Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. NorthHolland, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....(rather than the structure of the universe) Note 7. It is interesting to note that an analogue of the concept of domainindependence has also independently been introduced in the literature on set theory. This is the notion of absoluteness, which is crucial for independence proofs (see, e.g. [Ku80]) Indeed, a formula is ( AS ; FAS ) d.i. with respect to ; see the second example after De nition 6) i it is absolute according to the literature on set theory. Again we see here an interesting di erence between what has been taken to be important in database theory and in set theory: while ....

K. Kunen, Set Theory: an Introduction to Independence Proofs, North-Holland, Amsterdam, 1980.

....the original proof makes the formalization unusually long, and not entirely satisfactory: two parts of the proof do not fit together. It seems impossible to solve these problems without formalizing the metatheory. However, the present development follows a standard textbook, Kunen s Set Theory [6], and could support the formalization of further material from that book. It also serves as an example of what to expect when deep mathematics is formalized. 3 4 CONTENTS Contents 1 Introduction 6 2 Proof Outline 7 2.1 The Problem With Class Models . 7 2.2 ....

Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

....is conservative to add a bijection of the ACL2 universe with the natural numbers. 2.3 Total Orders From a Set Theoretic Perspective In this section we review some well know facts about total orders in a purely set theoretic setting. Good references include the books by Kunen, Devlin, and Halmos [11, 3, 4] and Part B of the Handbook of Mathematical Logic [1] especially chapter B.2 on the Axiom of Choice [5] This section can be skipped without impacting readability of the rest of this paper. Recall the Axiom of Choice, which (among many equivalent formulations) can be stated as follows: every set ....

K. Kunen. Set Theory - an Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, Amsterdam, 1980.

....and firmly rooted understanding of the symmetries. This is also the reason for the unproportional size of Section 2 compared to the other sections. 2. DEFINITIONS AND NOTATIONS 2.0. Set theoretic notation. Most of our set theoretic notation is standard and can be found in textbooks like [Je78] Ku83] or [BaJu95] For the definitions and some basic facts concerning the projective hierarchy we refer the reader to [Kan94, x12] We shall consider the set [ as the set of real numbers. For the Turing join of two reals x and y (i.e. coding two reals into one) we use the standard notation x ....

Kenneth Kunen: "Set Theory, an Introduction to Independence Proofs." [Studies in Logic and the Foundations of Mathematics 102], North Holland, Amsterdam (1983).

....same reason) Is there again an analogue of the concept of domain independence, which has independently been introduced in the literature on set theory Yes, there is. This is (as the name we have chosen suggests) the notion of absoluteness, which is crucial for independence proofs (see, e.g. [Ku80]) 28 . Indeed, a formula is (oe AS ; FAS ) Gammaabsolute (see the third example after Definition 16) iff it is absolute according to the literature on set theory. Again we see here an interesting difference between what has been taken to be important in relational database theory and what has ....

K. Kunen, Set Theory: an Introduction to Independence Proofs, NorthHolland, Amsterdam, 1980.

....is M generic over Q 1 . Fix such a condition S. Let Q 0 2 be the collection of elements T 2 Q 2 such that T is compatible with S. So we have that Q 1 ae Q 0 2 ae Q 2 . Below we will use the notion of a complete embedding from one partial order into another. See Definition 7. 1 on page 218 of [Kunen]. Claim 1. Q 1 is a complete suborder of Q 0 2 . Proof of Claim 1. Let G be M generic over Q 0 2 . Then G is M generic over Q 2 , and S 2 G. So G Q 1 is M generic over Q 1 . In summary we have that whenever G is generic over Q 0 2 , G Q 1 is generic over Q 1 . It is an easy exercise to ....

....o: P) can be completely embedded into r: o: Q) Proof of Claim 2. Work in M[G 1 ] Let fl = card Gamma P(r: o: P) Delta . As ffi 2 is inaccessible in M [G 1 ] fl ffi 2 . We claim that Q collapses fl to be countable. For let G be M[G 1 ] generic over Q. By exercise (D4) on page 244 of [Kunen], G is M generic over Q 0 2 , and M [G 1 ] G] M [G] It then follows that G is M generic over Q 2 . By chapter 9 of [Ma1] ffi 2 = 1 ) M[G] So fl is countable in M [G] M [G 1 ] G] In other words, Q collapses fl to be countable. Our conclusion easily follows from this. For let H be a ....

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Kenneth Kunen, Set theory: An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102, North Holland, New York, N.Y., 1980.

....S just gives new names for things which must already exist in a model for MT . Restricting oneself to conservative extensions of MT does not have any significant impact on the power of Z, since MT gives us the same mathematical power as first order Zermelo set theory and, to quote an exercise in [2], 99 of mathematics can be carried out in Zermelo set theory . It is the purpose of this paper to show that free type definitions in Z do not live in the missing 1 . 2 Indeed, an approach in which consistency is demonstrated in small steps, each of which demonstrates the conservativeness of a ....

....with two place predicates, Phi (Y ; X ) say, which are functional in Y and then use appropriate circumlocutions for assertions involving OE. For example, OE(A) B would be viewed as an abbreviation for Phi (A; B ) Standard technical devices exist to justify the terminology with OE, see e. g [2]. 9 i.e. Condition taken from Spivey, or, if you must, continuous with respect to the topology on the ZF universe in which the open sets are the sets which are inaccessible from countable directed unions. To make the second interpretation precise would require us to make precise the idea of a ....

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North Holland, 1980.

....continuum. To investigate partition filters, a useful tool is missing: the dualization of Ramsey ultrafilters. The aim of this paper is to fill this gap. 1 Partition filters 1. 1 Notations and definitions Most of our set theoretic notation is standard and can be found in textbooks like [Je78] Ku83] or [BJ95] So, we consider a natural number n as an ordinal, in particular n = fk : k ng and 0 = and consequently, the set of natural numbers is denoted by . For a set S, P(S) denotes the power set of S. The notation concerning partitions is not yet standardized. However, we will use the ....

Kenneth Kunen: "Set Theory, an Introduction to Independence Proofs." North Holland [Studies in Logic and the Foundations of Mathematics], Amsterdam (1983).

....provides for higher order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS [6] HOL [8] and Coq [5] We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC) Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abrial and Laffitte [1] Grabczewski has mechanized the first two chapters of Rubin and Rubin s famous monograph [24] proving equivalent eight forms of the Well ordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of ZermeloFr nkel (ZF) ....

....x:A.P[x] # x#A .P[x] bounded quantifier Figure 1. ASCII notation for ZF You need not understand the details of how this is used in order to follow the paper. 1 Not many set theory texts cover such material well. Elementary texts [9, 27] never get far enough, while advanced texts such as Kunen [12] race through it. But Kunen s rapid treatment is nonetheless clear, and mentions all the essential elements. The desired result (1) follows fairly easily from Kunen s Lemma 10.21 [12, page 30] ## # X # ## # # # X # ## This, in turn, relies on the Axiom of Choice and its consequence ....

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

....dominating family. Then it is not hard to see that the family fD : g ( where each D is constructed from d like D from d in the proof of Theorem 3.1, is an orthogonal family. a Let i be the least cardinality of an independent family (a definition and some results can be found in [Ku]) then 10 LEMMA 4.2 O i. PROOF: Let I [ be an independent family of cardinality i. Let I 0 : fr 2 [ r = T A n S Bg, where A; B 2 [I] A 6= A B = and r = x means j(r nx) xnr)j . It is not hard to see that jI 0 j = jIj = i. Now let I = I 1 [I 2 where ....

K. Kunen: "Set Theory, an Introduction to Independence Proofs." North Holland, Amsterdam 1983.

....theory. Furthermore, Frege seems not to have appreciated why Hilbert and 1 For an accessible discussion of Hilbert s life and contributions, see Constance Reid s excellent biography [Rei70] 2 For a similar analysis, see Weyl[Wey44] 3 For an accessible presentation of the proof, see [Kun80]. 2 others were interested in demonstrating that any contradiction in Euclidean geometry must reduce to one in the axioms of arithmetic. Concerning axioms of Euclidean Geometry, Frege says: I call axioms propositions that are true but that are not proved because our understanding of them ....

Kunen, Kenneth, Set Theory: An Introduction to Independence Proofs, Amsterdam; New York: North-Holland Pub. Co., 1980.

....Theorem. The Schroeder Bernstein Theorem says that for any sets a and b, if there is a one to one function from a to b and also a one to one function from b to a, then there is a one to one correspondence between a and b. We followed the proof sketch given in Exercise 8 of Chapter 1 of [7]. The following axiom introduces our assumptions. constrain fa and fb are one one (rewrite) and ; fa is one to one (implies (and (a x) a y) not (equal x y) not (equal (fa x) fa y) fb is one to one (implies (and (b x) b y) not (equal x y) not (equal (fb x) fb y) the ....

Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, New York, 1980.

....zf axioms with some of the quanti ers restricted to sets. And there is a faithful interpretation of zf in nbg [23, 26, 29] which implies that zf is consistent i nbg is consistent. However, nbg is nitely axiomatizable ( 14] and [19] present nite axiomatizations of nbg) while zf is not (see [16] or [21] for a proof) nbg is derived as follows from nbg. First, the underlying logic of nbg, ordinary rst order logic, is replaced with Partial First Order Logic (pfol) a variant of rst order logic in which operator symbols may denote partial functions and terms may be unde ned. The 1 ....

K. Kunen. Set Theory: An Introduction to Independence Proofs. NorthHolland, 1980.

....is equivalent to proving that feg is a submonoid of M . A standard interpretation that xes the primitive constants of the source theory but restricts the universe of the source theory is called a relativization. Relativizations are commonly used in set theory to establish relative consistency [19]. 4 PF PF is a version of simple type theory with partial functions and subtypes. This section gives a quick introduction to the syntax and semantics of PF . The next section introduces a user friendly version of PF called lutins. See [9] for the full, de nitive presentation of ....

K. Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

....zf axioms with some of the quanti ers restricted to sets. And there is a faithful interpretation of zf in nbg [32, 37, 40] which implies that zf is consistent i nbg is consistent. However, nbg is nitely axiomatizable ( 21] and [28] present nite axiomatizations of nbg) while zf is not (see [24] or [30] for a proof) As a version of nbg, stmm satis es DG1 and DG2 and has the same expressive power as nbg. stmm has additional machinery which makes it much more appropriate for mechanized mathematics than ordinary nbg. The additional machinery includes: 1. Support for unde ned terms (so ....

K. Kunen. Set Theory: An Introduction to Independence Proofs. NorthHolland, 1980.

....concept as well as reviewing and expanding upon two existing approaches familiar from the literature. Genericity concepts were rst employed in recursion theory when several analogies between priority methods and the forcing method in set theory were noted. In set theoretic forcing (cf. 2] [5]) a Cohen generic set may intuitively be thought of as possessing all of the properties which are suciently simple so as be recognizably dense in their extension within a countable ground model M of ZFC. This is to say that in generic extension M [G] over the ground model M via a M generic set ....

K. Kunen, Set Theory : An Introduction to Independence Proofs, North Holland, Amsterdam, 1980.

....a dual form of it, and will give some symmetries between them. Finally we give some relationships between the dual Mathias forcing and the dual Ramsey property. 1 Notations and definitions Most of our set theoretic notations and notations of forcings are standard and can be found in [Je 1] or [Ku] An exception is that we will write A B for the set of all functions from B to A, instead of B A because we never use ordinal arithmetic. A is the set of all partial functions f from to A such that the cardinality of dom(f) is finite. First we will give the definitions of the sets we ....

K. Kunen: "Set Theory, an Introduction to Independence Proofs." North Holland, Amsterdam 1983.

..... R(a) w a . 1 0 2 Figure 2: WF universe defined recursively in terms of ordinals This universe of WF is depicted in Figure 2 which bears a resemblance to Figure 1. This is justified by the common acceptance of the statement that the universe of ZF is equivalent to the universe of WF (Kunen 1980). Let us now recall Russell s Paradox. We let r be the set whose members are all sets x such that x is not a member of x. Then for every set x, x 2 r if and only if x 62 x. Substituting r for x, we obtained the contradiction. With the preceding discussion of WF the explanation is not difficult. ....

....S 2 a : For any infinite cardinal m, the set H(m) is defined as H(m) fx : jTC(x)j mg. The elements of H(m) are said to be hereditarily of cardinality less than m. fH(n) n = 1; 2; g is the set of hereditarily finite sets. Hence, every element of a hereditary set is a hereditary set (Kunen 1980). ....

Kunen, K. (1980). Set Theory: An Introduction to Independence Proofs . North-Holland, Amsterdam.

...., we have that are cardinals such that cov( Then there exists a notion of forcing P 2 M which preserves cofinalities and cardinalities , such that if G is P generic over M then M [G] j= MNP( Proof. We will use some standard forcing results, all of which can be found in [6]. We use a two stage iterated forcing construction. Working in M , and using the notation of [6] let P 0 : Fn( 2; let P 1 : Fn( 2) and let P : P 0 Theta P 1 . Let G be P generic over M . Note that both P 0 and P 1 , and hence also P, preserve cofinalities and ....

....P 2 M which preserves cofinalities and cardinalities , such that if G is P generic over M then M [G] j= MNP( Proof. We will use some standard forcing results, all of which can be found in [6] We use a two stage iterated forcing construction. Working in M , and using the notation of [6], let P 0 : Fn( 2; let P 1 : Fn( 2) and let P : P 0 Theta P 1 . Let G be P generic over M . Note that both P 0 and P 1 , and hence also P, preserve cofinalities and cardinalities . By standard forcing arguments, M [G] j= 2 0 = 2 = By Theorem 1.2, it ....

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K. Kunen, Set Theory: An Introduction to Independence Proofs, (North Holland, Amsterdam, 1980).

....theory and VDM notation a dynamic VDM is presented for constructing the specification of a whole software system. Keywords: Dynamic set, Formal specification, VDM, Structured method . Introduction In classical set theory a set represents a static collection of objects, as described in [1] 2] 3][4]. However, in many circumstances in the real world, changeable collections of objects must be modeled. Take the students of Y university as an example. If we use S to represent the students of Y university, then S represents a changeable collection of students since S represents different ....

Kenneth Kunen, "Set Theory: An Introduction to Independence Proofs", Elsevier Science Publishers B.V., 1983.

....in Sect. 3. Let #, # range over ordinals and #, over limit ordinals. The cumulative hierarchy of sets is traditionally defined by cases: V 0 = 0, V # 1 = P(V # ) and if is a limit ordinal, V = S # V # . More convenient is the equivalent definition V # # [ # # P(V # ) Kunen (1980), Chapter III, is useful background reading; he writes R(#) for V # . Here are some well known facts. Lemma 2.4. If # is an ordinal and is a limit ordinal then # # V # V # V # # V # 2 V V # V V V # V The set V is closed under the formation of variant tuples and ....

....Thus A = S # (A# V # ) If A # V # # B for all # then S # (A # V # ) # B and the result follows. ## Using this lemma requires some facts about intersection with V # . Definition 2.8. A set A is transitive if A # P(A) Lemma 2.9. V # is transitive for every ordinal #. Proof. See Kunen (1980), page 95. ## Now we can go down the cumulative hierarchy as well as up. Lemma 2.10. If #a, b# # V # 1 then a # V # and b # V # . Proof. Suppose #a, b# # V # 1 ; this is equivalent to a , a, b # P(V # ) Thus a, b # V # and since V # is transitive a, b # V # . ....

Kunen, K. (1980). Set Theory: An Introduction to Independence Proofs. North-Holland.

....objects, and (iii) to give the rules of the game to be played with newly introduced symbols [36] In general, ZF is the basic axiomatization used heavily in mathematics. The origin and the underlying mathematical properties and results of its axioms were extensively discussed in the literature [9, 13, 15, 16, 17, 18, 19, 20, 30, 31, 33, 35, 36, 37]. The axioms are defined in first order logic and only the membership relation (2) is considered to be a basic relation. The axioms of ZF are Extensionality, Null Set , Pair Set , Union, Infinity, Power Set , Separation (Subset) Replacement , and Foundation (Regularity) Choice is not an ....

K. Kunen. Set Theory: An Introduction to Independence Proofs . North-Holland, Amsterdam, 1980.

....this fact for adding more clauses to definition 15. Is there again an analogue of the concept of domain independence, which has independently been introduced in the literature on set theory Yes, there is. This is the notion of absoluteness, which is crucial for independence proofs (see, e.g. [Ku80]) 15 . If we take (oe; F ) to be as in example 3 above then a formula is (oe; F ) Gammad.i. w.r.t. iff it is absolute according to the literature on set theory. Again we see here an interesting difference between what has been taken to be important in relational database theory and in set ....

K. Kunen, Set Theory: an Introduction to Independence Proofs, NorthHolland, Amsterdam, 1980.

....surprises. Without going into too many of the details, let us assume that we start to develop set theory taking membership and equality as primitive predicates. The usual notation for sets (pairing, product, comprehension etc. can be introduced by conservative extension (see, for example, [7], 8] or the chapters on set theory in [1] let us assume that that has been done, so that we can write expressions such as (x; y) x Theta y, fx : y j OE(x)g and so on. Let us assume that functions are represented in the usual way as sets of pairs and that function application is denoted by a ....

Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North Holland, 1980.

....provides for higher order syntax. Other recent systems that have been used for mechanizing mathematics include IMPS [6] HOL [8] and Coq [5] We describe below machine proofs concerning cardinal arithmetic and the Axiom of Choice (AC) Paulson has mechanized most of the first chapter of Kunen [12] and a paper by Abrial and Laffitte [1] Grabczewski has mechanized the first two chapters of Rubin and Rubin s famous monograph [24] proving equivalent eight forms of the Well ordering Theorem and twenty forms of AC. We have conducted these proofs using an implementation of ZermeloFr nkel (ZF) ....

....: P [x] 9 x2A : P [x] bounded quantifier Figure 1. ASCII notation for ZF You need not understand the details of how this is used in order to follow the paper. 1 Not many set theory texts cover such material well. Elementary texts [9, 27] never get far enough, while advanced texts such as Kunen [12] race through it. But Kunen s rapid treatment is nonetheless clear, and mentions all the essential elements. The desired result (1) follows fairly easily from Kunen s Lemma 10.21 [12, page 30] 8 ff jX ff j j S ff X ff j This, in turn, relies on the Axiom of Choice and its consequence ....

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

....on x, an object of the form fx j OE(x)g, if the set cannot be formed (if the object is too big ) is a proper class for example, the paradoxical Russell Set fx j x = 2 xg. As Kenneth Kunen puts it, they do not formally exist and are best thought of as abbreviations for other expressions [Kunen, 1980]. Various axiomatic systems of set theory have been developed; one standard is that called ZermeloFrankel Set Theory with the Axiom of Choice, abbreviated ZFC. Its nine axioms follow, as given in [Aczel, 1988, page 117] who has avoided the use of abbreviations; see also [Kunen, 1980] for a ....

....expressions [Kunen, 1980] Various axiomatic systems of set theory have been developed; one standard is that called ZermeloFrankel Set Theory with the Axiom of Choice, abbreviated ZFC. Its nine axioms follow, as given in [Aczel, 1988, page 117] who has avoided the use of abbreviations; see also [Kunen, 1980] for a comprehensive discussion with helpful intermediate definitions, and [Partee et al. 1990, page 218 ff. for a slightly different exposition) For all sets a and b: Extensionality: A set is determined by its membership; a and b are equal if z 2 a implies z 2 b and z 2 b implies z 2 a. ....

[Article contains additional citation context not shown here]

Kenneth Kunen. Set Theory: An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, New York, 1980.

.... all the quantifiers in OE as follows: 8x: x] becomes 8x: x 2 R ) x] and 9x: x] becomes 9x: x 2 R [x] Now the reflection schema states that for any formula OE with free variables x 1 : xn : 8ff: 9fi: fi ff 8x 1 : xn 2 V fi : OE j OE V fi For a proof, see for example [Kun80] or [Kri71] In the special case where OE is a sentence then we see that there exist arbitrarily large ordinals fi such that OE is interdeducible with its relativization to V fi . Among the most simple consequences we see that ZF cannot be finitely axiomatizable in first order logic. Indeed if a ....

Kenneth Kunen. Set Theory: An Introduction to Independence Proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland, 1980.

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. NorthHolland, 1980.

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. NorthHolland, 1980.

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

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K. Kunen, Set theory: An introduction to independence proofs, North-Holland, 1980.

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K. Kunen, Set Theory: An Introduction to Independence Proofs, North Holland, Amsterdam, 1980.

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. North-Holland, 1980.

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Kunen, K., Set Theory: An Introduction to Independence Proofs, North-Holland, 1980

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Kenneth Kunen. Set Theory: An Introduction to Independence Proofs. NorthHolland, 1980.

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