aa r X i v : . [ m a t h . L O ] J a n YABLO’S PARADOX AND FORCING
SHIMON GARTI
Abstract.
We discuss the problem of self-reference in Yablo’s paradoxfrom the point of view of the relationship between names and objects.For this end, we introduce a forcing version of the paradox and tryto understand its implication on the self-referential component of theparadox.2010
Mathematics Subject Classification.
Key words and phrases.
Yablo’s paradox, Prikry forcing, self-reference. Introduction
Yablo introduced in [9] a paradox which allegedly belongs to the liarparadox family yet contains no self-reference. The paradox consists of aninfinite family of sentences S = { S n : n ∈ ω } where each S n is the statement ∀ ℓ > n, S ℓ is untrue. A moment of perusal leads to a contradiction whichforms the paradox. A central theme concerning Yablo’s paradox is the cor-rectness of the title of Yablo’s paper. According to the title, this paradox isnot self-referential. Whether this is true or not is still under debate, see forexample [1], [8] and [10].The main claim of the present paper is that this question depends on an-other important philosophical issue, the relationship between mathematicalobjects and their names. We will try to examine Yablo’s paradox from thepoint of view of forcing theory. Being a theory which enables us to sepa-rate objects from their names in an accurate way, we believe that it mayshed light on the paradox in general and on the self-referential componentconnected with it. We shall use Prikry forcing, which appeared in [5]. Wesuggest [3] as a source for background on Prikry forcing and we employ theJerusalem forcing notation as done in this monograph.1. Some remarks on forcing
Paul Cohen introduced the method of forcing in [2], in order to provethe consistency of the failure of the continuum hypothesis and the axiom ofchoice. The modern practice of forcing owes a lot to Shoenfield, [6]. In thissection we describe, shortly, the basic idea of this method.We begin with a universe of set theory which is called the ground model and denoted by V . We define, in V , a partial order P . The set P is called aforcing notion and the elements of P are called conditions. Each conditiongives a small piece of information about a mathematical object that we tryto force.This object does not belong to V , but its possible existence does notcontradict the axioms of set theory. From V we can imagine the desiredobject, give it a name and describe it to some extent. In the next step wechoose a generic set G ⊆ P , which does not belong to V (unless P is trivialin some sense). Now we extend the universe V by adding G and also manyother sets.The resulting model is called the generic extension , and denoted by V [ G ].There are two ways to describe the process of enlarging V to V [ G ]. Theaxiomatic description says that we add G and then close the universe underthe axioms, so we must add subsets of G , applications of the replacementaxiom in which G is involved, and so on. An alternative and more practicalway to describe V [ G ] is through the concept of names . Every object S ∈ V [ G ] has a name S ˜ in the ground model V . A recursive definition providesthe ability to interpret names using the generic set G . Thus V [ G ] is simply ABLO’S PARADOX AND FORCING 3 the interpretation of all the P -names from the ground model according tothe generic set G .This process is somewhat parallel to the geometric approach of the ancientGreeks. Let F be the field of numbers constructible by a ruler and a pairof compasses, the ground model of the Greeks. We know that π / ∈ F ,though the Greeks did not know this. Nonetheless, they could name it anddescribe it quite vividly as the area of a circle whose radius is 1. Moreover,they could give partial information about π using areas of bounded (andbounding) polygons. The most important thing is that an extension of theiruniverse embodies π as a real number.This is quite similar to the forcing process. One begins with a groundmodel and a desired mathematical object. This object is not an element ofthe ground model, but it can be described from the ground model. In theofficial terminology, it has a name in the ground model V . By extending V this name can be interpreted and become a real object in the genericextension V [ G ].The object that we would like to force is the set of sentences S whichforms Yablo’s paradox. In the ground model we will define conditions (anda partial order between them), each of which gives only partial informationabout S . This construction will be depicted in the next section.2. Prikry forcing and Yablo’s paradox
Consider the statement S n which says ∀ ℓ > n, ¬ S ℓ . Although S n doesnot refer to S n , it mentions the index n . The explicit reference to the indexseems inevitable in any formulation of Yablo’s paradox. We are asking,therefore, whether such a reference is a self-reference.From this point of view, the question of self-reference in Yablo’s paradoxis a reflection of the substantial issue of names and objects. If one arguesthat Yablo’s paradox is not self-referential then one claims that the name ofan object differs significantly from the object itself. Therefore, a referenceto n , the name of S n , is not necessarily a self-reference. If one identifiesnames and objects by claiming that a (mathematical) object is nothingbut the union of its names, then one concludes that Yablo’s paradox is self-referential indeed. Observe that this point also appears in other formulationsof Yablo’s paradox, like the set-theoretical version of Goldstein in [4].We suggest below a formal angle from which this aspect of self-referencein Yablo’s paradox can be examined. This will be done, basically, by anapplication of forcing. We shall force the existence of the set S , and we wishto say something about self-reference with respect to this set.It has been claimed that though each S n is not self-referential, the entirecollection S contains a self-reference. In order to build S one has to defineevery S n . In order to define S n one has to say something about an end-segment of S . Thus the existence of S captures some circularity, and herelies the quintessential self-reference of the paradox, see for example the short SHIMON GARTI observation of Smith in [7]. We indicate that this point is also a centralissue when one tries to understand why Yablo’s paradox does not lead to acontradiction in formal systems of set theory like
ZFC .Our goal is to phrase a version of Yablo’s paradox in which the statement S n does not refer to S , or to an end-segment of S . In our formulation, each S n mentions only a finite set of S ℓ s. To gather all the information we willextend our universe of set theory. Definition 2.1.
Yablo’s paradox forcing notion.Let U be a normal ultrafilter over a measurable cardinal κ . We define aPrikry-type forcing notion P .( ℵ ) A condition p ∈ P is a triple ( s, S, A ) = ( s p , S p , A p ) where s ∈ [ κ ] <ω , A ∈ U and max( s ) < min A . If s = { α , . . . , α n − } and i The forcing notion ( P , ≤ , ≤ ∗ ) is κ + -cc, satisfies Prikry propertyand ( P , ≤ ∗ ) is κ -closed. Hence all cardinals are preserved in the genericextension by P .Proof .We commence with the chain condition. If p, q ∈ P and s p = s q thennecessarily S p = S q and it follows that p k q , so the size of any antichainis bounded by the number of stems which is κ . Prikry property can beproved using Rowbottom’s theorem as done with the usual Prikry forcing.The closure degree of ( P , ≤ ∗ ) is κ since U is normal and in particular κ -complete. Combining all the above statements we see that all cardinals arepreserved, as required. (cid:3) . G ⊆ P be generic over V and let ( ρ n : n ∈ ω ) be the associatedPrikry sequence added by the generic set. Define S = S { S p : p ∈ G } . Onecan verify that S is a system of ω -many sentences { S ρ n : n ∈ ω } whichexemplifies Yablo’s paradox in V [ G ].Let us add a few words about the connection between S ∈ V [ G ] and theconditions of P . Every p ∈ P contains a finite approximation S p of S . Foreach p , the set S p has a last element and hence S p forms no paradox. If q is another condition then possibly p ⊥ q , so q contains a different (mayhapcontradictory) information about S . However, G is directed and hence picksonly compatible conditions for creating S in the generic extension. ABLO’S PARADOX AND FORCING 5 The set S is an object in V [ G ], but not in the ground model V . In thenext section we shall discuss the meaning of this fact with respect to self-reference in Yablo’s paradox. For the present section we just indicate that S is not an object but it has a name S ˜ in V , hence the above generic versionof Yablo’s paradox highlights the name-object aspect of the self-referenceissue. 3. The ground model and the generic extension Our main claim in this section is that the forcing version of Yablo’s para-dox is free from self-reference even if one argues that the original paradox isself-referential. Remark that S ∈ V [ G ] and therefore if we study the para-dox in V [ G ] then it is the usual Yablo’s paradox. In particular, the debateand the various arguments concerning circularity and self-reference are inthe same position.But let us try to consider the paradox from the ground model’s pointof view. It seems that in V there is no self-reference at all, as each S n refers only to finitely many statements, all of which appear explicitly in thecondition p to which S n belongs. In particular, there is no appeal to S orsome end-segment of S . One may argue, however, that from the groundmodel’s point of view there is no paradox at all. As far as the set S doesnot exist we cannot obtain the desired contradiction, and this set does notexist in V . One may agree that we have a name S ˜ for this set and hence wehave a name of a paradox but not an actual paradox.Here we touch upon the issue of the definition of a paradox. In its basicform, a paradox is a collection of statements which lead to a contradiction.The forcing version introduced in the previous section certainly leads to acontradiction, and hence should be regarded as a paradox. In order to geta contradiction one has to invoke some set-theoretical arguments, includingthe mathematical principles which establish the generic extension by P . Yetthere is no essential difference between these principles and the parallelprocess rendered in deriving a contradiction at any other paradox. Thoughwe have only a name S ˜ in V , we know how to describe a formal process inwhich we choose a generic object G , interpret all the names including S ˜ , andreach to a contradiction. We conclude, therefore, that even in V we have aparadox, this time with no self-reference.4. Acknowledgement I thank the referee of the paper for his/her work, especially for suggestingthe incorporation of the second section into the paper. SHIMON GARTI References 1. Jc Beall, Is Yablo’s paradox non-circular? , Analysis (Oxford) (2001), no. 3, 176–187. MR 18372712. Paul J. Cohen, Set theory and the continuum hypothesis , W. A. Benjamin, Inc., NewYork-Amsterdam, 1966. MR 02326763. Moti Gitik, Prikry-type forcings , Handbook of set theory. Vols. 1, 2, 3, Springer,Dordrecht, 2010, pp. 1351–1447. MR 27686954. Laurence Goldstein, A Yabloesque paradox in set theory , Analysis (Oxford) (1994),no. 4, 223–227. MR 13248095. K. L. 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