Set Theory, Forcing and Real Line

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1 Set Theory, Forcing and Real Line Giorgio Laguzzi March 21, 2013 Abstract We give a very brief survey on ZFC theory (Zermelo-Fraenkel Set Theory) and we present an intuitive introduction to the method of forcing and some applications to the real line. Our exposition will be very informal, without any claim of completeness and rigour. The idea is just to give a very intuitive idea of what Set Theory and forcing-method are, and why they are interesting and useful from the viewpoint of Mathematics. In the final part we also expose some results recently obtained, concerning regularity properties of the real numbers, such as Lebesgue measurability, Baire property, Ramsey property and Silver measurability. Axioms of ZFC First of all, we would like to point out that the present paper really consists of a very informal introduction to Set Theory, and it is intended to just give a gentle and not too pedant presentation. For a very detailed and technical introduction to ZFC and forcing method, we refer the reader to [3], which we consider the most rigorous book on this topic. On the contrary, for a less demanding presentation, though still very precise and complete, one can see [6] and [1]. ZFC is a first order logic theory with equality and a unique binary relation symbol (intuitively thought as membership ). 1. Set existence: x(x = x). 2. Extensionality: x y( z(z x z y) x = y). 3. Comprehension Scheme: For every formula ϕ with free variables among x, z, v 1,..., v n, z v 1... v n y x(x y x z ϕ) 4. Foundation: x[ y(y x) y(y x z(z x z y))]. 5. Pairing: x y z(x z y z). 6. Union: F A Y x(x Y Y F x A). 7. Replacement Scheme: For every formula ϕ with free variables among x, y, A, v 1,..., v n, A v 1... v n ( x A!yϕ Y x A y Y ϕ). 1

2 On the basis of Axiom 1-7 one can define (subset), (empty set), S (ordinal successor, i.e., S(x) = x {x}), and the notion of well-ordering (the latter being an antisymmetric, reflexive, transitive binary relation R on a set X such that: x, y X(Rxy Ryx) and A X x A(x is R-minimal)). We can then complete our list of axioms as follows: 8. Infinity: x(0 x y x(s(y) x)). 9. Power Set: x y z(z x z y). 10. Choice: A R(R well-oders A). The intuitive meaning of the Comprehension Scheme is that given a set z and a formula ϕ, one can define a set y z of those elements in z satisfying ϕ. Note that it does not mean that given any formula, one can define a set consisting of those x satisfying such a formula. As a counterexample, one may consider the formula ϕ(x) x = x. In this case, we get the class V = {x : x = x}. (In this expression the symbol = has to be meant in two different ways; the former occurs as an equality in the metatheory, the latter inside the theory. This kind of confusion between symbols of the theory and symbols of the metatheory will come further, and one should keep in mind this difference). Obviously V cannot be a set itself, for otherwise we would have V V, contradicting the axiom of foundation. Such kind of object is called proper class, i.e., a class which is not a set. Any proper class can be described in ZFC, just via the corresponding formula, but it is not an object of the theory; that means the variables run only over sets, and not over proper classes. Some popular notions from the viewpoint of ZFC Ley us now try to become familiar with some common objects definable from ZFC. Natural numbers are identified with sets recursively constructed starting from the empty set : 0 = 1 = { } (note: 1 = { } = S(0)) n + 1 = n {n}. The set of all natural numbers is denoted by ω; note that ω can be even viewed as the union of all natural numbers. Since the successor operator S can be applied to any set, we can then define S(ω) = ω {ω}, which is also denoted by ω + 1, and we can then recursively define ω + 2 = S(ω + 1) = ω {ω} {ω + 1}. ω + n + 1 = S(ω + n) = ω {ω} {ω + 1} {ω + n}. ω + ω = {ω + n : n ω}. (ω + ω is denoted by ω 2 as well.) we can keep on such a construction in order to define the ω 2 + n s and then their limit ω 3 = {ω 2 + n : n ω}. 2

3 one can recursively get ω n and then ω ω = {ω n : n ω} (ω ω is denoted by ω 2.)... Remark 1. Note that 0, 1,..., n,... and ω + 1, ω + n,..., ω 2, ω ω can all be defined by recursion theorem. The unique object previously mentioned having a different flavour is just ω. In fact, for instance, if we look at ω 2, we realize that it is defined by using elements which are previously constructed (the ω + n s) plus replacement and union (running over ω, which we assume to be already defined): ω 2 = {ω + n : n ω}. On the contrary, for ω the situation is quite different; as limit, we would like to write it as a union, i.e., ω = {n : n ω}!!! But this is rather senseless, for the union defining ω runs over ω itself, which we want to define... This is somehow a symptom of the fact that we really need the axiom of infinity to ensure the existence of such a set. Note that all the sets α just defined are countable. i.e., there is a bijection j : α ω. A natural question which arises is whether in ZFC one can prove the existence of uncountable sets, i.e., sets that are not in bijection with ω. The answer is positive; in fact, let us define U = {α : j : α ω bijection} and then let ω + = {α : α U}. It is clear that ω + is uncountable, for if it were not then ω + would be in U and so ω + ω +, contradicting the axiom of foundation. This sets is usually denoted by ω 1. In a similar fashion one can then define ω 2, ω 3,..., ω ω and so on. Remark 2. All these sets that we have introduced have the particular feature of being transitive (i.e., every element of the set is also a subset) and well-ordered by. These sets are called ordinals. When dealing with ordinals, we can easily define an ordering in the following way: α < β iff α β. Another important notion used in Set Theory is that of cardinal: an ordinal α is a cardinal iff there is no β < α which is in bijection with α itself. So for example, ω, ω 1, ω n are cardinals, whereas ω + ω, ω ω are ordinals, but not cardinals. Finally, we introduce the notion of regular cardinal: given two ordinals α and β, we say that α is cofinal in β iff there is a function f : α β such that for every γ 1 β there is γ 0 α such that f(γ 0 ) > γ 1. a cardinal κ is said to be regular iff there is no λ < κ such that λ is cofinal in κ. Examples of regular cardinals are the ω n s, with n ω, whereas a counterexample is ω ω, since the function f : ω ω ω, such that f(n) = ω n, is cofinal. Non-regular cardinals are called singular. 3

4 The class of ordinal numbers is denoted by ON; more precisely, ON = {x : x is an ordinal}. Note that ON is transitive and well-ordered by. Nevertheless, ON is not an ordinal, since it is not a set, for otherwise we would have ON ON. Analogously, the class of all cardinals CN is proper. Models of ZFC In the previous section, we introduced some popular notions and we also show how one can recover some elementary notions (like natural numbers) under the viewpoint of Set Theory. It is not hard to show that actually many other common notions can fall into the range of action of this theory, such as those of function, real number, Boolean algebra, group, and so on. Even more, one can show that ZFC is so powerful that it can define the notion of structure, model and the language of ZFC too, so that it can somehow study itself. This self-reference could sound very suspicious, at first. Further, note that in the previous section we somehow cheat; when we said from the viewpoint of ZFC, what we actually meant? Formally ZFC is nothing more than a theory expressed in the first order language. In the previous section, the meaning from the viewpoint of ZFC must be interpreted as from the viewpoint of the intuitive interpretation of ZFC, i.e., the natural structure where we interpret the symbol as membership and we think of variables x, y, z as ranging over sets. The intuitive model that we had (and we have) in mind can be presented in the following way. Consider the recursive construction N 0 = N α+1 = P (N α ) (i.e., power set of N α ) N λ = λ = {N α : α < λ}. Finally put N = {N α : α ON}. Such formal definition of N provides the model that we intuitively had in mind. Nevertheless, an interpretation of our language with could be completely different. For instance, if we consider the structure having as universe the real line R and we interpret as < then we get a legal interpretation of our language. Obviously, one can immediately note that not all the axioms of ZFC are satisfied by this structure; for instance, the statement x(x = ) can be easily derived from ZFC, but it is not true in the structure (R, <), since the latter satisfies the statement x y(y x). The structure we just mentioned is not a model of ZFC and so one can object that it is not interesting for our purpose. Nevertheless, our intention was just to show how our language can be seen under a completely different (but still admissible) light. With the first attempt we have been unfortunate. So the question which arises is to find different structures which are models of ZFC. But, what we mean with the word model? Here, one has to be careful, since there are two formally different ways to define such a notion in ZFC. 1 - Relativization. let M be any class and E be a binary class-relation (intuitively, M is the universe and E is the interpretation of ). For every 4

5 formula φ the relativization φ M,E is recursively defined (on the complexity of φ) as follows: (x = y) M,E is x = y, and (x y) M,E is E(x, y); (ψ χ) M,E is ψ M,E χ M,E ; xψ is x(m(x) ψ M,E ). This definition means that, given some formula φ, one can produce a formula φ M,E which involves both the formula with one free variable defining the class M, say θ M (x,... ) (where... represent possible parameters), and the formula with two free variables defining the class E, say θ E (x, y,... ). Hence, one formally obtains a formula φ M,E expressed in the first order language of ZFC, and one can therefore, for instance, ask whether or not ZFC φ M,E. So, given a set of sentences S, one says that M, E is a model for S iff for every φ S, one has ZFC φ M,E. 2 - Model Theory. Another way to express the notion of model is by formalizing within ZFC all the notions of Model Theory. For instance, one can assign to each sentence φ a natural number n φ via the so called Gödelization. Further, one can also formalize the notion of set-structure A = A, E and the satisfactory relation, say =. In such a way, we can obtain the formula θ(a, E, n φ ) A, E = n φ. These two methods are formally very different. Nevertheless, a fairly standard proof on the complexity of a sentence φ shows the following metatheorem. Theorem 3. Let φ be a formula in the language of ZFC. Then ZF M, E (φ M,E θ(m, E, n φ )). When M, E are proper classes, we cannot use the universal quantification in the last formula. Nevertheless, we could get something similar, by fixing the two formulae θ M and θ E representing them. Theorem 4. Let φ be a formula in the language of ZFC, and fix two proper classes M and E described via the formula θ M and θ E, respectively. Then ZF φ M,E θ M,E (n φ ). Note that, in the right side, the formula θ M,E (n φ ) denotes a kind of reformulation of θ(m, E, n φ ), since M, E cannot be parameters any longer. By virtue of such (meta)theorems, from now on, we will adopt a sort of compromise, by using the notation M, E = φ to denote φ M,E (we omit E when this is meant as the standard interpretation of membership). Throughout this paper, all of the occurrences of statements like M, E = φ will be meant in the sense of the relativization. Let us now make some interesting and basic remarks. By Gödel s incompleteness theorem, there are statement φ such that neither φ nor φ are provable in ZFC. So in this case, if we pick M = ZFC+φ and N = ZFC + φ, we get M and N not being isomorphic. Except for 5

6 the usual artificial statement introduced by Gödel, many other natural examples have come out along the years (natural w.r.t. the intuitive interpratation of ZFC). Among these, let us consider the popular Continuum Hypothesis CH, asserting that the set of subsets of ω has cardinality ω 1, briefly 2 ω = ω 1. Gödel proved that the universe of constructible sets L satisfies CH, whereas Cohen in the 1960s invented the method of forcing to build a model for ZFC+ CH. Both L and the method of forcing will be briefly introduced in the coming pages. In some sense one can show that in ZFC there is a countable model of ZFC itself. Firstly, what does model mean here? By Gödel s second incompleteness theorem, ZFC cannot prove its own consistency, and so it cannot formally show that there is a model of itself. More precisely, ZFC M = ZF C, Hence the meaning of the statement ZFC proves there is a countable model of ZFC should be differently meant. In fact, the real meaning will be: given a finite list φ 0,..., φ n of axioms of ZFC, then ZFC M(M = φ 0 φ n ). This result is known as Reflection Theorem. Moreover, by Mostovsky s Collapse, one can make such a model being transitive. Secondly, what does countable mean? In the previous section we have shown that in ZFC one can prove that there are ordinals which are uncountable, like ω 1. Hence, one would say that any transitive model of ZFC has to contain ω 1 as an element and so also any of its element; but since ω 1 is uncountable we should have uncountably many such elements inside the model! What s the trick? One can actually recognize two inaccuracies in the above argument, which are somehow the two sides of the same coin. Firstly, we implicitly assumed the interpretation of being as membership, which is obviously not the unique possibility. In fact, as we mentioned at the beginning of this section, the interpretation of the binary relation symbol may vary, so that its meaning can be faraway from the intuitive one of membership. Secondly, the ω 1 that we are picking is the first uncountable cardinal from the viewpoint of M, but it could be anything else from an external viewpoint. Looking at that from a different viewpoint, this amazing fact does not seem so... amazing. In fact, if one thinks of the usual construction to prove Gödel s completeness theorem, it turns out that such a model has the same cardinality of the language in which the theory is expressed, therefore it is countable for ZFC. The intuitive reason for which such Gödel s model can be faraway from a standard natural interpretation is mainly because it is somehow built on the syntax. Note that, if we work inside a stronger theory than ZFC, we could really get a set model of ZFC. For example, let us assume the statement I κ > ω(κ is a cardinal α < κ(2 α < κ)) 6

7 (such kind of cardinal is called inaccessible.) One may then show that ZFC+I N κ = ZFC. Nevertheless, we are not interested in this kind of strong results. In fact, the aim of Set Theory is to study the consistency of a certain theory T relatively to some other theory T ; or in other words, assuming T being consistent, what can one say on the consistency of T? Obviously, the way to really build inside T a model M for T can be done only when T is strictly weaker than T, because of Gödel s second incompleteness theorem. It is then useless for our question, since the consistency of a weaker theory (T ) is simply implied by the consistency of the stronger one (T ). Hence, our question requires a different method. The following well-known fact sheds light on such results of relative consistency. Lemma 5. Let T and T be two sets of sentences. Assume that in T one can prove, for some class M, for every φ T, M = φ. Then Consistency of T Consistency of T. The latter implication is usually denoted with Con(T ) Con(T ). The idea is simply that if one can derive a contradiction from T, then some contradiction could also be derived in T. The idea of the proof is very simple; assume we can derive a contradiction from T, i.e, some formula of the form φ φ. Hence, by our assumption, we also get M = φ φ, indeed M = φ and M = φ. But that means T M = φ and M = φ. which is a contradiction. Let us conclude this section with two well-known examples of relative consistency results and with a mysterious remark. Example 6. We briefly introduce Gödel s constructible universe L. Firstly, remind that a set y is said to be definable over a structure A iff there exists a formula ϕ(x) in the language of A such that z y iff A = ϕ(z). For any set x, put Def(x) = {y x : y is definable over x, }. We then define L by recursion as follows: L 0 = L α+1 = Def(L α ) L λ = α<λ L α. We finally define the constructible universe as L = α ON L α. Gödel proved that such a structure is a model for ZFC, and it satisfies CH as well. It is not hard to show that L can be defined inside ZF, without any need of AC. Moreover, one can also check that for any φ ZFC, we have ZF L = φ. Hence, the following consistency result holds: Con(ZF) Con(ZFC + CH) 7

8 Example 7. One can show that Von Neumann s hierarchy, previously introduced, can be defined inside ZF, where the latter is the theory of Zermelo- Fraenkel without axiom of choice and foundation. Rather amazingly, one can show that the axiom of foundation is however satisfied by N, and so Con(ZF ) Con(ZF). Remark 8. We would like to show a strange thing turning out from our results. Let {φ n : n ω} be the list of axioms of ZFC. Moreover, put F n = φ 0 φ 1 φ n, for every n. By the reflection theorem we mentioned above, it follows that, for every natural number n, ZFC M = F n. So we would say ZFC n ω M = F n and then, by compactness, we get Contradiction... What went wrong? Real Line - Part I ZFC M = ZFC. In Analysis, the real line R is usually meant as the set consisting of rational and irrational numbers. Sometimes, it is also confused with the open interval (0, 1), which shares many topological and measure theoretical properties with R, since they are homeomorphic. However, in Set Theory, when one deals with problems regarding the real line, this is usually meant in a different flavour. The spaces which are largely investigated in the burgeoning area called set theory of the real line are the Cantor space and the Baire space, viewed as set theoretical privileged substitutes of R. Definition 9. The Cantor Space 2 ω is the set consisting of functions x : ω 2, endowed with the topology generated by basic open sets [s] = {x 2 ω : s x}, with s 2 <ω, i.e., the set of finite sequences of 0s and 1s. Analogously, one may define the Baire Space ω ω as the set of functions x : ω ω, endowed with the topology generated by [s] = {x ω ω : s x}, with s ω <ω, i.e., the set of finite sequences of natural numbers. One can also define a notion of countably additive measure on such spaces. This measure is induced by the measure m on basic open sets defined as follows. Cantor space: for every s 2 <ω, let m([s]) = 2 s, where s denotes the length of s. Baire space: for every s ω <ω, let m([s]) = j< s 2 (s(j)+1). From such m, one can construct the usual outer measure with the corresponding Carathedory s family, so to obtain a notion of measure and the corresponding family of measurable sets via a pretty standard method. Furthermore, once we have a topology and a notion of measure, we can introduce the two corresponding regularity properties. Definition 10. Let X 2 ω. 8

9 X is said to be measurable iff there is a Borel set B such that X B is null; X is said to satisfy the Baire property iff there is a Borel set B such that X B is meager. Analogously, one can define such notions on the Baire space. Many other notions of regularity can be introduced in a similar fashion. In the first chapter of my PhD thesis ([4]), I use a general method to introduce notions of regularity, of which Lebesgue measurability and Baire property turn out to be special cases. (We acknowledge that a similar approach has been independently used by Daisuke Ikegami and Yurii khomsky.) We said above that 2 ω and ω ω are somehow similar to the real line R. Let us now see what we meant. Two topological spaces X, Y are Borel isomorphic iff there is a function f : X Y such that for any Borel set B Y we have f 1 [B] is Borel. Fact 11. R, 2 ω and ω ω are Borel isomorphic. Even more, one can show that those spaces are homeomorphic up to a countable set. As an example, one may pick the function f : ω ω 2 ω \ Q (Q are the rationals, which are those elements in 2 ω which are eventually equal 0) such that x 0 {}}{{}}{{}}{ x = x 0 x 1... x n... f(x) = Borel isomorphisms are interesting from our viewpoint because questions regarding regularity properties are invariant between such spaces. More precisely, given two topological spaces X, Y, f Borel isomorphism between them, and X X one has X has the Baire property f[x] has the Baire property. (The same holds for Lebesgue measurability, perfect set property and any other regularity property introduced over the years.) For a detailed and extensive study of the real line with set theoretical methods, we refer the reader to [2]. Forcing Definition A forcing notion P is a triple (P,, 1), where P is called the set of conditions, is a partial order on P, without any minimal element (i.e., for every p P there is q P such that q p.), and 1 represents the maximum w.r.t.. Sometimes we abuse notation by writing that a condition is in P, by identifying the forcing with its set of conditions. Given two conditions p, q P such that q p we say that q is stronger than p. 2. A subset D P is said to be dense iff p P q D(q p). 3. A subset F P is said to be a filter iff: x 1 x n 9

10 (a) p, q F r F (r p r q); (b) if p F then q p(q F ). 4. A filter G is said to be generic iff for every dense subset D of P, one has G D. Further, we say that two conditions p, q P are incompatible (p q) iff there is no r P such that r p and r q. The existence of a generic filter is not generally provable. following result is what we need for our purpose. However, the Fact 13. Let D = {D n : n ω} be a countable family of dense subsets of P. Then there is a filter F P such that for every n ω, F D n. Proof. Consider the following recursive construction: pick p 0 D 0, and for every n ω, pick p n+1 p n such that p n+1 D n+1. Finally let F be the filter generated by these p n s, i.e., F = {q P : n ω(q p n )}. Then F satisfies the required property. Hence, the following is a simple corollary. Corollary 14. Let M be a countable transitive model of ZFC (i.e., some finite fragment of ZFC), and let P M be a forcing notion. Then there is G P generic filter over M, i.e., for every dense D P, if D M then G D. The following result shows a sufficient condition for which G / M. Fact 15. Let M be a transitive countable model and P be a forcing notion such that p P q 0, q 1 P(q 0 p q 1 p q 0 q 1 ). If G is P-generic over M, then G / M. The proof is by contradiction. Assume G M and put D = P \ G. Then D M and it is easy to check that D is dense. But, G D =, contradicting G being generic. The method of forcing was invented by Paul Cohen to build a model for ZFC, in which CH fails. Roughly, it says what follows: ( ) Let M be a countable model for ZFC n, where the latter is a finite conjunction φ 0 φ n of some axioms of ZFC. Further, let P M be a forcing notion and G P a generic filter over M. Then we can add G to the model M in order to define a structure M[G] as the smallest one extending M and containing G as an element. We also obtain the following: M[G] = ZFC n M[G] ON = M ON. Moreover, depending on the combinatorial properties of P, one can show that some further specific statements are true in M[G]. That means that different filters G s can decide differently about the truth of a given statement. But, how this model M[G] looks like? Very brief and rough presentation of the construction of M[G]. Our presentation is nothing more than a very rough and informal compendium of 10

11 Kunen s introduction to the method of forcing (see [3]), which represents, in our opinion, one of the most precise and detailed expositions written so far. First of all, we have to show how the elements of this new model are made. Given a forcing P, people living in M will not be generally able to know what an element in M[G] is, since that depends on the choice of the generic G. However, what people in M may do is to have a name for elements in M[G]. Formally, a P- name can be defined recursively as follows: τ is a P-name iff τ is a relation and and every (σ, p) τ is in τ and p P. The class of P-names in M is denoted by M P. As we said, a name does not definitively decide an element in M[G], but it is just a way to give a description, generally depending on the generic G, about how it will look like in the extension M[G] ( extension is used because we somehow extend the model M adding the new object G). Hence, we need a notion of evaluation of such names: Definition 16. The evaluation of τ via G is denoted by τ G and it is defined as follows: τ G = {σ G : p G((σ, p) τ)}. We then put M[G] = {τ G : τ M P }. Let us now see some easy examples. The empty-set is trivially a P-name and its evaluation is empty, by definition. Hence 0 = 0 G, for any generic G. Now, let τ = {(0, p)}, for some p P. In such a case, the evaluation of τ strictly depends on G; in fact, τ G = 0 if p / G, and τ G = {0} = 1 if p G. In some very particular cases, the evaluation is independent of G; for instance we just mentioned the case 0 = 0 G. Actually, one can note that if τ = {(0, 1)}, then τ G does not depend on G, since any filter contains the greatest element 1. More generally, if τ = {(σ j, 1) : j J} then τ G = {σ G : j J}. As a consequence, one obtains that there are standard names for elements of M which are independent of G; for x M there is a name ˇx defined recursively by: ˇx = {(ˇy, 1) : y x}. By the previous argument, ˇx is independent of the generic G P. There is also a canonical name for the generic G itself, i.e., Γ = {(ˇp, p) : p P}. In fact, by definition, Γ G = {ˇp G : p G} = {p : p G} = G. Definition 17. Given a condition p P and some sentence φ, we define as follows: p φ for every generic G(p G M[G] = φ). Note that, by definition, if q p and p φ, then also q φ. That explains why we use the word stronger when q p; in fact, this is meant in the sense that q contains more information than p, since it forces the same statements forced by p, and possibly some more sentences. 11

12 At first sight, it could seem that the decision whether or not some formula φ holds in M[G] depends on the knowledge of all generic G containing p as an element. Nevertheless, the following two facts, which are the key basic results of the theory of forcing, say something surprising: Fact 1. One can decide within M whether or not p φ. Fact 2. For any formula φ, if M[G] = φ then there is p G such that p φ. Let us see some concrete examples of forcing notions. Example 18. Let P(ω, ω) be the forcing consisting of finite sequence of natural numbers, ordered by extensions. Formally, P(ω, ω) = {p : p is a function n(dom(p) = n) ran(p) ω}, ordered by extension, i.e., q p iff dom(q) dom(p) and q dom(p) = p. Let G be generic over M and put x G = {p : p G}. Claim. x G belongs to ω ω, i.e., it is an infinite sequence of natural numbers. Proof. We have to show two things: 1. for every n ω, n dom(p), and 2. for every n ω, n ran(p). But, 1 and 2 simply follows by noting that D n = {p P(ω, ω) : n dom(p)} and D n = {p P(ω, ω) : n ran(p)} are dense, for every n ω. Remark that 2 is not strictly necessary to get x G ω ω, but it serves mostly to check that x G is surjective. A sceptic reader may now object that x G is simply a real (or better, an element of ω ω ) belonging to the model M, and what we are doing is just a strange way to define such a real. Let us now show that this is not the case. Fact 19. M[G] = z ω ω M n ω(z(n) < x G (n)). Before proving it, let us note that this is sufficient to show that x G / M, for otherwise z = x G would contradict the statement. So that means that the forcing technique is useful to add new reals to a model of ZFC. Proof of Fact 11. It suffices to note that, for every z ω ω M, is dense in P. D z = {p P(ω, ω) : n(n dom(p) p(n) > z(n))} Example 20. Define P(ω 2 ω, 2) in a similar fashion, i.e., P(ω 2 ω, 2) = {p : p is a function (α, n) ω 2 ω((α, n) dom(p)) ran(p) 2}. Note that, for every α, β ω 2, n ω, D n α,β = {p P(ω 2 ω, 2) : k n((α, k) dom(p) (β, k) dom(p) p(α, k) p(β, k)} 12

13 is dense. Hence, P(ω 2 ω, 2) adds ω 2 many new pairwise different elements of 2 ω. This would seem to directly imply that we have proved that the extension M[G] = 2 ω = ω 2, for any G P(ω 2 ω, 2)-generic over M. Nevertheless, one has to be more precise. In fact, what we have just shown is simply that M[G] = 2 ω = (ω 2 ) M, where (ω 2 ) M represents the ω 2 of M, which may be collapsed to some smaller cardinal in M[G] (As an example of forcing notion which certainly collapses ω 2 to ω, one may think of P(ω, ω 2 ), similarly defined.) However, this is not the case of P(ω 2 ω, 2). In fact, one can show that this forcing satisfies a property called countable chain condition, which in particular ensures such forcing does not collapse ω 2, or in other words (ω 2 ) M = (ω 2 ) M[G]. De facto, this forcing provides the famous result obtained by Paul Cohen in the 1960s, to build a model for ZFC+ CH. Real Line - Part II We now look at the real line with a particular emphasis on forcing method. In the first example of forcing that we introduced in the previous section, we have seen that P(ω, ω) adds a new real, or better, an element of ω ω. This brief section is devoted to introduce, in a fairly general way, several notions of forcing adding new reals. At this aim, we introduce the notion of tree. Remind that: 2 <ω denotes the set of finite sequences (arbitrarily large) of 0s and 1s. ω <ω denotes the set of finite sequences (arbitrarily large) of natural numbers. Definition 21. A set T ω <ω is said to be a tree iff T is closed under initial segments, i.e., t T n ω(n < t t n T ), where t n is the restriction of t having domain equals n. A tree T is said to be perfect iff any of its node is extendible to a splitting node, i.e., s T t T (s t i, j(i j t i T t j T )). A node t as above is called splitting. Note that a perfect tree has necessarily infinite height. Given a tree T, one says that x ω ω is a branch through T iff n ω(x n T ). The set of all branches through T is denoted by [T ] and it is called the body of T. (Note that all of these definitions can be similarly introduced for 2 ω as well.) The notion of tree plays an important role, since it is equivalent to the notion of closed set. More precisely, if T is a tree, then [T ] is closed. And, if C is any closed set, then there is a tree T such that [T ] = C (note that we did not specify which space we are working with, since this is true both for the Baire and for the Cantor space). Using the notion of tree, one can now introduce a rather general notion of forcing. 13

14 Definition 22. A forcing P is said to be arboreal iff every T P is a perfect tree and t T (T t P), where T t := {s T : s t t s}. Example Forcing P(ω, ω) can be viewed as an arboreal forcing. In fact, given a condition p P(ω, ω), one can define T p = {t ω <ω : p t}. Then let C = {T p : p P(ω, ω)}, ordered by inclusion, i.e., T T iff T T. It is straightforward to check that C and P(ω, ω) are isomorphic, and hence they yield the same forcing extension. (This fact can be rather easily proved.) Letter C to denote such forcing comes from the fact that this is nothing more than the forcing introduced by Cohen in the 1960s. It turns out that C is strictly connected with the Baire property. 2. Consider the forcing R = {T 2 <ω : [T ] has strictly positive measure}, ordered by inclusion. This forcing is usually called random forcing, and its raison d être consists of the strict connection with Lebesgue measurability. 3. Sacks forcing S = {T 2 ω : T is a perfect tree}, ordered by inclusion. 4. Silver forcing V = {T S : s, t T ( t = s (t i T s i T ))}, ordered by inclusion. 5. Miller forcing M = {T ω ω : T is a perfect tree whose splitnodes have infinitely many successors}, ordered by inclusion. One can also introduce a notion of smallness for subsets of the real line associated with a certain arboreal forcing P, and thus its corresponding notion of regularity. Definition A set of reals X is said to be P-null iff T P T P(T T [T ] X = ). Then, let J P = {X : X is P-null}, and I P be the closure of J P under countable unions, i.e., I P = {X : X = n ω X n, for X n s in J P }. 2. A set of reals X is said to be P-measurable iff T P T P(T T ([T ] X I P [T ] \ X I P )). One can prove (rather surprisingly?) that C-measurability coincides with the Baire property, while R-measurability coincides with Lebesgue measurability. Historical Background and recent results The interest of Mathematics around regularity properties is rather old. The precursors may be considered the Lebesgue measurability and the Baire property. Assuming AC (Axiom of Choice), Vitali was able to construct a non-measurable set, and the same proof also provided a set without Baire property. So a natural question was to understand whether or not AC was really necessary to obtain such a nasty set. In the 1960s, Solovay resolved this problem, by using the method of forcing. Solovay s forcing may be viewed as a generalization of Cohen forcing previously introduced. Given two cardinals γ < λ, one defines P(γ, λ) = {p : p is a function α < γ(dom(p) = α ran(p) λ}, 14

15 ordered by extension. One can show that P(γ, λ) collapses λ to γ. More generally, given κ > γ, one can define the product forcing P(γ, < κ) = λ<κ P(γ, λ). In the generic extension via this product forcing, one can show that any cardinal strictly below κ is collapsed to γ, and so κ becomes γ +. Hence, denoting by V[G] the extension via P(ω, < κ), with κ inaccessible, Solovay established the following result: V[G] = all sets of reals in L(R) are Lebesgue measurable and so in particular L(R) V [G] = all sets of reals are Lebesgue measurable. The same also held for the Baire property. Solovay s model, introduced in [9], represented (and still represents nowadays) one of the corner-stones of set theory of the reals. Furthermore, Solovay s model turned out to be the tip of the iceberg of a rich research field. In the 1980s, Shelah resolved the most intriguing problem risen from Solovay s work. In [7], Shelah introduced a really profound construction, called amalgamation, to build strongly homogeneous algebras, by mean of which he was able to establish the following result: starting from a ground model V without inaccessible cardinals, one can build a Boolean algebra B, such that V[G] = all sets of reals in L(R) have the Baire property, where G is B-generic over V. On the contrary, yet in the same paper, Shelah also established: if we assume all Σ 1 3 sets are Lebesgue measurable, then one can prove that for every x ω ω, L[x] = ω 1 is inaccessible. Hence, Shelah s results marked a huge difference between the behaviour of the Baire property and that one of the Lebesgue measurability, by showing that, whereas the latter does need an inaccessible, the former does not. Note also that, as a consequence, one obtains that in Shelah s model where all sets of reals have the Baire property, there exists a non-measurable set. Such a result was the first one regarding separation of regularity properties on the family of all sets of reals; furthermore, just one year later (in [8]) Shelah also showed the reverse separation, by constructing a model where all sets of reals are Lebesgue measurable, but there exists a set without Baire property. In the wake of these results, in [4], Friedman and I constructed a model where all sets are Silver-measurable; there is a set which is not Miller-measurable; ω 1 is inaccessible by reals. This results has been then improved in [5], by introducing the notion of unreachability, where I obtained a model for all sets are Silver-measurable; there is a non-miller-measurable set; there is a non-lebesgue-measurable set; ω 1 is inaccessible by reals. 15

16 References [1] Alessandro Andretta, Dispense corso Istituzioni di Logica Matematica. [2] Tomek Bartoszyński, Haim Judah, Set Theory-On the structure of the real line, AK Peters Wellesley (1999). [3] Kenneth Kunen, Set Theory-An introduction to independence proof, Elsevier, [4] Giorgio Laguzzi, Arboreal forcing notions and regularity properties of the real line, PhD thesis, Universität Wien, 2012, written under the supervision of Sy D. Friedman. [5] Giorgio Laguzzi, On the separation of the regularity properties of the real line, submitted. [6] Gabriele Lolli, Dagli insiemi ai numeri, Bollati Boringhieri, 1995 [7] Saharon Shelah, Can you take Solovay s inaccessible away?, Israel Journal of Mathematics, Vol. 48 (1985), pp [8] Saharon Shelah, On measure and category, Israel Journal of Mathematics, Vol. 52 (1985), pp [9] Robert M. Solovay, A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, Vol. 92 (1970), pp