Concept

Constructible universe

Overview

In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by

{\displaystyle L,}

is a particular class of sets that can be described entirely in terms of simpler sets.

{\displaystyle L}

is the union of the constructible hierarchy

{\displaystyle L_{\alpha }}

. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis". In this paper, he proved that the constructible universe is an inner model of ZF set theory (that is, of Zermelo–Fraenkel set theory with the axiom of choice excluded), and also that the axiom of choice and the generalized continuum hypothesis are true in the constructible universe. This shows that both propositions are consistent with the basic axioms of set theory, if ZF itself is consistent. Since many other theorems only hold in systems in which one or both of the propositions is true, their consistency is an important result.

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

Possible passages

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The exercise of logical reason is always concerned with these absolutely general conditions. In its broadest sense, the discovery of mathematics is the discovery that the totality of these general abstract conditions, which are concurrently applicable to the relationships among the entities of any one concrete occasion, are themselves inter-connected in the manner of a pattern with a key to it. This pattern of…
Alfred North Whitehead, Science and the modern world
States what Constructible universe is.
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The notion of continuity depends upon that of _order_, since continuity is merely a particular type of order. Mathematics has, in modern times, brought order into greater and greater prominence. In former days, it was supposed (and philosophers are still apt to suppose) that quantity was the fundamental notion of mathematics. But nowadays, quantity is banished altogether, except from one little corner of Geometry…
Bertrand Russell, Mysticism and Logic and Other Essays
States what Constructible universe is.
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One of the most successfully cultivated branches of philosophy in our time is what is called inductive logic, the study of the conditions under which our sciences have evolved. Writers on this subject have begun to show a singular unanimity as to what the laws of nature and elements of fact mean, when formulated by mathematicians, physicists and chemists. When the first mathematical, logical and natural…
William James, Pragmatism
States what Constructible universe is.
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Philosophers who discussed Constructible universe

Hilary Putnam

Related ideas

Place in the tradition

Part of Outline of logic.

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