Deep Sequent Systems for Modal Logic
... Labelled systems are formulated in a hybrid language which not only contains modalities but also variables and an accessibility relation. There are some concerns about incorporating the semantics into the syntax of a proof system in this way. Avron discusses them in [1], for example. However, even w ...
... Labelled systems are formulated in a hybrid language which not only contains modalities but also variables and an accessibility relation. There are some concerns about incorporating the semantics into the syntax of a proof system in this way. Avron discusses them in [1], for example. However, even w ...
Generalized Cantor Expansions - Rose
... life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is uniq ...
... life. A less common way to represent a number is the so called Cantor expansion. Often presented as exercises in discrete math and computer science courses [8.2, 8.5], this system uses factorials rather than exponentials as the basis for the representation. It can be shown that the expansion is uniq ...
On the equation ap + 2αbp + cp = 0
... reduction at 2. Then the indicated irreducibility follows from a stronger statement which is proved by Diamond and Kramer in [8]: Let I be an inertia subgroup of Gal(Q/Q) for the prime 2; then the action of I on E[l] is irreducible if l ≥ 3. Since the proof of this statement is quite elementary, we ...
... reduction at 2. Then the indicated irreducibility follows from a stronger statement which is proved by Diamond and Kramer in [8]: Let I be an inertia subgroup of Gal(Q/Q) for the prime 2; then the action of I on E[l] is irreducible if l ≥ 3. Since the proof of this statement is quite elementary, we ...
EMBEDDING AN ANALYTIC EQUIVALENCE RELATION IN THE
... space and E0 ⊂ Ω × Ω is a Σ11 equivalence relation, then there are ∆11 equivalence relations E1 and E2 and a ∆11 subset Z of Ω, such that E1 ∪ E2 (Z × Z) is Borel equivalent to E0 . If Ω is the Baire space, then E1 and E2 can be taken to be closed, Z can be taken to be open, and the Borel equivale ...
... space and E0 ⊂ Ω × Ω is a Σ11 equivalence relation, then there are ∆11 equivalence relations E1 and E2 and a ∆11 subset Z of Ω, such that E1 ∪ E2 (Z × Z) is Borel equivalent to E0 . If Ω is the Baire space, then E1 and E2 can be taken to be closed, Z can be taken to be open, and the Borel equivale ...
In this lecture we will start with Number Theory. We will start
... a complete list of primes in p1 , p2 , . . . , pN so we can test everyone of them. But as n gives remainder 1 when we divide with any of the numbers p1 , p2 , . . . , pN , not a single one of them can divide n. But this is a contradiction because each number must be divisible by some prime number. W ...
... a complete list of primes in p1 , p2 , . . . , pN so we can test everyone of them. But as n gives remainder 1 when we divide with any of the numbers p1 , p2 , . . . , pN , not a single one of them can divide n. But this is a contradiction because each number must be divisible by some prime number. W ...
Elementary methods in the study of the distribution of prime numbers
... analysis was purged [Eda, Fog]. And, most telling, the theorem soon took its place alongside the aforementioned theorem of Euclid in many books on number theory, e.g. [Gio, Cha 2, HW, Nag, Sch]. It was now natural to ask, How good an error term in the P.N.T. could be produced by elementary methods? ...
... analysis was purged [Eda, Fog]. And, most telling, the theorem soon took its place alongside the aforementioned theorem of Euclid in many books on number theory, e.g. [Gio, Cha 2, HW, Nag, Sch]. It was now natural to ask, How good an error term in the P.N.T. could be produced by elementary methods? ...
On the chromatic number of the lexicographic product and the
... V (G) onto INn = {1, 2, . . . , n}, such that xy ∈ E(G) implies f (x) 6= f (y). The smallest number n for which an n–coloring exists is the chromatic number χ(G) of G. G is called χ–critical if χ(G − v) < χ(G) for every v ∈ V (G). Every nontrivial graph contains a χ–critical subgraph with the same ...
... V (G) onto INn = {1, 2, . . . , n}, such that xy ∈ E(G) implies f (x) 6= f (y). The smallest number n for which an n–coloring exists is the chromatic number χ(G) of G. G is called χ–critical if χ(G − v) < χ(G) for every v ∈ V (G). Every nontrivial graph contains a χ–critical subgraph with the same ...
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.