Completeness Theorem for Continuous Functions and Product
... Of course, there are many concrete interpretations of Q (Q∞ “there exist infinitely many"”, Qℵ1 “there exists uncountable many” [10], Q>r “to have probability at least r” [9, 12], Qopen “to be open” [14], etc.). Combining a consistency property argument and the Henkin construction yields an Lω1 ω (Q) ...
... Of course, there are many concrete interpretations of Q (Q∞ “there exist infinitely many"”, Qℵ1 “there exists uncountable many” [10], Q>r “to have probability at least r” [9, 12], Qopen “to be open” [14], etc.). Combining a consistency property argument and the Henkin construction yields an Lω1 ω (Q) ...
Proof Theory - Andrew.cmu.edu
... I will assume the reader is familiar with the language of first-order logic. Contemporary logic textbooks often present formal calculi for first-order logic with a long list of axioms and a few simple rules, but these are generally not very convenient for modeling deductive arguments or studying the ...
... I will assume the reader is familiar with the language of first-order logic. Contemporary logic textbooks often present formal calculi for first-order logic with a long list of axioms and a few simple rules, but these are generally not very convenient for modeling deductive arguments or studying the ...
CONGRUENCE PROPERTIES OF VALUES OF L
... Theorem 1 together with Kolyvagin’s theorem implies: Corollary 1. If E/Q is a modular elliptic curve, then the number of |D| ≤ X for which ED has rank zero is ÀE X/ log X. While Theorem 1 is very strong in that it applies to a general F , it falls short of the ‘positive proportion’ estimates predict ...
... Theorem 1 together with Kolyvagin’s theorem implies: Corollary 1. If E/Q is a modular elliptic curve, then the number of |D| ≤ X for which ED has rank zero is ÀE X/ log X. While Theorem 1 is very strong in that it applies to a general F , it falls short of the ‘positive proportion’ estimates predict ...
On Elkan`s theorems: Clarifying their meaning
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
CONDITIONAL INDEPENDENCE 1. Introduction
... A collection of random ï-variables is called mutually symmetric if each of them, X, is symmetrically distributed around 0 on each set S taken from the field Vx °f s e t s m t n e probability space defined by the other random variables, i.e. for each such X and S, \SX is symmetric around 0. A collect ...
... A collection of random ï-variables is called mutually symmetric if each of them, X, is symmetrically distributed around 0 on each set S taken from the field Vx °f s e t s m t n e probability space defined by the other random variables, i.e. for each such X and S, \SX is symmetric around 0. A collect ...
A Primer on Mathematical Proof
... smaller claims, each followed by its own proof. Mathematicians often call minor or preliminary statements Propositions or Lemmas, major results are Theorems, and consequences of the main result are Corollaries. (There is no formal distinction between the terms proposition, lemma, theorem, and coroll ...
... smaller claims, each followed by its own proof. Mathematicians often call minor or preliminary statements Propositions or Lemmas, major results are Theorems, and consequences of the main result are Corollaries. (There is no formal distinction between the terms proposition, lemma, theorem, and coroll ...
L11
... because of the definition of . ● Since P Q R is false for 14 entries out of 16, we are left only with two entries to be tested for which is true. ...
... because of the definition of . ● Since P Q R is false for 14 entries out of 16, we are left only with two entries to be tested for which is true. ...
Theorem
In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems—and generally accepted statements, such as axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called hypotheses or premises. In light of the interpretation of proof as justification of truth, the conclusion is often viewed as a necessary consequence of the hypotheses, namely, that the conclusion is true in case the hypotheses are true, without any further assumptions. However, the conditional could be interpreted differently in certain deductive systems, depending on the meanings assigned to the derivation rules and the conditional symbol.Although they can be written in a completely symbolic form, for example, within the propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being ""trivial"", or ""difficult"", or ""deep"", or even ""beautiful"". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem.