omitted in MA, we first show that for countable sets J, Fn(I,J) satisfies ccc, and for this, we first prove the following powerful combinatorial result. LEMMA 14.3 (∆-SYSTEM LEMMA). Let E be an uncountable family of finite sets. Then there exists an uncountable family C E and a finite set ∆ such that for any pair of distinct elements x, y C : x y =⊆ ∆. ∈ ∩ Proof. Notice first that either there exists an uncountable E ′ E such that for every a E ′, x E ′ : a x is countable, ⊆ ∈ { ∈ ∈ } S or for every uncountable E ′ E there exists an a E ′ such that x E ′ : a x is uncountable⊆ .
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