NI FOL Into Df3 Proof

8/4/2019 NI FOL Into Df3 Proof

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Formalizing set theory in diagonal-free cylindric

algebras, searching for the weakest logic with

Godels incompleteness property.

Draft version!

Hajnal Andreka and Istvan Nemeti

2011. July

Abstract

We show that first-order logic can be translated into a very simple andweak logic, and thus set theory can be formalized in this weak logic. Thisweak logical system is equivalent to the equational theory of diagonal-free 3-dimensional cylindric algebras, i.e., Boolean algebras with threecommuting complemented closure operators. Equivalently, set theory canbe formulated in propositional logic with 3 commuting S5 modalities (i.e.,in the multimodal logic [S5,S5,S5]). There are many consequences, e.g.,free finitely generated Df3s are not atomic and [S5,S5,S5] has G odelsincompleteness property.

1 Introduction

Tarski in 1953 [13, 14] formalized set theory in the theory of relation algebras.

Why did he do this? Because the equational theory of relation algebras (RA)corresponds to a variable-free logic, in other words, to a propositional logic.Tarski got the surprising result that a propositional logic is strong enough toexpress mathematics, to be the playground for mathematics. The classicalview before this result was that propositional logics in general were weak inexpressive power, decidable, uninteresting in a sense. By using the fact that settheory can be built in it, Tarski proved that the equational theory of RA is un-decidable. This was the first propositional logic shown to be undecidable. Thisis why the title of the book [15] is Formalizing set theory without variables.

This contains full proofs for the two main theorems. We make this preprint available inthis draft version because so many people expressed strong interest in the proofs.

Research supported by the Hungarian National Foundation for scientific research grantsT81188.

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From the above it is clear that changing RA in Tarskis result to a weakerclass of algebras is an improvement of the result and it is worth doing. For moreon this see Tarski-Givant [15, pp.892 904 and footnote 17 on p.90].

Relation algebras are halfway between CA3 and CA4, the classes of 3-dimen-sional and 4-dimensional cylindric algebras, respectively. We sometimes jokinglysay that RA is CA3.8. Why is RA stronger than CA3? Because, the relation

algebra reduct of a CA3 is not necessarily an RA, e.g., associativity of relationcomposition can fail in the reduct. See [5, sec 5.3], and for more in this linesee Nemeti-Simon [9]. Why is CA4 stronger than RA? Because RaCA4 RA.However, RA = SRaCA4 (Maddux, see [5, sec 5.3]), so the equational theories ofRA and RaCA4 coincide. Thus Tarski formulated Set Theory, roughly, in CA4.

Draft paragraph. Finite variable fragment hierarchy of FOL. What it saysabout FOL. Monks result: For all finite n there is a 3-variable formula whichis valid but which can be proved (in FOL) with more than n variables only.Intuitively this means that during the proof there are steps when we have touse n independent data (stored in the n variables as in n machine registers).Recent solution of Problem 2.12, related. For example, the associativity ofrelation product (of binary relations) as formalized in the Ra-reduct of a CAncan be expressed with 3 variables but can be proved only with 4 variables. L3 is

essentially incomplete, it does not have a finite Hilbert-style inference system.The inference system |

3belonging to L3 expresses CA3.

Tarskis main idea in [15] is to use pairing functions to form pairs, and so tostore two pieces of data in one register. He used this technique to translate usualinfinite-variable first-order logic FOL in the three variable fragment of it. Fromthen on, he used that any three-variable FOL-formula about binary relationscan be expressed by an RA-equation, [5, sec 5.3]. He needed two registers forstoring the data belonging a binary relation and he had one more register formaking computations belonging to a proof.

Nemeti [6], [7] formalized set theory in CA3 (hence also in the class SAof semi-associative relation algebras), thus improving Tarskis result. The mainidea for this improvement was using the paring functions to store all data always,during every step of a proof, in one register only and so one got two registersto work with in the proofs. In this approach one represents binary relations asunary ones (on pairs). For the execution of this idea see sections 3-5 of thepresent paper.

One of the uses of equality in FOL is that it can be used to express (simulate)substitutions of variables. The reduct SCA3 of CA3 forgets equality dij butretains substitution in the form of the term-definable operations sij . The logicbelonging to SCA3 is weaker than 3-variable fragment of FOL. Zalan Gyenis [4]improved parts of Nemetis result by using SCA3 in place of CA3.

Df3 is much weaker than CA3. Without equality or substitutions, if onehas only binary relations, one cannot really use the third variable for anything.Therefore we need at least one ternary relation symbol in order to use the thirdvariable (which we have to use in a formalization of set theory, because the two-

variable fragment is decidable). Therefore, while in formalizing set theory in thethree-variable fragment of FOL (in CA3) we could do with one binary relation

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symbol, we did not have to change vocabulary during the formalization, in thepresent equality- and substitution-free case we have to change vocabulary, andwe have to pay attention to this new feature of the translation function.

What does it mean to formalize set theory? Why set theory? Have we learntanything about set theory?

2 A simple logical system: three variable logic

without equality or substitutions

The language of this simple system contains three variable symbols, x, y, z, oneternary relational symbol P, and there is one atomic formula, namely P(x, y, z).The logical connectives are , , x, y, z. We denote the set of formulas byFmd3. We will use the derived connectives , , , , too. The proof system|d is a Hilbert style one with the following logical axioms and rules.

The logical axioms are the following. Let , Fmd3 and v, w {x, y, z}.((1)) , if is a propositional tautology.((2)) v( ) (v v).((3)) v.((4)) vv v.((5)) v( ) (v v).((6)) vv v.((7)) vw wv.The inference rules are Modus Ponens ((MP), or detachment), and Generaliza-tion ((G)).

This proof system is a direct translation of the equational axiom system ofDf3. Axiom ((2)) is needed for ensuring that the equivalence relation definedon the formula algebra by |d be a congruence with respect to(w.r.t.) the operation v. It is congruence w.r.t. the Boolean connectives , byaxiom ((1)). Axiom ((1)) expresses that the formula algebra factorized with is a Boolean algebra, axiom ((5)) expresses that the quantifiers v are operators

on this Boolean algebra (i.e., they are additive), axioms ((3)),((4)) express thatthese quantifiers are closure operations, axiom ((6)) expresses that they arecomplemented closure operators (i.e., the negation of a closed element is closedagain). Together with ((5)) they imply that the closed elements form a Booleanalgebra, and hence the quantifiers are normal operators (i.e., the Boolean zero isa closed element). Finally, axiom ((7)) expresses that the quantifiers commutewith each other.

We define Ld3 as the logic with formulas Fmd3 and with proof system |d .The logic Ld3 inherits a natural semantics from first-order logic (FOL). Theproof system |d is sound with respect to this semantics, but it is not com-plete. Moreover, there is no finite Hilbert-style inference system which wouldbe complete and sound at the same time w.r.t. this semantics (because thequasi-equational theory of RDf3 is not finitely axiomatizable, see [5] and [2]).

We note that in the above system, axiom ((6)) can be replaced with the

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following ((8)):((8)) v( v) (v v).

Restricted 3-variable FOL is introduced in [5, Part II, p.157], with proof sys-tem |r . If we restrict this system |r to formulas not containing the equality= then we get a system equivalent to our Ld3. Lets call this system restricted

3-variable FOL without equality. That is, the formulas are those of restricted3-variable FOL which contain no equality, and we leave out from the axioms of|r the axioms which contain equality. This way we get a proof system withModus Ponens and Generalization as deduction rules and the following axioms:

((V1)) , if is a propositional tautology.((V2)) v( ) (v v).((V3)) v .((V4)) v, if v does not occur free in .

Lets call this1 equality-free |r . Rule ((V4)) in this system essentially usesvariables in its using the notion of free variables of a formula. On the otherhand, no axiom in |d needs to use the structure of a formula occurring in

a rule, it is essentially variable-free. So, an advantage of |d to equality-free|r is that it is more algebraic, more like propositional logic. On the other

hand, equality-free |r contains fewer axioms (it contains only ((V1))-((V4))as axioms).

The logic Ld3 has a neat modal logic form: three commuting S5 modalities.This is denoted as [S5,S5,S5], see [3, p.379, lines 15-20].

One can present this logic in yet one different form: Equational logic as thebackground logic, and the defining axioms of Df3 as logical axioms.

The following theorem says that this simple logic Ld3 is strong enough fordoing all of mathematics in it. It says that we can do set theory in Ld3 as follows:in place of formulas of set theory we use their translated versions Tr() in

Ld3, and then we use the proof system |d

of Ld3 between the translatedformulas in place of the proof system of FOL between the original formulas ofset theory. Moreover, for sentences in the language of set theory, and Tr()mean the same thing (are equivalent) modulo a bridge between the twolanguages. We need this bridge because the language L of set theory containsonly one binary relation symbol and equality, and the language Ld3 containsonly one ternary relation symbol P. When f : A B is a function and X A

then f(X)d= {f(a) : a X} denotes the image of X under this function f.

Theorem 2.1. (Formalizability of set theory in Ld3) There is a recursive trans-lation function Tr from the language L of set theory into Ld3 for which the

following are true for all sentences in the language of set theory:

1We note that we also omitted ((V9)) of [5] because v abbreviates v in our approach,

so ((V9)) is not needed.

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(i) ZF |= iff Tr(ZF) |d Tr().

(ii) ZF + |= Tr(), where

d= xyz[(P(x, y, z) (x = y = z (x, y))].

Theorem 2.1 is proved in section 5.

The next theorem is a partial completeness theorem for Ld3.Theorem 2.2. (Partial completeness theorem for Ld3) There is a recursive subsetK Fmd3 and there is a recursive function tr mapping all FOL-formulas intoK such that the following are true:

(i) |= iff |d for all K.

(ii) |= iff |= tr() for all FOL sentences .

According to the above theorem, the proof system |d is complete within K.But is K big enough? Yes, we can prove any valid FOL-formula by translatingit into K and then proving the translated formula by |d . We know that |d isnot strong enough to prove all valid Fmd3 formulas. However, we can formulateeach sentence in a slightly different form, namely as tr() so that this versionof can be now proved by |d iff it is valid.

Theorem 2.3. (Godel style incompleteness theorem for Ld3) There is a formula Fmd3 such that no consistent recursive extensionT of is complete.

Discussion 2.1. In Theorems 2.1-2.3, at least one at least ternary relationsymbol P is needed in the target-language Ld3, the axiom of commutativity((7)) is needed in the proof system |d (because omitting ((7)) from |d resultsdecidability of the so obtained proof system, see [7]). We do not know whethercomplementedness of the closure operators ((6)) is needed or not. Also, twovariables do not suffice because the satisfiability problem of the two-variablefragment of FOL is decidable.

3 Finding QRA-reducts in Df3

In this section we begin the proof of Theorem 2.1. We show that every Df3contains lots of QRAs in them. We do this by defining relation algebra typeoperations in the term language of Df3 and proving that these operations formQRAs in appropriate relativizations. Since QRAs are representable, this willamount to a partial representation theorem for Df3s, and to partial com-pleteness theorem for Ld3. We will work in Ld3 in place of Df3.

There will be parameters in the definitions to come. These will be formu-las in Fmd3, namely xy, xz with free variables {x, y} and {x, z} respectively,together with two other formulas p0, p1 with free variables {x, y}. Thus, if youchoose xy, xz, p0, p1 with the above specified free variables then you will arrive

at a QRA-reduct of any Df3 corresponding to these. We get the QRA-reduct

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by assuming some properties of the meaning of these formulas, this will be ex-pressed by a formula Ax. In section 5 then we will choose these parameters sothat they fit set theory, which means that the formula Ax built up from themis provable in set theory. Intuitively, the formulas xy, xz stand for equalityx = y, x = z and p0, p1 will be arbitrary pairing functions.

So, choose formulas xy, xz, p0, p1 with the above specified free variables

arbitrarily, they will be parameters of the definitions to come. To simplifynotation, we will not indicate these parameters.

We now set ourselves to defining the above relation algebra type operationson Fmd3. To help readability, we often write just comma in place of conjunctionin formulas, especially when they begin with a quantifier. E.g., we write x(, )in place of x( ). Further, True denotes a provably true formula, say

Trued= xy xy. First we introduce notation to support the intuitive

meaning of the parameters xy, xz as equality.

Definition 3.1. (Simulating equality between variables)

x.

= yd= xy,

x.

= zd= xz,

y.

= zd

= x(x.

= y x.

= z),y

.= x

d= x

.= y,

z.

= xd= x

.= z,

z.

= yd= y

.= z,

x.

= xd= True,

y.

= yd= True,

z.

= zd= True.

Definition 3.2. (Simulating substitution with (simulated) equality)

((x, y))d= ,

((x, z))

d

= y(y

.

= z ),((y, z))

d= x(x

.= y ((x, z))),

((y, x))d= z(x

.= z ((y, z))),

((z, x))d= y(y

.= z ((y, x))),

((z, y))d= x(x

.= z ),

((x, x))d= y(x

.= y ),

((y, y))d= x(x

.= y ),

((z, z))d= x(x

.= z ((x, x))).

Remark 3.3. In FOL, ((y, z)) is semantically equivalent with the formula weget from the formula by replacing x, y with y, z everywhere simultaneously,

when xy, xz mean true equality. This is Tarskis trick to simulate substitu-

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tions. Etc. Merry-go-round semantically valid but not provable, even with trueequality (i.e., in CA3).

Next we introduce notation supporting intuition about the pairing functionsp0, p1. First we define some auxiliary formulas. We will use the notation 2 ={0, 1}, to make the text shorter. Let 2 denote the set of all finite sequencesof 0, 1 including the empty sequence as well. If i, j 2 then ij denotestheir concatenation usually denoted by ij, and |i| denotes the length of i.Further, if k 2, then we write k instead of < k > for the sequence < k > oflength 1. Accordingly, 00 denotes the sequence < 0, 0 >.

We are going to define Fmd3-formulas ui.

= vj for u, v {x, y, z} and i, j 2.The intuitive meaning of ui0...in

.= vj0...jk is that if p0, p1 are partial functions

then pin . . . pi0u = pjk . . . pj0v. As usual in the partial algebra literature, theequality holds if both sides are defined and are equal. E.g., the intuitive meaningof x0

.= y01 is that all of p0x, p0y, p1p0y exist and p0x = p1p0y.

Definition 3.4. (Simulating projections)Let {u, v, w} = {x, y, z}, i, j 2 and k 2.

(u.

= v)d= u

.= v,

(uk.

= v)

d

= pk((u, v)),(uik

.= v)

d= w(ui

.= w, pk((w, v))) if i =,

(ui.

= vj)d= w(ui

.= w, vj

.= w) if j =,

(xi.

= xj)d= y(x

.= y, xi

.= yj),

(yi.

= yj)d= x(x

.= y, xi

.= yj),

(zi.

= zj)d= x(zi

.= x, zj

.= x).

We will omit the index in formulas ui.

= vj , i.e., we write ui.

= v andu

.= vi for ui

.= v and u

.= vi resp. if i 2.

So far we did nothing but introduced notation supporting the intuitive mean-ing of the parameters xy, xz, p0, p1 as equality and partial pairing functions.

Almost any of the concrete formulas supporting this would do, we only had tofix one of them since our proof system |d is very weak, it would not proveequivalence of most of the semantically equivalent forms. Now we write up astatement Ax about the parameters using the just introduced notation. Let

Hd= {i 2, |i| 3}.

Definition 3.5 (pairing axiom Ax). We define Ax Fmd3 to be the conjunctionof the union of the following finite sets (A1),...,(A4) of formulas:

(A1) {ui.

= vj , vj.

= wk ui.

= wk : u, v, w {x,y,z}, i , j, k H}

(A2) {ui.

= vj , uik.

= uik uik.

= vjk : u, v {x, y, z},ik,jk H, k 2}

(A3) {ui.

= ui, vj.

= vj w(w0.

= ui, w1.

= vj) : u,v,w {x, y, z}, w /

{u, v}, i , j H}

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(A4) {w u.

= w : u, w {x, y, z}}.

We are ready to define our relation-algebra type operations on Fmd3. Theywill have the intended meaning on formulas with one free variable x. Let Fmd13denote the set of formulas in Fmd3 with at most one free variable x.

Definition 3.6 (relation algebra reduct Dra of Fmd3). Let , Fmd3.

paird= yp0 yp1.

uid= x(x

.= ui, ) if u {y, z} and i 2

.

d= y(y0, y1, x0

.= y00, y01

.= y10, y11

.= x1), see Figure 3,

d= y(y,y0

.= x1, y1

.= x0),

1d= x0

.= x1,

d= pair ,

+ d= .

Drad= { Fmd3 : Ax |d

1 for some Fmd13},

Drad

= Dra, +, ,,

,

1.

Figure 3. Illustration of

In the definition of Dra, intuitively

1 means that we close the meaningof with the equivalence relation of equivalent pairs. I.e., let us define x

yd= x0

.= y0, x1

.= y1, then

1 means closed with this equivalence relation.If we assume uniqueness of pairs (like in SAx later) then we can write pair

in place of

1 in the definition of Dra.The next theorem is the heart of formalizing set theory in Fmd3. Let us

define the equivalence relation T on Fmd3 by

T d= T |d ,

where T Fmd3.

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Theorem 3.7. (i) Dra is an algebra, i.e., the operations +, ,, ,

1 do notlead out of Dra.

(ii) Dra { : , Fmd13}.

(iii) Dra/ Ax is a relation algebra.

(iv) The images of the formulas x1 .= x00 and x1 .= x01 form a quasi-projectionpair inDra/ Ax.

Proof. The proof of the analogous theorem in [7, Thm.9, p.43] goes throughwith some modifications. We indicate here these modifications.

The proof explanations (CA),. . . , (KV) introduced on [7, p.44] can be usedin our proof, too, except that we always have to check whether the explanationuses only the Df-part of (CA). If it uses axiom C7 ofCA, then we have to give analternate proof. Let (Df) denote the part of (CA) which does not use C7. Wecan use (UV) because it can be derived from Ax and (Df). Of course, throughoutwe have to change = to

.=.

We now go through the proof given in [7, pp.46-64] and indicate the changesneeded for our proof. All the statements on pp.48-54 beginning with (A1) and

ending with (S13) follow from Ax and (Df). In fact, we could just add thesestatements to Ax since they do not contain formula-schemes denoting arbitraryformulas (such as, e.g., (0) on p.54 does), and so they amount to finitely manyformulas only. Because of this, we do not indicate the changes needed in theproofs of these items.

We have to avoid statement (0) at the end of p.54 by all means, because ituses axiom C7 of (CA) essentially. Fortunately, we do not really use (0) in theproof, changing to x in some steps will suffice for eliminating (0).

The proof of (2/a) on p.55 has to be modified, and it can be done as follows.

Recall from [7] that has at most one free variable x, and ud= x(x

.= u, )

when u is different from x while xd= y(y

.= x,y).

Assume x / {u, v}. The proof for u,u.

= v |d v is as on [7, p.55]. Then

x,x.

= z |d by definition of xy(y

.= x,y), x

.= z |d

y(y.

= x,y,x.

= z) |d by Axy(y

.= z,y) |d by the case when x / {u, v}

y(z) |d

z.

x,x.

= y |d by Axz(z

.= x,x,x

.= y) |d by Ax

z(z.

= y,x,x.

= z) |d by the previous casez(z

.= y,z) |d by the case when x / {u, v}

y.

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y,y.

= x |d

y(y.

= x,y) |d by definition of xx.

z,z.

= x |d by Axy(y

.= z, z

.= x,z) |d by Ax

y(y .= x, z .= y,z) |d by the case when x / {u, v}y(y

.= x,y) |d

x.

On p.58, in the last line of the proof of (9) we have to write x in place of. Similarly, on p.59, in lines 7 and 8 we have to change to x and then wecan cross out reference to (0). Thats all! QED

4 Finding CA-reducts in Df3

Simon [12] defines a CAn-reduct in every QRA, for all n , and also proves

that these reducts are representable. We will use, in this paper, the CA3-reductof our QRA defined in the previous section, Dra/ Ax.We will use the following stronger form of Ax, just for convenience:

Definition 4.1. [strong paring axioms SAx] We define SAx Fmd3 to be theconjunction of the union of the finite sets (A1),...,(A4) of Def.3.5 together withthe following:

(A5) { x0.

= y0, x1.

= y1 x.

= y, x0.

= x0 x1.

= x1}.

The formulas in (A5) express that pairs are unique, and that p0 is definedexactly when p1 is defined. We use (A5) for convenience only, this way formulaswill be shorter. We could omit (A5) on the expense that formulas will be longerand more complicated. If we assume SAx then pair Dra for all Fmd13.

Every QRA has a CA3-reduct, which is representable, see Simon [12]. Thefollowing definition is recalling this CA3-reduct from [12] in our special caseof Dra/ SAx. The definition below is simpler than in [12] because we willassume uniqueness of pairs in SAx, which is not assumed in [12]. We will usethe abbreviation

xid= y(y

.= xi, y), when Fmd

13 and i 2

.

Beware: |d x usually is not true for Fmd13. (For the definition ofyj see Def.3.6.)

Definition 4.2. [cylindric reduct Dca of Fmd3]

Tripletd= x11

.= x11, see Figure,

(0)d= 0, (1)

d= 10, (2)

d= 11, and for all i, j < 3 and , Fmd1

3Ti

d= Tripletx0 Tripletx1

{x0(j)

.= x1(j) : i = j < 3},

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cid= Ti,

dijd= Tripletx1 x1(i)

.= x1(j),

d= Tripletx1 ,

+ d= ,

Dcad= { Fmd3 : SAx |d x1 Tripletx1 for some Fmd

13},

Dca d= Dca, +, , ci, dij : i,j < 3.

Theorem 4.3. The set Dca is closed under the operations defined in Def.4.2and the algebraDca/ SAx is a representable CA3.

Proof. We show that the algebra Dca/ SAx is the CA3-reduct of the quasi-

projective relation algebra Dra/ SAx, as defined in [12]. Let pd= x1

.= x00,

qd= x1

.= x01. In the following we will omit referring to SAx, so we will look

at p as an element ofDra/ SAx, while only p/ SAx is such. Let

ed= Tripletx0 x0

.= x1,

(i)d= Tripletx0 x1

.= x0(i),

(i)d

= Tripletx0 Tripletx1 x0(i).

= x1(i).

Now, one can show that e is the same as (3) in [12, Def.3.1], i.e.,

SAx |d e (qqqq

1).

Similarly, one can show that (i), (i) are the same as the ones in [12], and

Triplet is the same as 1(3) in [12, Def.3.1], e.g.,

SAx |d (1) (eqp),SAx |d (1) ((1)

(1)),

SAx |d Triplet (paire).

Thus, our reduct is the same as the 3-reduct defined in [12], and then we can

use [12, Thm.3.2,Thm.5.2]. QED

5 Formalizing set theory in Ld3

The 3-variable restricted fragment L3 of L is defined as follows. Language:three variable symbols, x, y, z, one binary relational symbol , so there is oneatomic formula, namely (x, y). Logical connectives: , , x, y, z and u = vfor u, v {x, y, z}. We denote the set of formulas by Fm3. Derived connectivesare , , , . The proof system |

3of L3 is a Hilbert style one defined in [5,

Part II, p.157]. The word-algebra of L3 is denoted as Fm3, it is the absolutelyfree CA3-type algebra generated by the formula (x, y).

In this section we prove Theorem 2.1 stated in section 2. As a first step, wewill define a translation function h from Fm3 to Fmd3. This will be analogous

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to the one defined in [7], but here a novelty is that the vocabulary of Fm3 isdifferent (disjoint) from that of Fmd3, and we will have to pay attention to thisdifference. Namely, Fm3 contains one binary relation symbol and equalityu = v for u, v {x, y, z} while Fmd3 contains one ternary relation symbol Pand does not contain equality. The formula introduced in section 2 bridgesthe difference between the two languages. Namely, is stated in a language

which contains both Fm3 and Fmd3 and is a definition of P in Fm3, so itleads from Fm3 to Fmd3. But is a two-way bridge, because

xy x = y |=

[(x, y) zP(x, y, z)][x = y = z (P(x, y, z)zP(x, y, z))]

,

so the equivalent form of provides a definition of equality = and in Fmd3,and thus it is a bridge leading from Fmd3 to Fm3.

As usual, we begin with definitions. To start, we work in Fm3. Recall theconcrete definitions of p0, p1, and from [7, p.71, p.35]. Define

+d=

xy[z(p0((x, z)), p0((y, z))), z(p1((x, z)), p1((y, z))) x = y]

x(yp0

(x, y) yp1

(x, y)).

Then p0, p1, , + are formulas of Fm3. We note that the notation ((u, v)) in

[7] is the same as our ((u, v)) introduced in Def.3.2, except that instead of x.

= yetc. the notation ((u, v)) uses real equality x = y etc. available in Fmd3.

Next, we work in Fmd3 and we get the versions of these formulas that thedefinition of P provides. We will write out the details. First we fix theparameters xy, xz, p0, p1 occurring in the formula SAx of Fmd3.

Definition 5.1. [fixing the parameters xy, xz of Fmd3]

Ed= z P(x, y, z),

Dd= P(x, y, z) E,

xyd= zD,

xz d= yD.

The next definition fixes the parameters p0, p1 ofFmd3. To distinguish themfrom their Fm3-versions, we will denote them by p0, p1. The definition below isa repetition of the definition of the corresponding formulas on p.71 of [7] suchthat we write E and the above defined concrete xy, xz in place of and x = y,x = z. We will use the notation introduced in section 2 and we will use notationto support set theoretic intuition. Thus, if we introduce a formula denoted as,say, x

.= {y}, then u

.= {v} denotes the formula ((u, v)) (see Definitions 3.2,3.1).

Below, op abbreviates ordered pair (to distinguish it from the formula pairdefined earlier.).

Definition 5.2. [fixing the parameters p0, p1 of Fmd3]

uv d= E((u, v)), for u, v {x,y,z},

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x.

= {y}d= z(zx z

.= y),

{x}yd= z(z

.= {x},zy),

x.

= {{y}}d= z(z

.= {y}, x

.= {z}),

x yd= z(xz,zy),

op(x)d= yz({z}x y

.= z)

yz[(y x, {y}x,z x, {z}x) y .= z], yz(yx zy),

p0d= op(x) {y}x,

p1d= op(x) [x

.= {{y}} (y x, {y}x)].

It is not hard to check that

, xy x = y |= Ax, + SAx.(1)

We are ready to define our translation mapping h from Fm3 to Fmd3. Fora formula Fmd23 and i, j 2

we define

((xi, xj))

d

= yz(y

.

= xi, z

.

= xj , ((y, z))).Definition 5.3. [translation mapping h]

(i) h : Fm3 Fmd13 is defined by the following:

h is a homomorphism from Fm3 into Dca such that

h((x, y))d= E((x1(0), x1(1))).

(ii) SAxd= SAx x(Tripletx1 h

(+)).

(iii) We define the mapping h : Fm3 Fmd03 as

h()d= x(SAx Tripletx1 h

()).

We say that a translation function f is Boolean preserving w.r.t. |d iff forall sentences , Fmd3 we have that |d f( ) (f() f()) and|d f( ) (f() f()).

Theorem 5.4. Let be a sentence and T be a set of sentences of Fm3. Thenthe following are true.

(i) T {+} |= h(T) |d h.

(ii) + |= h().

(iii) h is Boolean preserving and SAx |d h() h().

Proof. Proof of (ii): Let M be a model of + . Let Vd= {x, y, z} and

Vald

= VM, the set of evaluations of the variables in M. Now, by M |= +

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we have that pi and pi have the same meaning in M, and they form a pair ofpairing functions. Thus one can show by induction that

M |= (ui = vj)[k] iff M |= (ui.

= vj)[k] iff k(u)i = k(v)j in M.(2)

We say that a M is a triplet iff a11 is defined. If a is a triplet, then we assign

an evaluation val(a) Val to a such that val(a) assigns to x,y,z the elementsa(0), a(1), a(2) respectively. We will prove by induction the following statement:For all Fm3 and k Val we have

M |= h()[k] iff

k(x)1 is a triplet and M |= [val(k(x)1)]

.(3)

By (3) and M |= + then we have M |= SAx. Thus, to show M |= hit is enough to show M |= x(Tripletx1 h

). When is a sentence, thisfollows from (3). This finishes the proof of (ii).

Proof of (iii): The proofs of (4) and (6) in [7, p.73], which prove that isBoolean-preserving w.r.t. |

3, work for showing that h is Boolean preserving

w.r.t. |d , because h has the same structure as . Similarly, the proof of (5)in [7, p.73] is good for proving the second statement of the present (iii).

Proof of (i): First we prove (i) for the special case when T is the empty set. Let be a sentence of Fm3, we want to prove |d h() implies

+ |= . So, assumethat |d h, i.e., |d x(SAxTripletx1 h

). Then SAx |d x(Tripletx1h(+) h()), i.e.,

SAx |d h(+ ).(4)

Recall that h is a homomorphism from Fm3 to Dca/ SAx, and the latter is a

representable CA3. Let d= + . By (4) we have that the image of is not

1 under h, therefore there is a homomorphism g from Dca/ SAx to a cylindric

set algebra C such that the image of h under g is not 1. Let fd= gh, then

f : Fm3 C and f() = 1.(5)

Let U be the base set ofC, let Rd= {s0, s1 : s f((x, y))} and define the

model M as U, R. Then for all Fm3 and s U3 we have that M |= [s]iff s f(). Thus M|= by (5), and we are done with showing + |= .

In the other direction, we have to show that |d h implies + |= . Bysoundness of the proof system |d we have |= h, then by (ii) we have + |=. Since P does not occur in and in +, this means that + |= and we aredone.

Next, assume that T is a set of sentences of Fm3. We want to show thatT {+} |= iff h(T) |d h(). Now,

T {+

} |= iff (by compactness of Fm3)T0 {+} |= for some finite T0 T, iff

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+ |=

T0 for some finite T0 T, iff(by first part of (i))|d h(

T0 ) for some finite T0 T. Then, by Boolean preserving of h

|d

h(T0) h(), and so by Modus Ponens we geth(T0) |d h(). Conversely, from this we get by the soundness of |d that

h(T0) |= h().(6)

From here on we have to deal with the difference between the two languages.Let M be an arbitrary model of L3, so M contains one binary relation, say

M = M, M. Define PM according to the definition , i.e., PMd= {a,a,a :

a M} { a,b,c M M M : a, b M}. Let M+d= M, M, PM

be the expansion ofM with this new relation, and let Md= M, PM be the

reduct of this expansion to the language Ld3.Assume now M |= T0 {+}. Then

M+ |= T0 {+ }. Thus by (ii) we have thatM+ |= h(T0), thenM |= h() by (6). ThusM+ |= h + , so by (ii)M+ |= , and soM |= and thus, since M was an arbitrary model of L3T0 {

+} |= and we are done. QED

We are almost done with proving Thm.2.1, all we have to do is to use aconnection between L and L3 established in [7], [6].

Proof of Thm.2.1: By [7, Lemma 3.1, p.35], or by [6, Lemma 2.2, Remark2.4, p.25, p.30] there is a recursive function f : Fm2 Fm

23 for which

|= f() , f() = f(), f( ) = f() f(),(7)

for all sentences , of L. Take such an f and extend it to Fm by letting

f() d= f() where is the universal closure of (ifn is the smallest numbersuch that the free variables of are among v0,...,vn, then

is v0 . . . vn).Then f has the same properties as f and it is defined on the whole of Fm,not only on Fm2. Define Tr : Fm Fmd3 by

Tr()d= h(f())

for all Fm. We show that this translation function satisfies the require-ments of Thm.2.1. First, Tr is recursive because both f and h are such. Wedefined the parameters p0, p1 so that

ZF |= +(8)

holds. Thus ZF |= ZF + +

|= f() by the chosen properties of f, andthen ZF+ |= ZF+ + |= hf by Thm.5.4(ii), so ZF+ |= Tr

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for all sentences of L. This is Thm.2.1(ii). To prove Thm.2.1(i), first weshow

ZF |= f(ZF) + + |= f().(9)

Indeed, assume ZF |= and let M |= f(ZF) + +. Then M |= ZF by (7), soM

|= +

+

by our assumption ZF |= , soM

|= f() by (7). This showsf(ZF) + + |= f(). Conversely, assume now the latter, and we want to proveZF |= . Let M |= ZF, then M |= ZF + + by (8), thus M |= f(ZF) + + by(7), then M |= f() + + by our assumption, and then M |= by (7) again.We have shown that (9) holds.

By combining this (9) with Thm.5.4(i) we get ZF |= iff Tr(ZF) |d Trwhich is Thm.2.1(i).

Later, in section 6, we will also need the following:

Tr is Boolean preserving and SAx |d Tr() Tr.(10)

Indeed, this follows from Thm.5.4(iii) and from (7): Let , Fm0. Then|d Tr( ), iff by the definition ofTr|d hf( ), iff by (7)

|d h(f f ), iff by Thm.5.4(iii)|d hf hf , iff by the definition ofTr|d Tr Tr.

Similarly,|d Tr( ), implies by the definition ofTr|d hf( ), implies by (7)|d h(f f ), implies by Thm.5.4(iii)|d hf hf , implies by the definition ofTr|d Tr Tr.

Finally, SAx |d h(f ) h(f ), by Thm.5.4(iii), so SAx |d h(f()) h(f ) by (7), and then SAx |d Tr() Tr(), as was to be shown.QED(Thm.2.1)

6 Free algebras

In this section we prove that the one-generated free Df3 is not atomic.

Theorem 6.1. The one-generated free Df3 is not atomic.

Proof. It is enough to show that the zero-dimensional part of the free Df3 isnot atomic by [5, 1.10.3(i)]:

ZdFr1Df3 is not atomic implies that Fr1Df3 is not atomic.(11)

Let us define as , i.e.

iff |d .

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It is easy to show that

Fr1Df3 is isomorphic to Fmd3/ and(12)

ZdFr1Df3 is isomorphic to Fmd03/ .(13)

There is a non-separable formula Fm0 which is consistent with + (with

our concrete pairing formulas), by [7, pp.69-71] (or equivalently by [6, Lemma2.7, p.34]). Define

d= SAx Tr.

Then Fmd03. We will show that there is no atom below / and the latteris nonzero in Fmd03/ . Assume the contrary, i.e., that

/ is an atom below / (14)

and we will derive a contradiction. Let

Td= { Fm0 : |

d Tr}.

Then T Fm0

. We will show that T is recursive and it separates the conse-quences of from the sentences refutable from which contradicts the choice of, namely that is inseparable. From now on let Fm0 be arbitrary. Now,/ being an atom implies

|d Tr |d Tr.(15)

From (15) we get that both T and the complement of T are recursively enumer-able, so T is recursive. Next we show

|= implies T.(16)

Indeed, assume that |= . Then |= . Then, in particular, + |= ,and so |d Tr( ) by Thm.2.1(i). Then |d Tr() Tr by (10). By Modus

Ponens then SAx

+ Tr() |d Tr, i.e., |d Tr. By (14) we have

|d (17)

so we have |d Tr, i.e., T as was desired. Next we show

|= implies / T.(18)

Indeed, assume |= . Then() T by the previous case (16), and this means|d Tr(). Now, by (17) and the definition of we have|d SAx, and thus by |d Tr() and (10)|d Tr(). Thus by (15) we get

|d

Tr, by the definition of T then / T.

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By the above we have shown that there is no atom below / . It remainsto show that the latter is nonzero. This follows from the fact that + hasa model. Let M be such that M |= +. Expand this model with PM sothat M+ |= also. Then M+ |= Tr() by Thm.2.1(ii), and also M+ |= SAx

by M+ |= + . Thus M+ |= , and so M |= where M is the reduct ofM+ to the language of . QED

References

[1] Andreka, H., Givant, S. R., Mikulas, Sz., Nemeti, I., and Simon, A., Notionsof density that imply representability in algebraic logic. Annals of Pure andApplied Logic 91 (1998), 93-190.

[2] Andreka, H., Nemeti, I., and Sain, I., Algebraic Logic. In: Handbookof Philosophical Logic Vol. 2, Second Edition, Kluwer, 2001. pp.133-296.http://www.math-inst.hu/pub/algebraic-logic/handbook.pdf

[3] Gabbay, D. M., Kurucz, A., Wolter, F., and Zakharyaschev, M., Many-

dimensional modal logics: theory and applications., Elsevier, 2003.

[4] Gyenis, Z., On atomicity of free algebras in certain cylindric-like varieties.Logic Journal of the IGPL 19,1 (2011), 44-52.

[5] Henkin, L., Monk, J. D., and Tarski, A., Cylindric Algebras Parts I and II,North-Holland, 1985.

[6] Nemeti, I., Logic with three variables has Godels incompleteness prop-erty - thus free cylindric algebras are not atomic., Mathematical Insti-tute of the Hungarian Academy of Sciences, Preprint No 49/1985, 1985.http://www.math-inst.hu/ nemeti/NDis/NPrep85.pdf

[7] Nemeti, I., Free algebras and decidability in algebraic logic., Dissertationwith the Hungarian Academy of Sciences, Budapest, 1986. In Hungarian.English summary is [8]. http://www.math-inst.hu/ nemeti/NDis/NDis86.pdf

[8] Nemeti, I., Free algebras and decidability in algebraic logic. Summary inEnglish. 12pp. http://www.math-inst.hu/ nemeti/NDis/NSum.pdf

[9] Nemeti, I., and Simon, A., Relation algebras from cylindric and polyacidalgebras. Logic Journal of the IGPL 5,4 (1997), 575-588.

[10] Sayed-Ahmed, T., Tarskian Algebraic Logic., Journal on Relation Methodsin Computer Science 1 (2004), pp.3-26.

[11] Sayed-Ahmed, T., Algebraic logic, where does it stand today?, The Bulletinof Symbolic Logic 11,4 (2005), pp.465-516.

[12] Simon, A., Connections between quasi-projective relation algebras and cylin-dric algebras., Algebra Universalis 56,3-4 (2007), 263-301.

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[13] Tarski, A., A formalization of set theory without variables., J. SymbolicLogic 18 (1953), p.189.

[14] Tarski, A., An undecidable system of sentential calculus., J. Symbolic Logic18 (1953), p.189.

[15] Tarski, A., and Givant, S. R., Formalizing set theory without variables.,AMS Colloquium Publications Vol. 41, AMS, Providence, Rhode Island, 1987.

Hajnal Andreka and Istvan NemetiRenyi Mathematical Research InstituteBudapest, Realtanoda st. 13-15H-1053 [email protected], [email protected]

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