With this development, the structure imposed by a formal language on the universe of discourse is characterized by a group of automorphisms of the model of the partial theory. 11 We can consider a formal language to give a certain structure to the universe of parts, and we characterize this structure by a group of automorphisms of the universe of parts. One of the most interesting of these is the possibility of formally defining our intuitive sense of the distance between languages.
The "property" to which a semantic category must correspond is the property of being a projection of the domain or image of a. If the model of set theory is pure, that is, the only individual is the empty set, then its there are no true automorphisms of the inodel; If a formal language is defined on the basis of a model of set theory, it seems reasonable to maintain that the semantic transformations of the language depend only on the set-theoretic structure of the model, that is, they are structural, and further that they Be constructive, as illustrated above.
A significant aspect of the correspondence between formal language and its syntax is the relationship between repeated applications of grammatical rules and the composition of semantic transformations. The diagram above also illustrates that we can consider a grammatical rule as an abstraction of the corresponding semantic transformation in the range and domain of the semantic transformation. In the first definition, an object can be context-free definable even though some of the semantic transformations in T used to define the object are not context-free semantic transformations.
PART THEORY
In words, c is a part of the conglomerate if every part of c satisfies some satisfying F(a) • This definition and the following axiom scheme occupy a position in the. Axioms 1, 2 and 9 together with axiom scheme 7 form the main set of axioms of the theory of parts. This defines the union of the parts b and c , but does not guarantee that b + c is a part of the universe.
This defines the intersection of two parts, which may not always exist as part of the universe. Zero is not part of the universe, so according to Theorem 13 it is not part of any entity. The theories from the previous section indicate that a model of the partial theory is also a model of a Boolean algebra.
Assuming there are no atoms, we can always divide any part of the universe into smaller parts.
These permutations are in the group of the formal language provided they retain all the structure determined by the formal language. P • G is the group of the formal language F if for every basic or derived semantic transformation, T , of F and every sequence (x1, ••• , xn} of referents in X such that. Under this definition, G is the group of the formal language F if every autonorphis1n in G commutes with every semantic transformation of F when the semantic transformation is defined.
Definition: Let GF be the group of the formal language F • If two parts are in the orbit sa1n e under GF , then they are indistinguishable from F. 34;author of Waverly" as another part, y • Now if there is an automorphism g in the set of the formal language such that g(x} = y. To say "Scott is the author of Waverly" is to assert that x and y are equivalent under the set of the formal language and x may or may not be a toy equal • Now suppose there is a phrase involving the word "Scott" that corresponds to the semantic transformation T in the sequence of referents.
The structure defined by a formal language can also be characterized by the collection of all semantic transformations invariant under the group of the formal language. The semantic transformation of the closure of an automorphism group is the collection of all semantic transformations that co-commutate with every automorphism in the group. If GF is the group of a formal language F, then LGF includes all basic and derived semantic transformations of F, when these transformations are considered to be restricted to definable parts.
Definition: Let F be a formal language < T, K, X> • Then let DF be the set of contextually definable parts relative to the formal language F • Let DLGF be the set of contextually definable parts relative to the class of referents X and the collection of sern.antic transformations LGF. Theorem: If DF is closed in the gauge topology, then DLGF is equal to the subalgebra generated by DF. 1, ••• , xn is in DF and y is not in the subalgebra generated by DF • Then we will show that there is an automorphism, g , in GF such that x.
That is, if we speak a fully formal language as described here, then our 'intuition' roughly corresponds to the indefinable transformations in LGF. • If this seems reasonable, equating DF with DLGF means that we have a language powerful enough to formally define anything we can. Given a formal language In this chapter we show that a set theory can be embedded in suitable models for part theory. Cohen [6 J has given a clear account of the constructive method of Zermelo-Fraenkel set theory, our presentation is formal, and completes the details of Cohen's presentation. Using the embedding to be presented, this predicate can also be written for part theory, and the parts that satisfy this predicate are the sets that satisfy all the axioms of set theory, including regularity and choice. By property (i) of s, the only singleton sets that are part of z are sx and sy. Z is also a set since it is the least upper bound of sx and sy. Then the statement of the power set axiom we prove here is Vx3:yVz[sz'!Ty-zcx]. This version of the power set axiom is much stronger than is required to show that the power set of every set exists. To describe the constructive set theory, we define the ordinals of the model of set theory. Since in our proof of the axiom of infinity we demonstrated the set of all integers, w, we need only note that w is an ordinal and that w E Mw+. We have developed some of the consequences of assuming that a formal language has a model of part theory as its universe. We will step back and reconsider what properties a model of part theory must have in order to obtain set theory. The factor group GF /N can be thought of as automorphisms of sets that leave F invariant. The only way to include non-atomic parts in set theory is to extend the definition of the predicate "Set" to. If a language of the form F defines only a finite number of parts, we can still find set theory under F • For each atom x of the algebra of definable parts, we can choose an infinite number of corresponding parts x that form an atomic Boolean algebra when relativized to x • Then GF has a normal subgroup , N , which fixes the chosen algebras, and LN has the set theory that defines the singular function. If the part-theoretic union, y, of the determinable parts does not reach the universal part, 1, then the intuition can be satisfied by developing set theory within the complement of y, using the algebra of the determinable parts as some finite sets. 1 is generated by all permutations of the singleton sets which leave [O} fixed since all automorphisms of a Boolean algebra fix 0 and if x-/:. The sets are the elements of the subalgebra generated by the parts that can be defined in either language. If there is room in the model for both the non-atomic subalgebra of parts and a set theory, then LN, if N is the group of set theory, will contain a function, f , from the non-atomic definable parts to sets such that if. The automorphism that takes x to y need not preserve the structure of x, since it is not in the group of the formal language, and so there is no requirement that the red part of x be mapped in any form. A change is perceptible because of the change in structure, where the structure is determined by the language. 34; indicates that the entity with the structure of a red pencil has been transformed into an entity with the structure of black soot without specifying the transformation of substructures such as the red part of the red pencil. In the second section, the results obtained by the decision method are used to prove the existence of the union of all elements of the direct product Boolean algebra corresponding to a given elementary formula. Using these results, we claim that any lower predicate calculus formula whose only predicate is 11 ~ 11 is equivalent to a formula in standard form. HS was defined such that any condition for the remaining members of R is satisfied by one of HS v. For the proof in the other direction, assume 3:x F. Then z as constructed above also satisfies F. It remains to show that any lower predicate formula. calculus with ~ as the only predicate is equivalent to a division of formulas of the form S. Assuming that the Boolean algebra is a subdirect product of an atomic algebra and an atomless Boolean algebra that possesses only the properties necessary to prove the elimination theorem of quantities, we prove that the Boolean algebra must be the direct product of its factors. B must possess this property for the proof of the quantifier elimination theorem to be valid. Since jr ~ y s for all s -f r and jr f- y s for all r and s , we have that the least upper bound of all satisfying the above conjunction is the intersection of the y. 0 , contradiction, Therefore z' must be in the complement of the union of all the y • In this case,. The Consistency of the Axiom of Choice and of the· Generalizecrl::ontinuum Hypotnesis, Prmcefon Un'!VerSTty Press: Princeton, N . Parikh, Rohit J, : "On Context-Free Languages"; Journal of the Association for Computing Machinery, October 1966, Vol.LGF 2
SET THEORY
CONCLUSIONS AND FURTHER RESEARCH Formal Language Definitions
