Concept
Complete theory
Overview
In mathematical logic, a theory of a language is complete if it is consistent and it proves every closed formula with which it is not inconsistent. That is to say, a consistent theory
T
{\displaystyle T}
is complete if, for every sentence
φ
{\displaystyle \varphi }
in the language, either
T
⊢
φ
{\displaystyle T\vdash \varphi }
holds or
T
∪
{
φ
}
{\displaystyle T\cup \{\varphi \}}
is inconsistent. Another common definition, that is equivalent if the formal system satisfies the principle of explosion, requires instead that either
φ
{\displaystyle \varphi }
or its negation
¬
φ
{\displaystyle \neg \varphi }
is provable from
T
{\displaystyle T}
. Using this definition, consistency of
T
{\displaystyle T}
follows automatically if "either" and "or" are read as exclusive disjunction, and it thus can be omitted from the definition. If
T
{\displaystyle T}
is furthermore deductively closed, completeness reduces to the concise condition that exactly one of
φ
{\displaystyle \varphi }
and
¬
φ
{\displaystyle \neg \varphi }
is contained in
T
{\displaystyle T}
for every sentence
φ
{\displaystyle \varphi }
. Recursively axiomatizable first-order theories that are consistent and rich enough to allow general mathematical reasoning to be formulated cannot be complete, as demonstrated by Gödel's first incompleteness theorem.
This syntactic sense of complete is distinct from the semantic notion of a complete formal system, which asserts that for every theory that can be formulated in the formal system, all semantically valid statements are provable theorems (for an appropriate sense of "semantically valid"). Gödel's completeness theorem is about this latter kind of completeness.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
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This is nothing else than the necessity of abstract logic, which is the presupposition involved in the very fact of interrelated existence as disclosed in each immediate occasion of experience. The key to the pattern means this fact:—that from a select set of those general conditions, exemplified in any one and the same occasion, a pattern involving an infinite variety of other such conditions, also exemplified in…
Alfred North Whitehead, Science and the modern world
States what Complete theory is.
In modern expositions of the axioms of geometry certain conditions are laid down which the relation of congruence between segments is to satisfy. It is supposed that we have a complete theory of points, straight lines, planes, and the order of points on planes--in fact, a complete theory of non-metrical geometry. We then enquire about congruence and lay down the set of conditions--or axioms as they are called--which…
Alfred North Whitehead, The Concept of Nature The Tarner Lectures Delivered in Trinity College, November 1919
States what Complete theory is; names Complete theory twice.
Meanwhile every mathematical proposition remains always a universal truth, which is valid for innumerable particular cases; and a graduated process from the simple to the complicated propositions which are to be deduced from them, is also essential to mathematics; therefore, in every respect mathematics is a science. The completeness of a science as such, that is, in respect of form, consists in there being as much…
Arthur Schopenhauer, The World as Will and Idea
States what Complete theory is.
Related ideas
- Formal language
- Gödel's completeness theorem
- Mathematical logic
- Model theory
- Sentence (mathematical logic)
- Completeness (logic)
- Modal logic
- Formal system
- Well-formed formula
- Classical logic
- Gödel's incompleteness theorems
- Principle of explosion
- Countable set
- Contradiction
- Kripke semantics
- Euclidean geometry
Place in the tradition
Part of First-order logic.