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Lettris is a curious tetris-clone game where all the bricks have the same square shape but different content. Each square carries a letter. To make squares disappear and save space for other squares you have to assemble English words (left, right, up, down) from the falling squares.
Boggle gives you 3 minutes to find as many words (3 letters or more) as you can in a grid of 16 letters. You can also try the grid of 16 letters. Letters must be adjacent and longer words score better. See if you can get into the grid Hall of Fame !
Change the target language to find translations.
Tips: browse the semantic fields (see From ideas to words) in two languages to learn more.
1.a harmonious uniformity or agreement among things or parts
2.(logic) an attribute of a logical system that is so constituted that none of the propositions deducible from the axioms contradict one another
3.the property of holding together and retaining its shape"wool has more body than rayon" "when the dough has enough consistency it is ready to bake"
4.logical coherence and accordance with the facts"a rambling argument that lacked any consistency"
Bound consistency • Cache consistency • Causal consistency • Consistency (database systems) • Consistency (disambiguation) • Consistency (knowledge bases) • Consistency (negotiation) • Consistency (statistics) • Consistency Theory • Consistency criterion • Consistency model • Convention of consistency • Data consistency • Delta consistency • Directional consistency • Eventual consistency • Fork consistency • Gentzen consistency proof • Gentzen's consistency proof • Hyper-arc consistency • Internal consistency • Internal consistency of the Bible • Novikov self-consistency principle • Omega consistency • PRAM consistency • Path consistency • Photo-consistency • Recovery Consistency Objective • Relational arc consistency • Relational consistency • Relational path consistency • Release consistency • Robinson's joint consistency theorem • Self-consistency • Self-consistency principle • Sequential consistency • Size consistency • Time consistency • Vector-field consistency • Weak consistency
union; connexion; junction[Classe]
propriété de ce qui constitue une unité (fr)[ClasseParExt.]
connectedness, connection, link[Hyper.]
In logic, a consistent theory is one that does not contain a contradiction. The lack of contradiction can be defined in either semantic or syntactic terms. The semantic definition states that a theory is consistent if and only if it has a model, i.e. there exists an interpretation under which all formulas in the theory are true. This is the sense used in traditional Aristotelian logic, although in contemporary mathematical logic the term satisfiable is used instead. The syntactic definition states that a theory is consistent if and only if there is no formula P such that both P and its negation are provable from the axioms of the theory under its associated deductive system.
If these semantic and syntactic definitions are equivalent for a particular logic, the logic is complete.[clarification needed] The completeness of sentential calculus was proved by Paul Bernays in 1918 and Emil Post in 1921, while the completeness of predicate calculus was proved by Kurt Gödel in 1930, and consistency proofs for arithmetics restricted with respect to the induction axiom schema were proved by Ackermann (1924), von Neumann (1927) and Herbrand (1931). Stronger logics, such as second-order logic, are not complete.
A consistency proof is a mathematical proof that a particular theory is consistent. The early development of mathematical proof theory was driven by the desire to provide finitary consistency proofs for all of mathematics as part of Hilbert's program. Hilbert's program was strongly impacted by incompleteness theorems, which showed that sufficiently strong proof theories cannot prove their own consistency (provided that they are in fact consistent).
Although consistency can be proved by means of model theory, it is often done in a purely syntactical way, without any need to reference some model of the logic. The cut-elimination (or equivalently the normalization of the underlying calculus if there is one) implies the consistency of the calculus: since there is obviously no cut-free proof of falsity, there is no contradiction in general.
In theories of arithmetic, such as Peano arithmetic, there is an intricate relationship between the consistency of the theory and its completeness. A theory is complete if, for every formula φ in its language, at least one of φ or ¬ φ is a logical consequence of the theory.
Presburger arithmetic is an axiom system for the natural numbers under addition. It is both consistent and complete.
Gödel's incompleteness theorems show that any sufficiently strong effective theory of arithmetic cannot be both complete and consistent. Gödel's theorem applies to the theories of Peano arithmetic (PA) and Primitive recursive arithmetic (PRA), but not to Presburger arithmetic.
Moreover, Gödel's second incompleteness theorem shows that the consistency of sufficiently strong effective theories of arithmetic can be tested in a particular way. Such a theory is consistent if and only if it does not prove a particular sentence, called the Gödel sentence of the theory, which is a formalized statement of the claim that the theory is indeed consistent. Thus the consistency of a sufficiently strong, effective, consistent theory of arithmetic can never be proven in that system itself. The same result is true for effective theories that can describe a strong enough fragment of arithmetic – including set theories such as Zermelo–Fraenkel set theory. These set theories cannot prove their own Gödel sentences – provided that they are consistent, which is generally believed.
is said to be absolutely consistent or Post consistent if and only if at least one formula of is not a theorem of .
is said to be maximally consistent if and only if for every formula , if Con () then .
is said to contain witnesses if and only if for every formula of the form there exists a term such that . See First-order logic.
Let be a maximally consistent set of -formulas containing witnesses.
Define a binary relation on the set of -terms such that if and only if ; and let denote the equivalence class of terms containing ; and let where is the set of terms based on the symbol set .
Define the -structure over the term-structure corresponding to by:
Let be the term interpretation associated with , where .
For all , if and only if .
There are several things to verify. First, that is an equivalence relation. Then, it needs to be verified that (1), (2), and (3) are well defined. This falls out of the fact that is an equivalence relation and also requires a proof that (1) and (2) are independent of the choice of class representatives. Finally, can be verified by induction on formulas.
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