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1. Axiomatization


Subject Philosophy

Key-Topics mathematics, science

DOI: 10.1111/b.9780631230205.2001.00004.x


Axiomatization is a formal method for specifying the content of a theory wherein a set of axioms is given from which the remaining content of the theory can be derived deductively as theorems. The theory is identified with the set of axioms and its deductive consequences, which is known as the closure of the axiom set. The logic used to deduce theorems may be informal , as in the typical axiomatic presentation of Euclidean geometry; semiformal , as in reference to set theory or specified branches of mathematics; or formal , as when the axiomatization consists in augmenting the logical axioms for first-order predicate calculus by the proper axioms of the theory. Although Euclid distinguished axioms from postulates , today the terms are used interchangeably. The earlier demand that axioms be self-evident or basic truths gradually gave way to the idea that axioms were just assumptions, and later to the idea that axioms are just designated sentences used to specify the theory. Axiomatization played a central role in positivistic philosophy of science, which analyzed theories as axiomatic systems. (See theories. ) Initially these systems were construed as purely syntactical systems which invoked no pre-axiomatic “meanings'” in specifying the content of the theory. On this view the axiomatization implicitly defined the key terms of the axiomatization (see craig's ... log in or subscribe to read full text

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