In 1920, Thoralf Skolem simplified a previous result by Leopold Löwenheim, leading to the Löwenheim – Skolem theorem and, in 1930, to the notion of a Herbrand universe and a Herbrand interpretation that allowed ( un ) satisfiability of first-order formulas ( and hence the validity of a theorem ) to be reduced to ( potentially infinitely many ) propositional satisfiability problems.

Gödel's first incompleteness theorem shows that for languages sufficient for doing a certain amount of arithmetic, there can be no effective deductive system that is complete with respect to the intended interpretation of the symbolism of that language.

A theorem whose interpretation is a true statement about a formal system is called a metatheorem.

In deductivism, the Pythagorean theorem is not an absolute truth, but a relative one: if you assign meaning to the strings in such a way that the rules of the game become true ( i. e., true statements are assigned to the axioms and the rules of inference are truth-preserving ), then you have to accept the theorem, or, rather, the interpretation you have given it must be a true statement.

In the Bayesian interpretation, Bayes ' theorem is fundamental to Bayesian statistics, and has applications in fields including science, engineering, economics ( particularly microeconomics ), game theory, medicine and law.

For more on the application of Bayes ' theorem under the Bayesian interpretation of probability, see Bayesian inference.

Bayes ' theorem under this interpretation is most easily visualized using tree diagrams, as shown to the right.

Under the Bayesian interpretation of probability, Bayes ' rule may be thought of as Bayes ' theorem in odds form.

Decades later John Bell proved Bell's theorem ( see p. 14 in Bell ), in which he showed that, if they are to agree with the empirical predictions of quantum mechanics, all such " hidden-variable " completions of quantum mechanics must either be nonlocal ( as the Bohm interpretation is ) or give up the assumption that experiments produce unique results ( see counterfactual definiteness and many-worlds interpretation ).

When a vector field represents force, the line integral of a vector field represents the work done by a force moving along a path, and under this interpretation conservation of energy is exhibited as a special case of the fundamental theorem of calculus.

Graphical interpretation of the theorem.

Laurie Kirby and Jeff Paris gave an interpretation of the Goodstein's theorem as a hydra game: the " Hydra " is a rooted tree, and a move consists of cutting off one of its " heads " ( a branch of the tree ), to which the hydra responds by growing a finite number of new heads according to certain rules.

The Kirby – Paris interpretation of the theorem says that the Hydra will eventually be killed, regardless of the strategy that Hercules uses to chop off its heads, though this may take a very, very long time.

One informal interpretation of the second recursion theorem is that any partial computable function can guess an index for itself.

Moreover, by the envelope theorem the optimal value of a Lagrange multiplier has an interpretation as the marginal effect of the corresponding constraint constant upon the optimal attainable value of the original objective function: if we denote values at the optimum with an asterisk, then it can be shown that

One then usually is looking for the smallest set that has the property of being a fixed point of the function f. Abstract interpretation makes ample use of the Knaster – Tarski theorem and the formulas giving the least and greatest fixpoints.

The theorem can be given a combinatorial interpretation in terms of partitions.

One interpretation of the theorem is that it computes homotopy 1-types.

This is the physical interpretation of the Borel-Weil theorem or the Borel-Weil-Bott theorem.

The interpretation of the theorem changed with the formulation of the theory of Emil Artin, around 1930.

The subjective view, that the wave function is merely a mathematical tool for calculating the probabilities in a specific experiment, has some similarities to the Ensemble interpretation in that it takes probabilities to be the essence of the quantum state, but unlike the ensemble interpretation, it takes these probabilities to be perfectly applicable to single experimental outcomes, as it interprets

This commentary tends to focus on the meaning of the mishnayot themselves, without as much reliance on the Gemara's interpretation and is, therefore, considered valuable as a tool for the study of Mishnah as an independent work.

Thus it can be seen that, according to Taimni's interpretation of the Yoga Sutras, samadhi is the main tool the yogin uses to achieve the end goal of yoga, the joining of the individual self with the Universal Absolute.

This revision was proposed in order “ to update archaic language, to clarify some theological ambiguities, to seek greater theological precision, to address new issues, to have a SOF that would be better suited to be used as a teaching tool in our churches .” Specific beliefs based on biblical interpretation can vary somewhat due to the congregational governance system that gives autonomy to individual local EFCA churches.

These interpretations propose that actual definite properties of a physical system " do not exist " prior to the measurement ; and the wavefunction has a restricted interpretation, as nothing more than a mathematical tool used to calculate the probabilities of experimental outcomes, hence in agreement with positivism in philosophy as the only topic that science should discuss.

The broad interpretation of " tool " coincided with that given by the designer, philosopher, and engineer Buckminster Fuller, though another thinker admired by Brand and some of his cohorts was Lewis Mumford, who had written about words as tools.

The modern Western interpretation of the karambit is far removed from the original agricultural tool.

Proper web log analysis tool configuration and report interpretation requires a bit of technical and business knowledge.

The modeller is free to use encapsulation as a conceptual tool, and it does not necessarily have any physical interpretation.

However, when that agency can be seen as a ' tool ' of the executive, and in turn a ' tool ' of the majority party, the only reasonable alternative is for the interpretation of the legislation to be left in the hands of the courts.

BMEWS intelligence is simultaneously flashed to NORAD ( North American Air Defense Command ) in Colorado Springs, Colorado, for interpretation ; ;

According to the official interpretation of the Charter, a member cannot be penalized by not having the right to vote in the General Assembly for nonpayment of financial obligations to the `` special '' United Nations' budgets, and of course cannot be expelled from the Organization ( which you suggested in your editorial ), due to the fact that there is no provision in the Charter for expulsion.

In view of the acceptance accorded the status of motor vehicles for tax purposes, in the absence of any specific provision it would seem entirely consistent to apply the same interpretation to boats or aircraft.

The purpose set forth at the beginning of this book was first to introduce the reader to a general background knowledge of the various types and capabilities of the forecasting methods already in use, so that he might then be in a position to evaluate for himself the validity of the rather astonishing empirical correlation that is to follow, and to appraise the forecast that its interpretation suggests for the future of farm prices over the years immediately ahead.

But over and beyond the compelling need for a binding precedent decision, I am convinced that the decisions of the Superior Courts which in effect nullify the Secretary's Regulation are not a correct interpretation of the Secretary's power under the Federal law.

This interpretation of the role of residence in the economy of middle-class culture could lead to various projections for the churches.

It will be recalled from the discussion in Section 7 that the position of the `` right '', as represented by Barth, rests on the following thesis: The only tenable alternative to Bultmann's position is a theology that ( 1 ) rejects or at least qualifies his unconditioned demand for demythologization and existential interpretation ; ;

Mr. Nagrin has described four `` places '', each with its scenery and people, added two `` diversions '', and concluded with `` A Toccata for the Young '', a refreshingly underplayed interpretation of rock'n'roll dancing.

His interpretation of the Pauline phrase is that we should seek the common good more than the private good, but this is because the common good is a more desirable good for the individual.

One interpretation of his motive is that it was in revenge for Achilles ' sacrilege in murdering Troilus, the god's own son by Hecuba, on the very altar of the god's own temple.

Others suggest the alphabet was developed in central Egypt during the 15th century BC for or by Semitic workers, but only one of these early writings has been deciphered and their exact nature remains open to interpretation.

Some linguists believe the case for either interpretation is about equally strong ; they have been called the " skeptics ".

This interpretation has since been revised, and the prize has been awarded to, for example, Dario Fo and José Saramago, who do not belong to the camp of literary idealism.

There was room for interpretation by the bodies he had named for deciding on the physical sciences and chemistry prizes, given that he had not consulted them before making the will.

This is sometimes expressed as " everything that is true is provable ", but it must be understood that " true " here means " made true by the set of axioms ", and not, for example, " true in the intended interpretation ".

It is worth noting Ian Woods advice that although Gregory provides the fullest information for this period, where it touches Merovingian affairs, he often " allowed his religious bias to determine his interpretation of the events.

According to this interpretation, Agathon is mocked in the Thesmophoriazousae not only for his notorious effeminacy, but also for the pretentiousness of his dress: " he seems to think of himself, in all his elegant finery, as a rival to the old Ionian poets, perhaps even to Anacreon himself.

The main source for the affair is a mythology à clef by Synesius of Cyrene, Aegyptus sive de providentia, ( 400 ) an Egyptianising allegory that embodies a covert account of the events, the exact interpretation of which continues to baffle scholars.

Therefore the interpretation of what kinds of sexual activity are acceptable for a layman is not a religious matter as far as Buddhism is concerned.

The symbolism of Revelation 9: 11 leaves the exact identification of Abaddon open for interpretation.

Labor arbitration comes in two varieties: interest arbitration, which provides a method for resolving disputes about the terms to be included in a new contract when the parties are unable to agree, and grievance arbitration, which provides a method for resolving disputes over the interpretation and application of a collective bargaining agreement.