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Extra resources for A brief course in the calculus, by William Cain.
And a sentence To give a few examples, the relation to: between a set W X E is extended to the cases when W is replaced by a sentence Y E or X is replaced by a set In case W = 0, we write "to: X" instead of "0 to: X"; the sentence X is then called logically valid or logically true. Two sets W, 0 are said to be semantically equivalent relative to a set if u W to: 0 and U 0 to: w; they are called simply semantically equivalent (and not semantically logically equivalent) in case = 0 .
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. 3; it is called the (first-order) theory of ll, in symbols 0pll. More generally, given a class K of such structures ll, the theory of K, 0pK, is the intersection of all 0pll for II E K. A sentence X E E is called a consequence of a set \[I <;::; E , and \[I is said to imply X, in symbols \[I F X , if every model of \[I is a model of X .
This includes, in particular, such contexts as "\II f- X". We also make an analogous stipulation concerning "=== " and " ===+ " . We may, of course, return to the original symbolism in some exceptional cases, so as to enhance the clarity of the text. Notice that 8q\ll cannot be identified with 8q+\II , even if \II E. 4. The equipollence of a system with an extension As we have said before, we are not interested in developing a general theory of formalisms. However, the notion of equipollence between two formalisms and, more generally, between two systems, is of such fundamental importance for our work that it seems proper to state some observations regarding this notion in a general and precise way.
A brief course in the calculus, by William Cain. by Michigan Historical Reprint Series