ABSTRACT
Now in practice it is difficult to represent relations between belief systems, or the results of merging belief systems, or the consequences of such relations and mergings, in L1Ar alone. Throughout Chapter 1, for example, we interspersed statements written in LtAr with lines of text written in a 'natural' language, which in this case happened to be a specialized, technical or even scientific English, but which we can more generally describe as argot. In fact, on the whole, we fmd that whenever we intersperse our symbolic text with text in argot, we are actually referring to what will become the control elements of O'll" subsequent computer code. H now we wished to formalize this connecting text written in argot, as we must do when writing a code in a programming language, then we find that we can always reduce it, under the same constraints as already introduced, to another frrst-()rder language, which we call the first-order language of set theory and represent by LtSeL There have not surprisingly been several competing schemes of representation for both LtAr and L1Set but we shall here follow the now rather standard practice of adapting the Peano form for L1Ar and the Zermelo-Fraenkel form for L1Set (Manin, 1977). If instead of 'first-()rder
languages'wespeakof'predicatelanguages',andsorepresentthesetofall suchlanguagesby~.thenthemannerinwhichtheschemeofsignification isinterpretedandusedbecomesitselfapredicatecalculus.Thesetof elementary,irreduciblesymbolsthatareusedinanysuchlanguageiscalled analphabet.InFig.2.1thegeneralalphabetofL1SetandL1A,.respectively areorderedaccordingtotheusualwayoftheirpartitioningintodisjoint subsets(takenfromManin,1977,p.6).
