Topological aspects of real-valued logic
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
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Large gaps between consecutive prime numbers
... If P and Q were distributed randomly inside the intervals (x/2, x] and (x/4, y] respectively, with cardinalities given by (2.2), then standard probabilistic arguments (using for instance the Chernoff inequality) would suggest that the answer to question (i) is ∼ logyr x , while the answer to questio ...
... If P and Q were distributed randomly inside the intervals (x/2, x] and (x/4, y] respectively, with cardinalities given by (2.2), then standard probabilistic arguments (using for instance the Chernoff inequality) would suggest that the answer to question (i) is ∼ logyr x , while the answer to questio ...
THE ARITHMETIC LARGE SIEVE WITH AN APPLICATION TO THE
... which up to the po (1) factor is the best known result today. Conjecturally, Ankeny showed that GRH gives an even better estimate than Vinogradov conjectured, showing GRH implies np (log p)2 . We will prove a result of Linnik which shows that Vinogradov’s conjecture holds for all but very few prim ...
... which up to the po (1) factor is the best known result today. Conjecturally, Ankeny showed that GRH gives an even better estimate than Vinogradov conjectured, showing GRH implies np (log p)2 . We will prove a result of Linnik which shows that Vinogradov’s conjecture holds for all but very few prim ...
Algorithmic Number Theory
... the number n = i=1 pi + 1. It is easy to see that none of the primes p1 , . . . , pk is a divisor of n and n is larger than any of them. Hence n must be a prime, contradicting the assummption. ...
... the number n = i=1 pi + 1. It is easy to see that none of the primes p1 , . . . , pk is a divisor of n and n is larger than any of them. Hence n must be a prime, contradicting the assummption. ...
Formal deduction in propositional logic
... Comments • In (2) and (3) of the preceding proof, we used a finite subset Σ0 to replace Σ because Σ may be infinite and accordingly not available in (Tr). • Suppose Σ0 = C1 , . . . Cn . Then (3) consists of n steps Σ, ¬¬A ` C1 , ...
... Comments • In (2) and (3) of the preceding proof, we used a finite subset Σ0 to replace Σ because Σ may be infinite and accordingly not available in (Tr). • Suppose Σ0 = C1 , . . . Cn . Then (3) consists of n steps Σ, ¬¬A ` C1 , ...
NUMBER THEORY
... Definition: Let G(k) be the least value of s for which all sufficiently large numbers; i.e. all numbers with at most a finite number of exceptions, are representable by s kth powers. In view of above, G(3) 6 7, it’s known that G(3) > 4 - see below - but it’s not known where G(3) = 4, 5, 6, 7. Exact ...
... Definition: Let G(k) be the least value of s for which all sufficiently large numbers; i.e. all numbers with at most a finite number of exceptions, are representable by s kth powers. In view of above, G(3) 6 7, it’s known that G(3) > 4 - see below - but it’s not known where G(3) = 4, 5, 6, 7. Exact ...
Mathematical proof
In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture.Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, and so-called folk mathematics (in both senses of that term). The philosophy of mathematics is concerned with the role of language and logic in proofs, and mathematics as a language.