That is, while demonstrating the existence of such a set, it was not a constructive proof — it did not display " an object " — but rather, it was an existence proof and relied on use of the Law of Excluded Middle in an infinite extension.

The cut-elimination theorem for a calculus says that every proof involving Cut can be transformed ( generally, by a constructive method ) into a proof without Cut, and hence that Cut is admissible.

# proof theory and constructive mathematics ( considered as parts of a single area ).

The constructive proof of the theorem leads to an understanding of the aliasing that can occur when a sampling system does not satisfy the conditions of the theorem.

It was Weierstrass who raised for the first time, in the middle of the 19th century, the problem of finding a constructive proof of the fundamental theorem of algebra.

In mathematics, the proof of a " some " statement may be achieved either by a constructive proof, which exhibits an object satisfying the " some " statement, or by a nonconstructive proof which shows that there must be such an object but without exhibiting one.

This theorem also has important roles in constructive mathematics and proof theory.

This proof may not be considered constructive, because at each step it uses a proof by contradiction to establish that there exists an adjacent vertex from which infinitely many other vertices can be reached.

Facts about the computational aspects of the lemma suggest that no proof can be given that would be considered constructive by the main schools of constructive mathematics.

He contributed to Hilbert's program in the foundations of mathematics by providing a constructive consistency proof for a weak system of arithmetic.

In constructive logic, a statement is ' only true ' if there is a constructive proof that it is true, and ' only false ' if there is a constructive proof that it is false.

Informally, this means that given a constructive proof that an object exists, then that constructive proof can be turned into an algorithm for generating an example of it.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

However, in certain axiom systems for constructive set theory, the axiom of choice does imply the law of the excluded middle ( in the presence of other axioms ), as shown by the Diaconescu-Goodman-Myhill theorem.

In mathematics, the Banach fixed-point theorem ( also known as the contraction mapping theorem or contraction mapping principle ) is an important tool in the theory of metric spaces ; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points.

However, Fred Richman proved a reformulated version of the theorem that does work .< ref > See Fred Richman ; 1998 ; < cite > The fundamental theorem of algebra: a constructive development without choice </ cite >; available from .</ ref >

* a theorem that can be proved by constructive logic ; see mathematical constructivism.

lim f ( x < sub > n </ sub >) for all ascending sequences x < sub > n </ sub >, then the least fixpoint of f is lim f < sup > n </ sup >( 0 ) where 0 is the least element of L, thus giving a more " constructive " version of the theorem.

The proof of a mathematical theorem exhibits mathematical elegance if it is surprisingly simple yet effective and constructive ; similarly, a computer program or algorithm is elegant if it uses a small amount of code to great effect.

However, as with the intermediate value theorem, an alternative version survives ; in constructive analysis, any located subset of the real line has a supremum.

A constructive version of " the famous theorem of Cantor, that the real numbers are uncountable " is: " Let

Every theorem proved with idealistic methods presents a challenge: to find a constructive version, and to give it a constructive proof.

Polynomials in Bernstein form were first used by Bernstein in a constructive proof for the Stone – Weierstrass approximation theorem.

" This theorem can be proven via a constructive proof, or via a non-constructive proof.

A constructive proof of the theorem would give an actual example, such as:

However, the theorem does not rely upon the axiom of choice in the separable case ( see below ): in this case one actually has a constructive proof.

The proof of the theorem is constructive: it demonstrates an algorithm for STCON, the problem of determining whether there is a path between two vertices in a directed graph, which runs in space for n vertices.

In a single evening, Scarf realized that he could directly translate the Lemke-Howson ’ s algorithm through a limiting process into an elementary and constructive proof of his core existence theorem.

At certain critical stages, and only for sound diagnostic reasons, it may be important to accompany family members in their use of these resources if their problem-solving behavior is to be constructive rather than defeating.

What ought to be, what is his potential role as a force for constructive social change??

In summary, while we stress that constructive engagement between anthropology and the military is possible, CEAUSSIC suggests that the AAA emphasize the incompatibility of HTS with disciplinary ethics and practice for job seekers and that it further recognize the problem of allowing HTS to define the meaning of “ anthropology ” within DoD.

In his view, " The dark age of pro and contra slogans, unfair polemics, and humiliations is not yet completely over and done with, but there seems to be some hope for a more constructive discussion " ( ib.

A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.

These foundational applications of category theory have been worked out in fair detail as a basis for, and justification of, constructive mathematics.

** Evaluation and conclusion – summary of process and results, including constructive criticism and suggestions for future improvements

Individual deists varied in the set of critical and constructive elements for which they argued.

After Locke, constructive deism could no longer appeal to innate ideas for justification of its basic tenets such as the existence of God.

Postulates 1, 2, 3, and 5 assert the existence and uniqueness of certain geometric figures, and these assertions are of a constructive nature: that is, we are not only told that certain things exist, but are also given methods for creating them with no more than a compass and an unmarked straightedge.

Through this, the exhibition was designed to inspire greater public enthusiasm and support for the constructive work and planning by engineers and public officials who had contributed so much toward improvement of streets and highways.

These renounce all outstanding territorial claims and lay the foundation for constructive relations.

Thus the psychopathologized individual for Freud was an immature individual, and the goal of psychoanalysis was to bring these fixations to conscious awareness so that the libido energy would be freed up and available for conscious use in some sort of constructive sublimation.

The " Technology Roadmap for Productive Nanosystems " aims to offer additional constructive insights.

Much constructive mathematics uses intuitionistic logic, which is essentially classical logic without the law of the excluded middle which states that for any proposition, either that proposition is true, or its negation is.

This definition corresponds to the classical definition using Cauchy sequences, except with a constructive twist: for a classical Cauchy sequence, it is required that, for any given distance, there exists ( in a classical sense ) a member in the sequence after which all members are closer together than that distance.

In the constructive version, it is required that, for any given distance, it is possible to actually specify a point in the sequence where this happens ( this required specification is often called the modulus of convergence ).

This constructive measure theory provides the basis for computable analogues for Lebesgue integration.

Traditionally, some mathematicians have been suspicious, if not antagonistic, towards mathematical constructivism, largely because of limitations they believed it to pose for constructive analysis.

Applications for constructive mathematics have also been found in typed lambda calculi, topos theory and categorical logic, which are notable subjects in foundational mathematics and computer science.