Constructibility
Encyclopedia
In mathematics
, there are several notions of constructibility. Each of the following is by definition constructible:
The term may also refer to:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, there are several notions of constructibility. Each of the following is by definition constructible:
- a pointPoint (geometry)In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
in the Euclidean plane that can be constructed with compass and straightedgeCompass and straightedgeCompass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
. Also, any complex numberComplex numberA complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
associated to such a point; see constructible numberConstructible numberA point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...
. - a regular polygonRegular polygonA regular polygon is a polygon that is equiangular and equilateral . Regular polygons may be convex or star.-General properties:...
that can be constructed with compass and straightedge; see constructible polygonConstructible polygonIn mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....
. - a theoremTheoremIn mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
that can be proved by constructive logic; see mathematical constructivism. - a set in Kurt GödelKurt GödelKurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
's universe L, which may be constructed by transfinite application of certain constructions in set theory; see constructible universeConstructible universeIn mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
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The term may also refer to:
- a processIndustrial processIndustrial processes are procedures involving chemical or mechanical steps to aid in the manufacture of an item or items, usually carried out on a very large scale. Industrial processes are the key components of heavy industry....
in construction design whereby plans are reviewed by others familiar with construction techniques and materials to assess whether the design is actually buildable. Often referred to as a "Constructability Review", the process usually occurs prior to the plans being put out for bid.