Basic law of the course of value

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Gottlob Frege's Basic Law of Value Processes (Basic Law V) is an axiom scheme from naive set theory .

In the basic laws of arithmetic formulated Gottlob Frege in Begriffsschrift an axiomatic system for arithmetic with six axioms. Of these, the fifth (Basic Law V) has a special role both formally and historically. In modern notation it reads: ⊢ (ext ε f (ε) = ext α g (α)) = (∀af (a) = g (a)). The system formed is strong enough to derive common theorems from it, but together with Frege's variant of the comprehension axiom, Basic Law V leads to Russell's antinomy . The system was inconsistent with this .

Formal description

Frege's assumption can be presented in such a way that the value curve of a function f forms a set of ordered pairs of function values f (ε) and corresponding arguments ε. Basic Law V now states that the value curve of two functions f and g identical iff. f and g map each object a to the same value. In the case of terms that are functions with the value range {true; false} for Frege ( truth value functions ), the following applies: the term scope (the extension , i.e. the set of objects that fall under a term) of the term F is identical to the term scope of the term G if and only if all objects that fall under F also fall under G (material equivalence).

Frege now also advocates a substitution principle, which states that for every predicate F with a free variable x there is a corresponding term or a set that includes all objects that fall under F. This is a variant of an unrestricted axiom of comprehension ( axiom schema of specification , also of separation or of comprehension ). The naive set theory of the 19th century usually allowed the existence or producibility of such sets. Taken together, however, both principles have the consequence that Russell's antinomy can be generated in Frege's system (e.g. as "a set of all sets that do not contain themselves").

Inconsistency and suggested solutions

Bertrand Russell had pointed out this problem to Frege with a postcard. At the beginning of the 20th century, different proposals were developed in order to arrive at a consistent set theory. The most widely used ever since by far the solution is in axiomatized amount teachings limited variant of a Komprehensions- or separation axiom (engl. Restricted comprehension ) to use.

According to a proposal by Crispin Wright , the feasibility of which has been formally proven by George Boolos and Richard G. Heck , the law V can be replaced by Hume's principle for an axiomatization of arithmetic , so that a consistent theory is also obtained. This is a fundamental building block of Wright and Bob Hale's intended revival of Frege's program of reducing arithmetic to logic - they are therefore called representatives of "neo- logicism ".

literature

Individual evidence

  1. ^ Basic laws of arithmetic. Volume 1, 1893 ( digitization at korpora.org ).
  2. ^ Crispin Wright : Frege's Conception of numbers as objects (= Scots Philosophical Monographs. Vol. 2). Aberdeen University Press, Aberdeen 1983, ISBN 0-08-025726-7 .
  3. ↑ See Richard G. Heck, Jnr .: On the consistency of second-order contextual definitions. In: Noûs. Vol. 26, No. 4, 1992, ISSN  0029-4624 , pp. 491-494.
  4. ↑ See, inter alia, Bob Hale : Reals by abstraction. In: Philosophia Mathematica. Vol. 2000, ISSN  0031-8019 , pp. 100-123.