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Sets, numbers and taxa. (English) Zbl 0544.03029

The ”taxonomic theory” of this paper provides a new theory of the foundations of set theory and arithmetic, in which there is only one undefined term, (”taxon”, replacing set or class), and one undefined relation, (”association”, replacing ”overlapping”). Each taxonomic ”system” of taxa related by association, has its own species of association, which is characteristic of the system. The only axioms are those which specify which pairs of taxa are associated. Russell’s paradox is automatically eliminated. Zermelo’s axioms for set theory are replaced by theorems, and so are Peano’s axioms for arithmetic. The usual connectives are explicitly defined in terms of the relation of ”association”, by the use of binary algebra as a formal language. Russell’s definition of ”number” is shown to imply the axiom of choice and is replaced by a generalised version of Dedekind’s theory of ”chains”. The origins of taxonomic theory are A. N. Whitehead’s theory of ”extensive abstraction”, Lesniewski’s axiomatic mereology, Lam’s taxonomic nomenclature, and G. Birkhoff’s theory of lattices.

MSC:

03E99 Set theory
92F05 Other natural sciences (mathematical treatment)
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