types, theory of

types, theory of
Russell's own reaction to his paradox of the class of all classes that are not members of themselves (see Russell's paradox ) was to suggest that the definition is ill-formed because it involves the illegitimate notion of ‘all classes’. If the entities of a theory are classified in a hierarchy, with individuals at the lowest level, sets of individuals at the next, sets of sets further up again, and if prohibitions are introduced against sets containing members of different types, then the paradox cannot be formulated.
When Russell published his solution he was also influenced by Poincaré's constructivist response to the contradictions of set theory . This suggested that in any application of the axiom of comprehension, (∃y )((∀x )(xy ↔ F x )), the variable x should not range over entities including the set y . If it does, the definition becomes impredicative, and offends against the vicious circle principle . To prohibit this, Russell sorted predicates into orders, again with restrictions on well-formedness. The result is called the ramified theory of types. Unfortunately it proved impossible to construct classical mathematics within the ramified theory, and Russell was forced to introduce the axiom of reducibility, whose effect is to collapse the hierarchy of predicates.
Although the non-ramified or simple theory of types has attracted much subsequent work, all type theory suffers from a problem of unintuitive duplication. Thus for Russell there exists not one but an infinite number of the number 2, since there are sets of all sets having two members at each level in the hierarchy. Expressions needed at each type level, such as ‘=’, are supposedly systematically ambiguous, and in some developments there are an infinite number of identity symbols.
The relative consistency of type theory has been proved within Zermelo–Fraenkel set theory , thereby also showing that the latter is stronger than the former. Few logicians now believe that there is any reason to build type restrictions into the formal systems within which set-theoretic and mathematical reasoning is best represented.

Philosophy dictionary. . 2011.

Игры ⚽ Нужен реферат?

Look at other dictionaries:

  • types, theory of — In logic, a theory introduced by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–13) to deal with logical paradoxes arising from the unrestricted use of propositional functions as variables. The type of a… …   Universalium

  • theory — /thee euh ree, thear ee/, n., pl. theories. 1. a coherent group of general propositions used as principles of explanation for a class of phenomena: Einstein s theory of relativity. 2. a proposed explanation whose status is still conjectural, in… …   Universalium

  • theory of types — See types, theory of …   Philosophy dictionary

  • Theory of Constraints — (TOC) is an overall management philosophy. Dr. Eliyahu M. Goldratt introduced the Theory of constraints in his 1984 book titled The Goal . It is based on the application of scientific principles and logic reasoning to guide human based… …   Wikipedia

  • Theory of justification — is a part of epistemology that attempts to understand the justification of propositions and beliefs. Epistemologists are concerned with various epistemic features of belief, which include the ideas of justification, warrant, rationality, and… …   Wikipedia

  • Theory of constraints — Part of a series of articles on Industry Manufacturing methods Batch production • Job production Continuous production Improvement method …   Wikipedia

  • Types of socialism — Since the 18th century, socialist ideas have developed and separated into many different types of socialism. Theoretically, socialism has a very broad definition, likewise, the arguments for or the reasons for advocating state or collective… …   Wikipedia

  • Theory of multiple intelligences — Human intelligence Abilities and Traits Abstract thought Communication · Creativity Emotional Intelligence Kn …   Wikipedia

  • Types of unemployment — Economists distinguish between four major types of unemployment, i.e., cyclical, frictional, structural and classical. (Another distinction, not discussed here, is between voluntary and involuntary unemployment.) Real world unemployment may… …   Wikipedia

  • Theory of Forms — Plato s Theory of Forms [The name of this aspect of Plato s thought is not modern and has not been extracted from certain dialogues by modern scholars. The term was used at least as early as Diogenes Laertius, who called it (Plato s) Theory of… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”