[permalink] [id link]
A consequence of this axiom is that Alfred Tarski's proof of the decidability of the ordered real field could be seen as an algorithm to solve any problem in Euclidean geometry.
from
Wikipedia
Some Related Sentences
consequence and axiom
Indeed, their existence is a non-trivial consequence of the axiom of choice.
To avoid this consequence, we can modify the axiom of extensionality to apply only to nonempty sets, so that it reads:
Because is usually a theorem or axiom, a consequence is that the negation of false () is true.
One such consequence is the existence of uncountable models of true arithmetic, which satisfy every first-order induction axiom but have non-inductive subsets.
Indeed, it can be argued that self-reference is a necessary consequence of Aristotle's Law of non-contradiction, a fundamental philosophical axiom.
This theorem, in turn, can be used to prove that Zorn's lemma is a consequence of the axiom of choice.
" The transitivity of congruence is an easy consequence of this axiom and Reflexivity.
consequence and is
It is the consequence of the system of ideas that constitutes the frame of our international -- and in some degree our domestic -- policy.
One consequence is the occurrence of occasional conflicts because private owners of some inholdings object to public programs of use on neighboring National Forest or other Federal land, or because such ownerships are developed for uses that are not compatible with use for the public of neighboring National Forest land.
In the event the total of rupees accruing to the Government of the United States of America as a consequence of sales made pursuant to this Agreement is different from the rupee equivalent of $1,276 million, the amounts available for the purposes specified in paragraph 1, Article 2, will be adjusted proportionately.
According to the theory proposed, this is a consequence of the severe condition of perceived threat that persists unabated for the anxious child in an ambiguous sort of school environment.
) Functionalism as a sociological credo is, therefore, not a direct consequence of observations, but rather an indirect consequence of philosophical inference and judgment.
The consequence of this is that the girls at Brooklyn College outnumber the boys and do somewhat better academically.
This last point is important because if high school pupils are aware that few, if any, graduates who have chosen a certain vocational program have obtained a job as a consequence of the training, the whole idea of relevance disappears.
Existence is created and willed by God and is not the consequence of a pre-existent rebellion or of a cosmic descent from eternity into history.
Nevertheless, the consequence of the policy proposed is everywhere subtly qualified: it is `` a possible result, however improbable '' ; ;
:: One consequence is that people are more cooperative if it is more likely that individuals will interact again in the future.
A simple illustration of such cause and effect is the case of experiencing the effects of what I cause: if I cause suffering, then as a natural consequence I will experience suffering ; if I cause happiness, then as a natural consequence I will experience happiness.
As a consequence, it is difficult to define concisely or precisely.
Each element has a specific set of chemical properties as a consequence of the number of electrons present in the neutral atom, which is Z ( the atomic number ).
A consequence of using waveforms to describe particles is that it is mathematically impossible to obtain precise values for both the position and momentum of a particle at the same time ; this became known as the uncertainty principle, formulated by Werner Heisenberg in 1926.
that is, for any statement that is a logical consequence of there actually exists a deduction of the statement from.
One consequence of the Hindu and Spiritist beliefs is that our current lives are both afterlife and a beforelife.
The outer surface of an agate, freed from its matrix, is often pitted and rough, apparently in consequence of the removal of the original coating.
As a consequence, the balance of anti-apoptotic and proapoptotic effectors is upset in favour of the former, and the damaged cells continue to replicate despite being directed to die.
consequence and Alfred
In the introduction Sagan presents his thesis — that " the mind ... a consequence of its anatomy and physiology and nothing more "— in reference to the works of Charles Darwin and Alfred Russel Wallace.
After the second court-martial-held as a consequence of the conclusions of the Supreme Court-Picquart resigned from the army but the exoneration of Alfred Dreyfus in 1906 also absolved Picquart, who was, by an act of the Chamber of Deputies, promoted to brigadier-general.
consequence and Tarski's
* Let us use the term “ deductive system ” as a set of sentences closed under consequence ( for defining notion of consequence, let us use e. g. Tarski's algebraic approach ).
consequence and proof
where is a metalogical symbol meaning that is a syntactic consequence when and are both in a proof ;
where is a metalogical symbol meaning that is a syntactic consequence if and are each on lines of a proof in some logical system ;
If some specific deductive system of first-order logic is sound and complete, then is it " perfect " ( a formula is provable iff it is a semantic consequence of the axioms ), thus equivalent to any other deductive system with the same quality ( any proof in one system can be converted into the other ).
It says that for any first-order theory T with a well-orderable language, and any sentence S in the language of the theory, there is a formal proof of S in T if and only if S is satisfied by every model of T ( S is a semantic consequence of T ).
It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.
The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions.
One can see from the proof that the Cauchy – Schwarz inequality is a consequence of the positivity and anti-symmetry inner-product axioms.
* Emperor Augustus is proclaimed Pater Patriae, or " father of the country " by the Roman Senate ; this bestowed title is the logical consequence and final proof of Augustus ' supreme position as princeps, the first in charge over the Roman state.
Actually, the correctness of the Bieberbach conjecture was only the most important consequence of de Branges's proof, which covers a more general problem, the Milin conjecture.
For example, as a consequence of this the form of proof known as reductio ad absurdum is suspect.
An even more elegant proof is a consequence of the law about total differentials, which states that the total differential,
The theorem is a syntactic consequence of all the well formed formulae preceding it in the proof.
A formula A is a syntactic consequence within some formal system of a set Г of formulas if there is a formal proof in formal system of A from the set Г.
The consequence ((( A ) B ) C ) = ( AC )(( B ) C ), C7 in LoF, enables an algorithm, sketched in LoFs proof of T14, that transforms an arbitrary pa formula to an equivalent formula whose depth does not exceed two.
A more elementary proof of Ribet's converse to Herbrand's theorem, a consequence of the theory of Euler systems, can be found in Washington's book.
The theorem is a syntactic consequence of all the well-formed formulas preceding it in the proof.
A consequence is that the Generalized Poincaré conjecture is true in PL for dimensions greater than four – the proof is to take a homotopy sphere, remove two balls, apply the h-cobordism theorem to conclude that this is a cylinder, and then attach cones to recover a sphere.
This second statement is a consequence of the first ( see proof below ).
1.016 seconds.