By Alfred Tarski, Steven Givant
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Extra resources for A Formalization of Set Theory without Variables (Colloquium Publications)
And A E n) iff the relation R denoted by A holds between Xk and Xl, that is, in set-theoretical notation, iff (Xk' Xl) E R. Similarly, X satisfies A B with A, BEn iff A and B denote the same relation. The remaining parts of the definition of satisfaction, referring to nonatomic formulas, are routine; cf. Tarski , p. 193. ,. , + . 3. , +. ,. Because of its form, each schema 8k, k = 1, ... ,. ,. 4(xiii) below and also Tarski , p. ) It is a matter of common belief that when passing from a formalism to its definitional extensions we do not enrich the means of expression and proof of the formalism involved.
N-l. (In the treatment of satisfaction for finite sequences we deviate slightly from the terminology of Henkin- Monk- Tarski , p. ) In terms of satisfaction other semantical notions are defined. The most important among them are the closely related notions of truth and model. A sentence Y E E is said to be true of II or to hold in II if every sequence x = (xo , ... , x n , . ) of elements of U satisfies Y. Under the same condition II is called a model of Y . More generally, II is a model of a set \[I <;::; E iff it is a model of every sentence X in \[I.
4(iii) It may seem dubious whether Definition (i) (a) (either in its syntactical or semantical form) really captures the intuitive content of equipollence in means of expression. We shall try to explain what we mean, using as an example the formal languages ,c and ,c+. As we know, ,c and ,c+ satisfy (i) (a) (with 8(1) = ,c and 8(2) = ,c+); thus one could say that every statement which can be formulated in one of these two languages can be formulated in the other as well. g. , that every binary relation definable in ,c + is also definable in ,c.
A Formalization of Set Theory without Variables (Colloquium Publications) by Alfred Tarski, Steven Givant