Modern Logic — A Survey: Historical, Philosophical and by J. M. Bochenski (auth.), Evandro Agazzi (eds.)

By J. M. Bochenski (auth.), Evandro Agazzi (eds.)

Logic has attained in our century a improvement incomparably more than in any earlier age of its lengthy heritage, and this has ended in such an enrichment and proliferation of its elements, that the matter of a few form of unified recom­ prehension of this self-discipline turns out these days unavoidable. This splitting into numerous subdomains is the normal final result of the truth that good judgment has meant to undertake in our century the prestige of a technology. This regularly signifies that the overall optics, lower than which a definite set of difficulties was once con­ sidered, breaks right into a lot of specialised sectors of inquiry, every one of them being characterised by means of the creation of particular viewpoints and of technical instruments of its personal. the 1st effect, that frequently accompanies the production of 1 of such really expert branches in a self-discipline, is that one has succeeded in setting apart the 'scientific center' of it, via proscribing the someway imprecise and redundant generality of its unique 'philosophical' configuration. yet, after it slow, it seems that a few of the discarded points are certainly vital and a brand new really good area of research is created to discover them. through follOwing this strategy, one ultimately reveals himself faced with this kind of number of self sufficient fields of study, that one wonders no matter if the very fact of labelling them below a typical denomination be not anything however the contingent impact of a natural historic tradition.

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The main result allows one to defme unambiguously the product of two complete theories, and from his proof one can see that with this product MODEL THEORY 53 the space of complete theories (for a given L) is a topological semigroup. If one then applies the theorem of topological algebra that a compact totally disconnected semigroup is a projective limit of fmite semigroups, one is led to the notion of autonomous system which Galvin used much later in an important paper [16]. Mostowski's theorem was the forerunner for similar theorems involving various operations on structures.

A theory is ,,-categorical if all its models of power" are isomorphic. Stray examples were known. For " > ~o, one had the Steinitz theorem for algebraically closed fields of fIXed characteristic. For" =~o, one had Cantor's theorem for dense linear order (topological characterization of Q). An extension of Cantor's theorem had been obtained by Hausdorff, for the so-called lla-sets. Quite amazingly, from the few clues in those examples, a rich metamathematical theory arose. 2. raproduct construction [51].

A prime extension of M to a model of T is an embedding M-M* of M into a model of T, so that for any other such M M# so that elementary map M* - M# we have an M-M* ~l# commutes. Morley proved that if T is w-stable then an embeddingM - M* always exists. Then a simple back and forth argument shows that if M is countable the embedding M M* is unique up to isomorphism over M. For generalM, no uniqueness statement was evident. Notice of course that when T is the theory of algebraically closed fields, MM* is the embedding of M in its algebraic closure.

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