PII. S0161171204307283 http://ijmms.hindawi.com © Hindawi Publishing Corp.
THE BEST POSSIBLE UNIFICATION FOR ANY COLLECTION
OF PHYSICAL THEORIES
ROBERT A. HERRMANN
Received 1 July 2003
We show that the set of all finitary consequence operators defined on any nonempty lan-guage is a join-complete lattice. This result is applied to various collections of physical the-ories to obtain an unrestricted standard supremum unification. An unrestricted hyperfinite ultralogic unification for sets of physical theories is also obtained.
2000 Mathematics Subject Classification: 03G10, 03H10, 03B80.
1. Introduction. A restricted hyperfinite ultralogic unification is constructed in [5]. The restrictions placed upon this construction were necessary in order to relate the con-structed ultralogic directly to the types of ultralogics used to model probability models [6]. In particular, the standard collections of consequence operators are restricted to a very special set of operatorsHX, whereXis itself restricted to the set of all significant members of a languageΛ. In this paper, all such restrictions are removed. For read-ers convincement, some of the introductory remarks that appear in [5] are repeated. Over seventy years ago, Tarski [8, pages 60–109] introducedconsequenceoperators as models for various aspects of human thought. There are two such mathematical the-ories investigated, thegeneraland thefinitary consequence operators [2]. LetLbe a nonempty language, ᏼ the power set operator, andᏲ the finite power set operator. There are three cardinality-independent axioms.
Definition1.1. A mappingC:ᏼ(L)→ᏼ(L)is a general consequence operator (or closure operator) if for eachX,Y∈ᏼ(L),
(1) X⊂C(X)=C(C(X))⊂L; (2) ifX⊂Y, thenC(X)⊂C(Y ).
A consequence operatorCdefined onLis said to befinitary(finite) if it satisfies (3) C(X)={C(A)|A∈Ᏺ(X)}.
Remark1.2. The above axioms (1), (2), and (3) are not independent. Indeed, (1) and (3) imply (2). Clearly, the set of all finitary consequence operators defined on a specific language is a subset of the set of all general operators. The phrase “defined onL” means formally defined onᏼ(L).
the general consequence operator, we are only interested in collections of finitary con-sequence operators. Dziobiak [1, page 180] statesTheorem 2.10below. However, the statement is made without a formal proof and is relative to a special propositional lan-guage.Theorem 2.10is obtained by using only basic set-theoretic notions and Tarski’s basic results for any language. Further, the proof reveals some interesting facts not pre-viously known. Unless noted, all utilized Tarski results [8, pages 60–91] are cardinality independent.
2. The lattice of finitary operators
Definition2.1. In all that follows, any set of consequence operators will be non-empty and each is defined on a nonnon-empty language. Define the relation≤on the setᏯ of all general consequence operators defined onLby stipulating that for anyC1,C2∈Ꮿ, C1≤C2if for everyX∈ᏼ(L), C1(X)⊂C2(X).
Obviously, the relation≤is a partial order contained inᏯ×Ꮿ. Our standard result will show that for the entire set of finitary consequence operatorsᏯf⊂Ꮿdefined onL, the structureᏯf,≤is a lattice.
Definition2.2. DefineI:ᏼ(L)→ᏼ(L)andU:ᏼ(L)→ᏼ(L)as follows: for each X⊂L, letI(X)=Xand letU(X)=L.
Notice thatIis the lower unit (the least element) andUthe upper unit (the greatest element) forᏯf,≤andᏯ,≤.
Definition2.3. LetC∈Ꮿ. A setX⊂Lis aC-system or simply a system ifC(X)⊂X and, hence, ifC(X)=X. For eachC∈Ꮿ, let(C)= {X|(X⊂L)∧(C(X)=X)}.
SinceC(L)=Lfor eachC∈Ꮿ, then each(C)= ∅.
Lemma2.4. For eachC1,C2∈Ꮿ,C1≤C2if and only if(C2)⊂(C1).
Proof. Let any C1,C2∈Ꮿ and C1≤C2. Consider any Y ∈(C2). Then C1(Y )⊂
C2(Y )=Y. Thus,C1∈(C1)implies that(C2)⊂(C1).
Conversely, suppose that(C2)⊂(C1). LetX⊂L. Then, since, by axiom (1),C2(X)∈ (C2), it follows, from the requirement thatC2(X)∈(C1), thatC1(C2(X))=C2(X). ButX⊂C2(X)implies thatC1(X)⊂C1(C2(X))=C2(X), from axiom (2). Hence,C1≤C2 and the proof is complete.
Definition2.5. For eachC1,C2∈Ꮿ, define the following binary relations inᏼ(L)× ᏼ(L). For each X⊂L, let(C1∧C2)(X)=C1(X)∩C2(X)and (C1∨wC2)={Y ⊂L| (X⊂Y =C1(Y )=C2(Y ))}. For finitely many members ofᏯ, the operators∧, ∨w are obviously commutative and associative. These two relations are extended to arbitrary Ꮽ⊂Ꮿby defining(Ꮽ)(X)=Ꮽ(X)={C(X)|C∈Ꮽ}and(wᏭ)(X)=wᏭ(X)=
{Y⊂L|X⊂Y =C(Y )for allC∈Ꮽ}[1, page 178]. Notice thatwᏭ(X)={Y⊂L| (X⊂Y )∧(Y∈{(C)|C∈Ꮽ})}.
Lemma2.6. LetᏭ⊂Ꮿ (resp.,Ꮿf) and = {X|(X⊂L)∧(X=wᏭ(X))}. Then
THE BEST POSSIBLE UNIFICATION FOR ANY COLLECTION 863 Proof. By Tarski’s [8, Theorem 11(b), page 71], which holds for finitary and general consequence operators, for eachX⊂LandC∈Ꮽ,X⊂wᏭ(X)=Y∈(C). Hence, ifY∈, then
wᏭ(Y)=Y∈(C)for each C∈Ꮽ. Thus ⊂{(C)|C∈Ꮽ}. Conversely, letY ∈{(C)|(C∈Ꮽ)}. From the definition ofw, wᏭ(Y )=Y and, hence,Y∈and this completes the proof.
Lemma2.7. Let the nonemptyᏮ⊂Lhave the property that for eachX⊂L, there exists someY ∈Ꮾsuch thatX⊂Y. Then the operatorCᏮ defined for eachX⊂Lby
CᏮ(X)={Y|X⊂Y∈Ꮾ}is a general consequence operator defined onL.
Proof. Assuming the hypothesis of the Lemma, it is obvious thatCᏮ:ᏼ(L)→ᏼ(L) andX⊂CᏮ(X). Clearly, ifZ⊂X⊂L, thenCᏮ(Z)⊂CᏮ(X); and, for eachY ∈Ꮾ,X⊂Y if and only ifCᏮ(X)⊂Y. Hence,CᏮ(CᏮ(X))={Y |CᏮ(X)⊂Y ∈Ꮾ} =CᏮ(X). This completes the proof.
Remark2.8. The hypothesis ofLemma 2.7can be weakened. However, our applica-tion does not require such a weakening.
Theorem2.9. With respect to the partial order relation≤defined onL, the structure
Ꮿ,∨w,∧,I,Uis a complete lattice with upper and lower units.
Proof. Let Ꮽ⊂Ꮿand Ꮾ={(C)|C ∈Ꮽ}. Since L∈Ꮾ, then by Lemma 2.7,
wᏭ=CᏮ∈Ꮿ. Moreover, by Lemmas2.4and2.6,CᏮ is the least upper bound forᏭ with respect to≤.
Next, letᏮ={(C)|C∈Ꮽ}. ForX⊂L,X⊂C(X)for eachC∈Ꮽ. For eachC∈Ꮽ, there does not exist aYC such thatYC∈(C),X=YC,YC=C(X), andX⊂YC⊂C(X). Hence,CᏮ(X)={Y|X⊂Y∈Ꮾ} ={C(X)|C∈Ꮽ} =Ꮽ(X). Hence,Ꮽ∈Ꮿand it is obvious thatᏭis the greatest lower bound forᏭwith respect to≤. This completes the proof.
Although the proof appears in error, Wójcicki [9] statedTheorem 2.9for a propo-sitional language. In what follows, we only investigate the basic lattice structure for Ꮿf,≤.
Theorem2.10. With respect to the partial order relation≤defined onᏯf, the struc-tureᏯf,∨w,∧,I,Uis a lattice with upper and lower units.
Proof. It is only necessary to consider two distinctC1,C2∈Ꮿf. As mentioned, the commutative and associative laws hold for∧ and ∨w, and by definition, each maps
ᏼ(L)intoᏼ(L). InᏯ,≤, usingTheorem 2.9, axioms (1) and (2) hold for the greatest lower boundC1∧C2and for the least upper boundC1∨wC2. Next, we have
C1∧C2(X)= C1(Y )|Y∈Ᏺ(X)∩ C2(Y )|Y∈Ᏺ(X)
= C1(Y )∩C2(Y )|Y∈Ᏺ(X)
= C1∧C2(Y )|Y∈Ᏺ(X)
(2.1)
Next, we show by direct means that for eachC1,C2∈Ꮿf, C1∨wC2∈Ꮿf. Let (the cardinality ofL)|L| =∆. For eachXi⊂L,i∈∆, letᏭ(Xi)= {Y |(Xi⊂Y ∈(C1)∩
(C2))∧(Y⊂L)}. Let{Y|Y∈Ꮽ(Xi)} =Yi. By Tarski’s [8, Theorem 11, page 71],Xi⊂ Yi∈(C1)∩(C2), and by definition,Yi=(C1∨wC2)(Xi). Hence,Yi∈Ꮽ(Xi)and is the least (⊂) element. ForXi⊂L, letᏭ(Xi)= {Y|(C1(Xi)⊂Y ∈(C1)∩(C2))∧(Y⊂L)}. SinceXi⊂Ck(Xi), k=1,2, thenᏭ⊂Ꮽ. SinceL∈Ꮽ(Xi), Ꮽ(Xi)= ∅. Indeed, let Y ∈Ꮽ(Xi). ThenXi⊂Ck(Y )=Y , k=1,2. Additionally,Xi⊂C1(Y )=Y implies that Xi⊂ C1(Xi)= C1(C1(Xi))⊂ C1(C1(Y ))=C1(Y )=Y. Hence, it follows that for any Xi⊂L, Ꮽ(Xi)=Ꮽ(Xi). For fixedXi⊂L, letXj∈Ᏺ(Xi). LetYjbe defined as above and, hence,Yjis the least element inᏭ(Xj)=Ꮽ(Xj). ConsiderᏰ= {Yj|Xj∈Ᏺ(Xi)}, and, forj=1,...,n, considerYj∈Ᏸand the correspondingXj⊂L. LetXk={Xj|j= 1,...,n} ∈Ᏺ(Xi). ThenYk={Y|Y∈Ꮽ(Xk)} ∈Ᏸ. IfY∈Ꮽ(Xk), thenY ∈Ꮽ(Xj), j= 1,...,n. Hence,Yj⊂Yk, j=1,...,n, implies thatY1∪ ··· ∪Yn⊂Yk. Tarski’s theorem [8, Theorem 12, page 71] implies thatY∗={Yj|Xj∈Ᏺ(Xi)} ∈(C1)∩(C2). Also, by definition, for allXj⊂L,Yj∈Ꮽ(Xj)implies thatC1(Xj)⊂Yj. The fact thatC1is finitary yieldsC1(Xi)⊂Y∗. Hence,Y∗ ∈Ꮽ(Xi). SinceC1(Xj)⊂C1(Xi), Xj∈Ᏺ(Xi), thenᏭ(Xi)⊂Ꮽ(Xj). ThusYj⊂Yi, Xj∈Ᏺ(Xi). Therefore,Y∗⊂Yi. But,Y∗∈Ꮽ(Xi) implies thatY∗=Yi. Restating this last result,{(C1∨wC2)(Xj)|Xj∈Ᏺ(Xi)} =(C1∨w C2)(Xi)and, therefore, axiom (3) holds for the binary relation∨w, andᏯf,∨w,∧,I,U is a lattice. This completes the proof.
Corollary2.11. LetᏭ⊂Ꮿ,Ꮿf ⊂Ꮿ, andᏭ∩Ꮿf= ∅. ThenwᏭ∈Ꮿf. The structure
Ꮿf,∨w,∧,I,Uis a join-complete lattice.
Proof. Simply modify the second part of the proof ofTheorem 2.10by substituting
{(C)|C∈Ꮽ}for(C1)∩(C2)and lettingC1∈Ꮽ∩Ꮿf. This completes the proof.
Remark2.12. It is known, sinceIis a lower bound for anyᏭ⊂Ꮿf, thatᏯf,∨w,I,U is actually a complete lattice with a meet operator generated by the ∨w operator. It appears that the meet operator ∧for infiniteᏭ need not correspond, in general, to the∨w-defined meet operator. Wójcicki [10] has constructed, for a set of consequence operatorsᏯ, an infiniteᏭ⊂Ꮿof finitary consequence operators with some very spe-cial properties. However, the general consequence operator defined for eachX⊂Lby
{C(X)|C∈Ꮽ}is not a finitary operator. Thus, in general,Ꮿf,∨w,∧,I,Uneed not be a meet complete lattice. This behavior is not unusual. For example, let infiniteX have an infinite topology ᐀. Then᐀,∪,∩,∅,Xis a join-complete sublattice of the latticeᏼ(X),∪,∩,∅,X. The structure᐀,∪,∅,Xis actually complete, but it is not a meet-complete sublattice of completeᏼ(X),∪,∩,∅,X.
THE BEST POSSIBLE UNIFICATION FOR ANY COLLECTION 865 In all that follows, a specific set of logic-system-generated consequence operators{Ci| i∈ ∆}defined on a specific set of languages {Li |i∈∆}will always be considered as trivially extended and, hence, defined by the insertion of hypotheses rule on the set
{Li|i∈∆}.In general, such a specific set of consequence operators is contained in the lattice of all finitary operators defined on{Li|i∈∆}.
A logic systemSand its corresponding consequence operator is atrivial extensionof a logic systemSdefined onLwhere, for a languageL⊃L,Sis the same asSexcept that the insertion rule now applies toL. The systemSand its corresponding consequence operatorCis a nontrivial extension if it is extended toLby insertion and some other n-ary relations that contain members ofL−Lare adjoined to those inS, or various originaln-ary relations inSare extended by addingntuples that contain members from L−L. For both the trivial and nontrivial cases and with respect to the languageL, it follows thatC≤C. In the trivial case, ifX⊂L, thenC(X)=C(X)=C(X∩L)∪(X−L). In practice, apractical logic systemis a logic system defined for the subsets of a finite languageLf. When a specific deduction is made from a set of hypothesesX, the setXis finite. If the logic system also includes 1-ary sets, such as the logical or physical axioms, the actual set of axioms that might be used for a deduction is also finite. Indeed, the actual set of all deductions obtained at any moment in human history and used by a science community form a finite set of statements that are contained in a finite language Lf. (Finite languages, the associated consequence operators, and the like will usually be denoted by a superscriptf.) The finitely manyn-ary relations that model the rules of inference for a practical logic system are finite sets.
Practical logic systems generate practical consequence operators, and practical con-sequence operators generate effectively practical logic systems in many ways. For ex-ample, the method found in [7], when applied to aCf, will effectively generate a finite set of rules of inference. The practical logic system obtained from such rules generates the original practical consequence operator. Hence, a consequence operatorCf defined onLf is considered apractical consequence operatoralthough it may not correspond to a previously defined scientific practical logic system; nevertheless, it does correspond to an equivalent practical logic system.
Our definition of aphysical theory is a refinement of the usual definition. Given a set of physical hypotheses, general scientific statements are deduced. If accepted by a science community, these statements becomenatural laws. These natural laws then become part of a science-community logic system.
In [5], a consequence operator generated by such a logic system is denoted bySN. From collections of such logic systems, theSNthey generate are then applied to specific natural-system descriptionsX. For scientific practical logic systems, the language and rules of inference need not be completely determinate in that, in practice, the language and rules of inference are extended.
Definition3.1. LetCbe a general consequence operator defined inL. LetX⊂L. (i) The setXisC-consistentifC(X)=L.
(ii) The setXisC-completeif for eachx∈L, eitherx∈XorC(X∪{x})=L. (iii) A set X⊂LismaximallyC-consistent if Xis C-consistent and whenever a set
Y=XandX⊂Y⊂L, thenC(Y )=L.
Notice that ifX⊂LisC-consistent, thenC(X)is aC-consistent extension ofXwhich is also aC-system. Further,C-consistentW isC-consistent with respect to any trivial extension ofCto a languageL⊃L.
Theorem3.2. Let the general consequence operatorCbe defined onL.
(i) The set X ⊂L is C-complete and C-consistent if and only if X is maximally C-consistent.
(ii)IfXis maximallyC-consistent, thenXis aC-system.
Proof. (i) Let X be maximallyC-consistent. Then X is C-consistent and, hence, C(X)=L. Hence, letx∈Landx∉X. ThenX⊂X∪ {x}implies thatX∪ {x}is not C-consistent. ThusC(X∪ {x})=L. Hence,X isC-complete. Conversely, assume that XisC-consistent andC-complete. ThenX=L. LetX⊂Y ⊂LandX=Y. Hence, there is somey∈Y−Xand fromC-completeness,L=C(X∪ {y})⊂C(Y ). Thus,Y is not C-consistent. Hence,Xis maximallyC-consistent and the result follows.
(ii) FromC-consistency,C(X)=L. Ifx∈C(X)−X, then maximalC-consistency im-plies that L=C(X∪ {x})⊂C(C(X))=C(X). This contradiction yields that X is a C-system.
The following easily obtained result holds for many types of languages [8, page 98], but these “Lindenbaum” constructions, for infinite languages, are not considered effec-tive. For finite languages, such constructions are obviously effeceffec-tive.
Theorem3.3. Let the practical consequence operatorCf be defined on arbitraryLf. IfX⊂Lf isCf-consistent, then there exists an effectively constructedY⊂Lf such that Cf(X)⊂Y andY isCf-consistent andCf-complete.
Proof. This is rather trivial for a practical consequence operator, and all of the construction processes are effective. Consider an enumeration forLf such thatLf =
{x1,x2,...,xk}. LetX⊂LfbeCf-consistent and defineX=X0. We now simply construct in a completely effective manner a partial sequence of subsets ofLf. Simply consider X0∪ {x1}. SinceX0is Cf-consistent, we have two possibilities. Effectively determine whetherCf(X0∪{x1})=Lf. If so, letX1=X0. On the other hand, ifCf(X0∪{x1})=Lf, then defineX1=X0∪ {x1}. Repeat this construction finitely many times. (Usually, if the language is denumerable, this is expressed in an induction format.) LetY =Xk. By definition,Y is Cf-consistent. Suppose that x∈Lf. Then there is some X
i such that either (a)x∈Xi or (b)Cf(Xi∪ {x})=Lf. For (a), sinceXi⊂Y, x∈Y. For (b), Xi⊂Y implies thatL=Cf(Xi∪{x})⊂Cf(Y∪{x})=Lf. Hence,Y isCf-complete and Xi⊂Y, for eachi=1,...,k. ByTheorem 3.2,Y is aCf-system. ThusX0⊂Y implies that Cf(X0)⊂Cf(Y )=Y, and this completes the proof.
THE BEST POSSIBLE UNIFICATION FOR ANY COLLECTION 867 extension of Cf(X)and, hence, also an extension of X, where Y is a maximallyCf -consistentCf-system.
Let the setΣp⊂Σconsist of all of the science-community practical logic systems defined on languagesLfi. Each member ofΣpcorresponds toi∈ |Σp|and to a practical consequence operatorCif defined onLfi. In general, the members of a set of science-community logic systems are related by a consistency notion relative to an extended language.
Definition3.5. A set of consequence operatorsᏯdefined onLissystem consistent if there existsY⊂L, Y=L, andY is aC-system for eachC∈Ꮿ.
Example3.6. LetᏯbe a set of axiomless consequence operators where eachC∈Ꮿ is defined onL. In [5], the set of science-community consequence operators is redefined by relativization to produce a set of axiomless consequence operators,SNV, each defined on the same language. Any such collectionᏯis system consistent since for eachC∈ Ꮿ, C(∅)= ∅ =L.
Example3.7. One of the major goals of certain science communities is to find what is called a “grand unification theory.” This is actually a theory that will unify only the four fundamental interactions (forces). It is then claimed that this will somehow lead to a unification of all physical theories. Undoubtedly, if this type of grand unification is achieved, all other physical science theories will require some type of restructuring. The simplest way this can be done is to use informally the logic-system expansion technique. This will lead to associated consequence operators defined on “larger” language sets.
Let a practical logic systemS0be defined onLf0,L={Lfi |i∈N}, withNthe set of natural numbers. LetL0⊂L1,L0=L1. (Note that the remaining members of{Lfi |i∈N} need not be distinct.) ExpandS0toS1=S0defined onLby adjoining to the logic system S0finitely many practical logic-system n-ary relations or finitely many additional n tuples to the originalS0, but where all of these additions only contain members from nonemptyL−Lf0. AlthoughS1should only be considered as nontrivially defined onLf1, ifL=L1, then theS1so obtained corresponds toC1, a consequence operator trivially extended toL. This process can be repeated in order to produce, at least, finitely many distinct logic systemsSi, i >1, that extendS0and a setᏯ1of distinct corresponding consequence operatorsCi.
Since these are science-community logic systems, there is anX0⊂Lf0 that is C f 0 -consistent. ByCorollary 3.4, there is an effectively defined setY⊂Lf0such thatX0⊂Y and Y is maximallyC0f-consistent with respect to the languageL
f
0. Hence,C f 0(Y )= Y ⊂Lf0 and C0f(Y )=L
f
0. Further, C f
0 is considered trivially extended to L. LetY= Y∪(L−Lf0). It follows that for eachCi, L−Lf0⊂Ci(L−L
f
0)⊂L−L f
0=L. By construction, for eachCi, Ci(Y )=Y; and for eachX⊂L, Ci(X)=C0(X∩Lf0)∪Ci(X∩(L−L
f 0)). So, letX=Y. Then for eachCi, Ci(Y)=C0(Y )∪(L−Lf
0)=Y∪(L−L f
0)=Y=L. Hence, the set of allCiis system consistent.
Letᐁbe a free ultrafilter onLand letx∈L. Then there exists someU∈ᐁsuch thatx∉
Usinceᐁ= ∅and∅∉ᐁ. LetB= {x}andᏯ= {P(U,B)|U∈ᐁ}, whereP(U,B)is the finitary consequence operator defined byP(U,B)(X)=U∪Xifx∈XandP(U,B)(X)=
Xifx∉X. (Note that this is the same operatorPthat appears in the proof of [5, Theorem 6.4].) There exists, at least, a sequenceS = {Ui|i∈N}such thatU0=Uand Ui+1⊂ Ui, Ui+1=Ui. It follows immediately from the definition thatP(Ui+1,B)≤P(Ui,B)and P(Ui+1,B)(B)=Ui+1∪B⊂Ui∪Bfor eachi∈N. Hence, in general,P(Ui+1,B) < P(Ui,B) for eachi∈N. LetY=L−{x}. ThenP(Ui,B)(Y )=Ui∪(L−{x})=L−{x} =Y,i∈N. Thus, the collection{P(Ui,B)|i∈N}is system consistent.
Theorem3.9. ConsiderᏭ⊂Ꮿf defined onLand the(≤)least upper boundwᏭ. ThenwᏭ∈Ꮿf, and ifᏭis system consistent, then there exists someY ⊂Lsuch that Y =wᏭ(Y )=C(Y )=Lfor eachC∈ᏭandwᏭ=U. Further, ifX⊂L,X=L, is a C-system for eachC∈Ꮽ, thenX=wᏭ(X)=C(X)=Lfor eachC∈Ꮽ.
Proof. Corollary 2.11 yields the first conclusion. From the definition of system consistent, there exists some Y ⊂L such that C(Y )=Y =Lfor each C ∈Ꮽ. From Lemma 2.6, for eachC∈Ꮽ, wᏭ(Y )=C(Y )=L. Hence,wᏭ=U. The last part of this theorem follows fromLemma 2.6and the fact thatXis also awᏭ-system. This completes the proof.
4. An ultralogic unification. Assume for nonemptyΣthat|Σ| ≤ ℵ0. LetᏯdenote a set of (logic-system) corresponding finitary consequence operators, each considered as defined on the languageL. There exists a surjectionf:N→Ꮿsuch thatf (i)is one of the members ofᏯ, and for eachC∈Ꮿ, there is somej∈Nsuch thatf (j)=C. For each i∈N, letf (i)=Cidenote the consequence operators inᏯ. As usual, for the following theorem, we use the boldface type convention [4, page 21], and for the caseᏯ,Ꮿwill denote boldface type.
Theorem4.1. LetLand{Ci|i∈N} =Ꮿbe defined as above. Suppose that every (nonempty) finite subset ofᏯis system consistent.
(i) Then there exists a hyperfinite ultralogicᐁ∈∗Ꮿfdefined on the set of all internal subsets of∗Lsuch that ᐁ=∗U, and an internalW ⊂∗Lsuch that, for eachCi∈Ꮿ, ∗Ci(W )=ᐁ(W )=W=∗L, whereᐁ(W )⊂∗L.
(ii) For each internalY ⊂∗L, {∗Ci(Y )|i∈N} ⊂ᐁ(Y )⊂∗L.
(iii) If finiteX⊂L, then {∗Ci(X)|i∈N} ⊂ᐁ(X), and if each member of Ꮿis a practical consequence operator, then{Ci(X)|i∈N} ⊂ᐁ(X).
(iv) LetX⊂L,X=L, be aC-system for eachC∈Ꮿ. Then∗X=∗Ci(∗X)=ᐁ(∗X)=∗L for each i∈N. IfX is finite,X=∗Ci(X)=ᐁ(X)for each i∈N. If forj ∈N,Cj is a practical consequence operator, thenX=Cj(X)=ᐁ(X)=ᐁ(X).
THE BEST POSSIBLE UNIFICATION FOR ANY COLLECTION 869 ᐅ that contains a nonstandard elementary extension∗ᏹ= ∗ᏺ,∈,= of the embed-dedᏹ. Notice that from our identifications, any standardX⊂Lhas the property that σX=X, and ifXis finite, then ∗X=X. Under our basic embedding, let g:N→Ꮿbe a surjection inᏺ that corresponds tof. Now consider the surjection∗g:∗N→∗Ꮿ. Let constanta∈N. Under our special Grundlagen embedding procedures,∗(g(a))= ∗g(∗a)=∗g(a)=∗Ca. Since∗gis a surjection, ana∈Ncorresponds to a member of σᏯand vice versa. Thus,∗grestricted to members ofσN=Nyields the entire setσᏯ.
Let nonemptyK⊂ᏼ(L)be the set of allX=Lthat ifX∈K, thenXis aC-system for eachC∈Ꮿ. ByTheorem 3.9, the definitions and the properties of the lattice structure onᏯf, for clarity, the unsimplified and redundantly expressed sentences
∀x(x= ∅)∧x∈Ᏺ(N) → ∃y∃w1y∈Ꮿf
∧(y=U)∧w1∈ᏼ(L)∧
∀z1∀v1∀v2v1∈x∧v2∈K∧v1∈N∧z1∈x∧z1∈N →gz1v2=
yv2=v2⊂L∧yv2=L∧∀v(v∈x)∧(v∈N) →
g(v)w1=yw1=w1=L∧∀z(z∈x)∧(z∈N) →g(z)≤y∧
∀ww∈Ꮿf
∧∀z1z1∈x∧z1∈N∧gz1≤w →(y≤w),
∀x∀yx∈Ꮿf
∧y∈Ꮿf
→(y≤x)←→
∀ww∈ᏼL →y(w)⊂x(w),
(4.1)
hold inᏹ. Hence, they hold under∗-transfer in∗ᏹ for objects in∗ᏺ. (Note that it is usually assumed that formal statements such as (4.1) can be made within a formal first-order language rather than expressing them explicitly.)
The set ∗Ꮿf is a collection of hyperfinite consequence operators, each defined on the internal subsets of ∗L. Let infinite λ∈∗N−N. Then ∗g[[0,λ]] is a hyperfinite subset of∗Ꮿ⊂∗Ꮿf. Hence, from∗-transformed sentences (4.1), there exists some hy-perfinite ᐁ∈∗Ꮿ
f defined on the set of all internal Y ⊂∗L with the properties that ∗g(i)(Y )⊂ᐁ(Y ) for eachi∈[0,λ] and, in particular fori∈N. Hence,{∗Ci(Y )| i∈N} ⊂ᐁ(Y ). Further, there exists an internalW⊂∗Lsuch that, for eachi∈[0,λ], ∗g(i)(W )=ᐁ(W )=W=∗L. In particular,∗g(i)(W )=∗Ci(W )=ᐁ(W )for eachi∈N. Sinceᐁ(W )=∗L, thenᐁ=∗U. Let finiteX⊂L. Then, due to our embedding proce-dures,∗(Ci(X))=∗Ci(∗X)=∗Ci(X)⊂ᐁ(X)=∗L. Hence, {∗Ci(X)|i∈N} ⊂ᐁ(X). (Note that in proofs such as this and to avoid confusion, we often, at first, use the nota-tion∗(Ci(X))to indicate the value (or name) of the result of applying∗to an object inᏺ which is a set such asCi(X)that contains additional operator notation. From a technical viewpoint,∗(Ci(X))=∗{Ci(X)} = {∗Ci(X)}and∗Ci(X)is the “name” for the set under the mapping∗. But using this procedure, there is confusion as to whether∗(Ci(X)) denotes the entire set or denotes the operator∗Ci applied toX. In these proofs,∗Ci always denotes the operator∗Ciapplied to internal subsets of∗L.) IfCiis a practical consequence operator, thenCi(X)is a finite set. Hence,∗Ci(X)=Ci(X). (iv) follows by
In [5], the set= {SV
Nj|j∈N}is the refined set of all relativized axiomless science-community consequence operators defined on a languageΛand they are used to unify, in a restricted manner, physical theory behavior. Moreover, for sequentially presented {SV
Nj|j∈N}, 1≤ |{S V
Nj|j∈N}| ≤ ℵ0.
Corollary4.2. LetΛdenote the languageLand= {SV
Nj|j∈N}.
(i) There exists a hyperfinite ultralogicᐁ∈∗Ꮿfthat is defined on all internalY⊂∗Λ such thatᐁ=∗U, and, for eachSV
Ni∈,∗S V
Ni(∅)=ᐁ(∅)= ∅. (ii) For each internalY ⊂∗Λ,{∗SV
Ni(Y )|i∈N} ⊂ᐁ(Y )⊂∗Λ. (iii) If finiteX⊂Λ, then{∗SV
Ni(X)|i∈N} ⊂ᐁ(X), and if each member of{SNVi|i∈
N}is a practical consequence operator, then{SV
Ni(X)|i∈N} ⊂ᐁ(X). (iv) Let X⊂Λ, X =Λ, be a C-system for each C ∈ {SV
N i(X)|i∈N}. Then ∗X= ∗SV
Ni(∗X)=ᐁ(∗X)=∗Λ, for eachi∈N.IfXis finite, then∗SVNi(X)=X=ᐁ(X)for each i∈N. If fori∈N,SV
Ni is a practical consequence operator, thenS V
Ni(X)=X=ᐁ(X).
5. Further applications. If theᏯin the hypotheses ofTheorem 4.1is restricted to a set of practical consequence operators, each defined on a finite languageLf, then it follows that the hyperfiniteᐁ corresponds to a hyperfinite logic systemthat is ∗-effectively∗-generated. If the effective notion is not required, then, in general, the ultralogicᐁcorresponds to a∗-logic system. AlthoughTheorem 4.1andCorollary 4.2 are mainly concerned with the original set of consequence operatorsᏯand{SV
Ni(X)| i∈N}, when|{SV
Ni(X)|i∈N}| = ℵ0, it is also significant for applications thatᐁunifies each “ultranatural relativized theory” ∗g(j), j∈[0,λ]−N. It follows that for each j∈[0,λ]−N,∗g(j)(∅)=ᐁ(∅)= ∅, and for internalY⊂∗Λ, ∗g(j)(Y )⊂ᐁ(W ). This also applies to the unrelativized case with the modifications that∅is replaced with W and each relativized consequence operatorSV
Ni is replaced with the physical theory consequence operatorSNi. Also note thatSNi andSNVi are usually considered practical consequence operators.
Depending upon the set Ꮿof consequence operators employed, there are usually manyX⊂L, X=L, such thatXis aC-system for eachC∈Ꮿ. For example, we assumed in [5] that there are two 1-ary relations for the science-community logic systems. One of these contains the logical axioms and the other contains a set of physical axioms; a set of natural laws. Let{S
Ni|i∈N}be the set of science-community corresponding conse-quence operators relativized so as to remove the set of logical theorems. Each member of a properly stated set of natural lawsNjused to generate the consequence operators
{S
Ni|i∈N}should be aC-system for each member of{S
Ni |i∈N}. As mentioned, the physical theories being considered here are not theories that produce new “natural laws.” The argument that the Einstein-Hilbert equations characterize gravitation fields, in general, leads to the acceptance by many science communities of these equations as a “natural law” that is then applied to actual physical objects. Newton’s Second Law of Motion is a statement about the notion of inertia within our universe. It can now be derived from basic laboratory observation and has been shown to hold for other physical models distinct from its standard usage [3]. The logic systems that generate the members of{S
THE BEST POSSIBLE UNIFICATION FOR ANY COLLECTION 871 system and applies the logic system toX. This gives a statement on how these natural laws affect, if at all, the behavior being described byX. It is this approach that implies that each properly describedNj=Lis aC-system for eachC∈ {SNi|i∈N}. Applying Theorem 4.1toᏯ= {S
Ni |i∈N}, where
{Ni|i∈N} ⊂K, leads to a result exactly likeCorollary 4.2, where results (i), (iii), and (iv) applied to members of{Ni|i∈N}are particularly significant.
At any moment in human history, one can assume, due to the parameters present, that there is, at least, a denumerable set of science-community logic systems or that there exists only a finite collection of practical logic systems defined on finite Lf. The corresponding set Ꮿf = {Cif |i=1,...,n} ⊂Ꮿff of practical consequence opera-tors would tend to vary in cardinality at different moments in human history. For the corresponding finite set of practical consequence operators, by Theorem 2.10, there is a standard (least upper bound) practical consequence operator ᐁ, and hence, “the best” practical logic system, that unifies such a finite set. The following result is the interpretation of Theorem 4.1 for such a finite set of practical consequence operators.
Theorem5.1. LetLf andᏯf be defined as above. Suppose thatᏯf is system consis-tent.
(i) Then there exists a practical consequence operator ᐁ1∈Ꮿff defined on the set of all subsets ofLf such thatᐁ
1=U, andW⊂Lsuch that, for eachCi∈Ꮿf,Ci(W )=
ᐁ1(W )=W=Lf, whereᐁ1(W )⊂Lf. (ii) For eachX⊂Lf,{C
i(X)|i∈N} ⊂ᐁ1(X)⊂Lf andᐁ1is the least upper bound inᏯff,∨w,∧,I,UforᏯf.
(iii) LetX⊂Lf,X=Lf, be aC-system for eachC∈Ꮿf. ThenX=Ci(X)=ᐁ1(X)=Lf for eachi∈N.
Letting finiteᏯf contain practical consequence operators of the typeSNi,S V
Ni, orSNi, exclusively, thenᐁ1would have the appropriate additional properties and would gen-erate a practical logic system. Corollary 2.11and Theorem 3.9 yield a more general unificationwᏭ,Ꮽ⊂Ꮿf, as represented by a least upper bound inᏯf,∨w,I,U, with the same properties as stated in Theorem 5.1. Thus, depending upon how physical theories are presented and assuming system consistency, there are nontrivial stan-dard unifications for such physical theories. Assuming that |Ꮽ| = ℵ0 and that Ꮽ is system consistent, then theCorollary 2.11unificationwᏭcorresponds to a (nontriv-ial) nonstandard ultralogic unification∗wᏭwith all of the same stated properties as those of theᐁ. It is obvious how these two ultralogic unifications model a higher in-telligence relative to general intelligent design theory. However,∗wᏭandᐁhave one significant difference. The ultralogicᐁis, with respect to internal subsets of∗L, a “least upper bound” of a hyperfinite collection, whereas∗wᏭneed not have this additional hyperfinite property. Further, system consistency is used only so that one statement in Theorem 4.1,Corollary 4.2,Theorem 5.1, and this paragraph will hold. This one fact is that each of these standard unifications of a collectionᏭ⊂Ꮿf is not the same as the upper unit if and only ifᏭis system consistent. Further, if anX⊂Lf (resp.,X⊂L) is
Remark 5.2. I mentioned that the nonstandard results obtained in Section 4are established by means of the most trivial methods used in Robinson-styled nonstandard analysis.
References
[1] W. Dziobiak,The lattice of strengthenings of a strongly finite consequence operation, Studia Logica40(1981), no. 2, 177–193.
[2] R. A. Herrmann,Nonstandard consequence operators, Kobe J. Math.4(1987), no. 1, 1–14. [3] ,Newton’s second law of motion holds in normed linear spaces, Far East J. Appl. Math.
2(1998), no. 3, 183–190.
[4] ,The Theory of Ultralogics. Part I, http://www.arXiv.org/abs/math.GM/9903081, 1999.
[5] ,Hyperfinite and standard unifications for physical theories, Int. J. Math. Math. Sci. 28(2001), no. 2, 93–102.
[6] ,Ultralogics and probability models, Int. J. Math. Math. Sci.27(2001), no. 5, 321–325. [7] J. Ło´s and R. Suszko,Remarks on sentential logics, Indag. Math.20(1958), 177–183. [8] A. Tarski,Logic, Semantics, Metamathematics. Papers from 1923 to 1938, Clarendon Press,
Oxford, 1956.
[9] R. Wójcicki,Some remarks on the consequence operation in sentential logics, Fund. Math. 68(1970), 269–279.
[10] ,On matrix representations of consequence operations of Łukasiewicz’s sentential calculi, Z. Math. Logik Grundlagen Math.19(1973), 239–247.
Robert A. Herrmann: Mathematics Department, United States Naval Academy, 572C Holloway Road, Annapolis, MD 21402-5002, USA
