Elliptic Curves and The Congruent Number Problem
... This paper will discuss progress that mathematicians have made towards solving the congruent number problem, including algorithms currently believed to solve the problem. While this unsolved problem is easy to state, understanding its solution likely requires an understanding of complex machinery an ...
... This paper will discuss progress that mathematicians have made towards solving the congruent number problem, including algorithms currently believed to solve the problem. While this unsolved problem is easy to state, understanding its solution likely requires an understanding of complex machinery an ...
M-rank and meager groups
... In this paper we prove that meager types have some of the properties of properly weakly minimal types. This sheds a new light also on the proofs of these properties in the weakly minimal case. Suppose p is a meager type. Since p is non-trivial, we get a clp -triangle. Given a clp -triangle {a0 , a1 ...
... In this paper we prove that meager types have some of the properties of properly weakly minimal types. This sheds a new light also on the proofs of these properties in the weakly minimal case. Suppose p is a meager type. Since p is non-trivial, we get a clp -triangle. Given a clp -triangle {a0 , a1 ...
Those Incredible Greeks! - The Saga of Mathematics: A Brief History
... This implies that b² is even and therefore so is b. Where are we then? It seems that both a and b are even. But didn’t we say that the fraction a/b was reduced to its lowest terms. This is impossible by Basic Fact #3 and we have obtained a contradiction. Thus, the original assumption – that it was r ...
... This implies that b² is even and therefore so is b. Where are we then? It seems that both a and b are even. But didn’t we say that the fraction a/b was reduced to its lowest terms. This is impossible by Basic Fact #3 and we have obtained a contradiction. Thus, the original assumption – that it was r ...
On perturbations of continuous structures - HAL
... There is another hurdle though, lying in the way we treat unbounded structures such as Banach spaces in continuous logic. The method of splitting such a structure into a many-sorted structure, each sort consisting of a bounded ball, as suggested in [BU], is unsatisfactory when it is the norm (and th ...
... There is another hurdle though, lying in the way we treat unbounded structures such as Banach spaces in continuous logic. The method of splitting such a structure into a many-sorted structure, each sort consisting of a bounded ball, as suggested in [BU], is unsatisfactory when it is the norm (and th ...
Reading
... THEOREM 1.9: For any abstraction principle AE, formula Φ, and cardinal κ, AE∇Φ is κ-satisfiable if and only if either AE is κ-satisfiable or R(Φ) is true on models of cardinality κ. PROOF: Straightforward, left to the reader. COROLLARY 1.10: For any formula Φ and cardinal κ, BLV∇Φ is κ-satisfiable ...
... THEOREM 1.9: For any abstraction principle AE, formula Φ, and cardinal κ, AE∇Φ is κ-satisfiable if and only if either AE is κ-satisfiable or R(Φ) is true on models of cardinality κ. PROOF: Straightforward, left to the reader. COROLLARY 1.10: For any formula Φ and cardinal κ, BLV∇Φ is κ-satisfiable ...
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF
... underlies the well-known translation from propositional modal logic with operators ◊ and , possibly indexed, into the first-order language over possible worlds models (van Benthem 1976, 1984). In this way, modal formalisms correspond to fragments of a full first-order (or sometimes higher-order) lan ...
... underlies the well-known translation from propositional modal logic with operators ◊ and , possibly indexed, into the first-order language over possible worlds models (van Benthem 1976, 1984). In this way, modal formalisms correspond to fragments of a full first-order (or sometimes higher-order) lan ...
Algebraic logic, I. Monadic boolean algebras
... that space, every open set is closed, or, equivalently, every closed set is open. (The proof is an easy application of Lemma 5.) In such a space the relation R, defined by writing x R y whenever x belongs to the closure of the one-point set {y}, is an equivalence whose associated quotient space is d ...
... that space, every open set is closed, or, equivalently, every closed set is open. (The proof is an easy application of Lemma 5.) In such a space the relation R, defined by writing x R y whenever x belongs to the closure of the one-point set {y}, is an equivalence whose associated quotient space is d ...
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.