You then descended one story, glommed a television set from the music room -- the only constructive feature of your visit, by the way -- and, returning to the ground floor, entered the master bedroom.

When he attended the Christian Anti-Communist Crusade school here about six months ago, Jim became convinced that an individual can do something constructive in the ideological battle and set out to do it.

Thus the axiom of choice is not generally available in constructive set theory.

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.

Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.

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.

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

Kronecker, now seen as one of the founders of the constructive viewpoint in mathematics, disliked much of Cantor's set theory because it asserted the existence of sets satisfying certain properties, without giving specific examples of sets whose members did indeed satisfy those properties.

Jürgen Schmidhuber, however, says the " set of mathematical structures " is not even well-defined, and admits only universe representations describable by constructive mathematics, that is, computer programs.

Constructivism also includes the study of constructive set theories such as IZF and the study of topos theory.

One trivial meaning of " constructive ", used informally by mathematicians, is " provable in ZF set theory without the axiom of choice.

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.

Some constructive set theories include weaker forms of the axiom of choice, such as the axiom of dependent choice in Myhill's set theory.

In algebra, for such entities as toposes and Hopf algebras, the structure supports an internal language that is a constructive theory ; working within the constraints of that language is often more intuitive and flexible than working externally by such means as reasoning about the set of possible concrete algebras and their homomorphisms.

Another constructive heuristic, Match Twice and Stitch ( MTS ) ( Kahng, Reda 2004 ), performs two sequential matchings, where the second matching is executed after deleting all the edges of the first matching, to yield a set of cycles.

The orientation of a particular set of sheets is identified by its three Miller indices ( h, k, l ), and let their spacing be noted by d. William Lawrence Bragg proposed a model in which the incoming X-rays are scattered specularly ( mirror-like ) from each plane ; from that assumption, X-rays scattered from adjacent planes will combine constructively ( constructive interference ) when the angle θ between the plane and the X-ray results in a path-length difference that is an integer multiple n of the X-ray wavelength λ.

It is generally considered uncontroversial, although constructive set theory prefers a weaker version to resolve concerns about predicativity.

Schmidhuber () uses this approach to define the set of formally describable or constructively computable universes or constructive theories of everything.

While a general proof can be given that almost all numbers are normal ( in the sense that the set of exceptions has Lebesgue measure zero ), this proof is not constructive and only very few specific numbers have been shown to be normal.

Intuitionistic type theory, or constructive type theory, or Martin-Löf type theory or just Type Theory is a logical system and a set theory based on the principles of mathematical constructivism.

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

For example, the problem of trisecting an angle with a compass and straightedge is one that naturally occurs within the theory, since the axioms refer to constructive operations that can be carried out with those tools.

This work was a precursor of a modern formulation in terms of constructive type theory.

In the 1980s, Per Martin-Löf developed intuitionistic type theory ( also called Constructive type theory ), which associated functional programs with constructive proofs of arbitrarily complex mathematical propositions expressed as dependent types.

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

* axiomatic, algebraic and constructive quantum field theory

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

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.

The pilot wave model, originally developed by Louis de Broglie and further developed by David Bohm into the hidden variable theory proposes that there is no duality, but rather a system exhibits both particle properties and wave properties simultaneously, and particles are guided, in a deterministic fashion, by the pilot wave ( or its " quantum potential ") which will direct them to areas of constructive interference in preference to areas of destructive interference.

The diffusion is most pronounced and most likely to become fixed, however, in those who have had no or very minimal opportunities to develop the autonomy and initiative that could have been directed into constructive expression and so served as sources of developing self-certainty.

It should be noted, however, that the constructive element of deism was not unique to deism.

While Burma's presence in ASEAN was seen as a test of the organisation's philosophy of constructive engagement, the presence of Burma in ASEAN however has started to be seen as an embarrassment to the organisation, because of Burma's human rights record and lack of democracy.

" He argued, however, that " the current discourses and concerns on multiculturalism in Korea " lacked " the constructive and analytical concepts for transforming a society ".

After several states had closed their banks in what became the banking crisis of 1933, Hoover issued a February 20, 1933, plea to the House of Representatives to pass the Glass bill as the “ first constructive step to remedy the prime weakness of our whole economic life .” On March 4, 1933, however, the lame duck session of the 72nd Congress adjourned without either the Glass bill or the House deposit insurance bill becoming law.

Such theorems are relatively unproblematic when all of the proofs involved are constructive ; however, the status of " pure existence metatheorems " is extremely unclear,

A weak Brouwerian counterexample does not disprove a statement, however ; it only shows that the statement has no constructive proof.

This débacle, however, had a constructive impact on Goehr: rather than dismissing criticism as the mere result of incompetence on the part of critics and performers, he genuinely faced the questions of the position of the avant-garde composer and his music:

Most publications in the field of interlinguistics are, however, not so constructive, but rather descriptive, comparative, historic, sociolinguistic, or concerned with translation by humans or machines.

Such is the story accepted by Teófilo Braga, but Delfim Guimarães shows that the first part is doubtful, and, putting aside the testimony of a contemporary and grave writer, Diogo do Couto, he even denies the title of poet to Cristóvão Falcão, arguing from internal and other evidence that Chrisfal is the work of Bernardim Ribeiro ; his destructive criticism is, however, stronger than his constructive work.

If, however, these same sinusoids have phases which change rapidly as the frequency changes, they will add incoherently, varying between constructive and destructive addition at different times.

Muravyov, however, was able to persuade the Chinese that Russia's intentions were peaceful and constructive.

Re Hagger 2 Ch held that the constructive trust comes into existence on the death of the first testator, however this approach was revised in Re Hobley which decided that it must come into existence before the death of the first testator to satisfy the requirement of certainty of subject matter.

It deserves to be noticed, however, that a certain obscurity exists in regard, to the relative duties of the director of naval construction and the director of dockyards touching constructive works in the yards.