In Herre & Schroeder-Heister's "Formal Languages and Systems", on p6,
It (i.e. a consequential operation) is called a deductive system, if the consequences of a set X can be obtained from a finite subset of X, i.e., if in addition to the three conditions mentioned, $$C(X) ⊆ \cup \text{ $\{C(Y ) : Y ⊆ X$, $Y$ finite $\}$ (compactness)}$$
Is it correct that in set theory, union $\cup$ is defined between finitely many sets, and not for infinitely many sets (uncountably many sets in particular)?
Does compactness mean that any $X$ has finitely many finite subsets $Y_i$'s, s.t. $C(X) \subseteq \cup_i C(Y_i)$?
Thanks.
p.s. Could you recommend other references that provides definitions of a consequence operator being inclusive, idemponent, monotonic, and compact, besides Herre's? Herre's is the only source I have seen these concepts.
No, there is no such restriction. The notation being used here is that if $X$ is a (arbitrary) collection of sets $\cup X$ is the union of all the sets in that collection, i.e. the collection of all elements of elements of $X.$
No, it just means that any consequence of $X$ is also the consequence of some finite subset of $X.$ It will only sometimes be the case that there is some fixed finite subset of $X$ such that all consequences of $X$ are consequences of that subset. (Or that there is some fixed finite collection of finite subsets such that all consequences of $X$ are consequences of one of these subsets.)
Sorry, don't know of any.