PDF - UNT Digital Library
... I wish to thank Dr. F.J. Navarro-Bermudez and Dr. Helge Tverberg for generously allowing me to make use of some of ...
... I wish to thank Dr. F.J. Navarro-Bermudez and Dr. Helge Tverberg for generously allowing me to make use of some of ...
Proof theory for modal logic
... deduction theorem. The latter states that if B is derivable from A and some other assumptions Γ, then A ⊃ B is derivable from the assumptions Γ alone. This roughly amounts to the fact that implication internalizes in the object language the notion of derivability. The rule of necessitation then requ ...
... deduction theorem. The latter states that if B is derivable from A and some other assumptions Γ, then A ⊃ B is derivable from the assumptions Γ alone. This roughly amounts to the fact that implication internalizes in the object language the notion of derivability. The rule of necessitation then requ ...
2. Ideals in Quadratic Number Fields
... If M and N are R-modules, then so is M ⊕ N = {(m, n) : m ∈ M, n ∈ N } via the action r(m, n) = (rm,√rn). In the following, K = Q( d ) is a quadratic number field, and {1, ω} is a basis of its ring of integers OK . Our first job is the classification of all Z-modules in OK . Proposition 2.7. Let M ⊂ ...
... If M and N are R-modules, then so is M ⊕ N = {(m, n) : m ∈ M, n ∈ N } via the action r(m, n) = (rm,√rn). In the following, K = Q( d ) is a quadratic number field, and {1, ω} is a basis of its ring of integers OK . Our first job is the classification of all Z-modules in OK . Proposition 2.7. Let M ⊂ ...
CHAP05 Distribution of Primes
... the smaller prime in such a representation but watch when we continue up to 100. ...
... the smaller prime in such a representation but watch when we continue up to 100. ...
Hilbert`s Program Then and Now - Philsci
... the foundations of mathematics in 1917. In September 1917, he delivered an address to the Swiss Mathematical Society entitled “Axiomatic Thought” [1918a]. It is his first published contribution to mathematical foundations since 1905. In it, he again emphasized the requirement of consistency proofs f ...
... the foundations of mathematics in 1917. In September 1917, he delivered an address to the Swiss Mathematical Society entitled “Axiomatic Thought” [1918a]. It is his first published contribution to mathematical foundations since 1905. In it, he again emphasized the requirement of consistency proofs f ...
Somewhat More than Governors Need to Know about Trigonometry1
... This is an example of the famous “casus irreducibilis” of the cubic formula, and it is well known that one cannot avoid using complex numbers in writing down these roots (see e.g., [2, §14.7] or keep reading this article). Since cos(20◦ ) is so ugly, we shouldn’t include it in our table. In fact, th ...
... This is an example of the famous “casus irreducibilis” of the cubic formula, and it is well known that one cannot avoid using complex numbers in writing down these roots (see e.g., [2, §14.7] or keep reading this article). Since cos(20◦ ) is so ugly, we shouldn’t include it in our table. In fact, th ...
EppDm4_04_02
... Starting Point: Suppose r and s are particular but arbitrarily chosen real numbers such that r and s are rational; or, more simply, Suppose r and s are rational numbers. Then ask yourself, “What must I show to complete the proof?” To Show: r + s is rational. ...
... Starting Point: Suppose r and s are particular but arbitrarily chosen real numbers such that r and s are rational; or, more simply, Suppose r and s are rational numbers. Then ask yourself, “What must I show to complete the proof?” To Show: r + s is rational. ...
a PDF file of the textbook - U of L Class Index
... The first of these deductions is very famous, but the second one is lame. It may seem odd to even call it a deduction, because the two hypotheses have nothing at all to do with the conclusion, but, given our definition, it does count as a deduction. However, it is is a very poor one, so it cannot be ...
... The first of these deductions is very famous, but the second one is lame. It may seem odd to even call it a deduction, because the two hypotheses have nothing at all to do with the conclusion, but, given our definition, it does count as a deduction. However, it is is a very poor one, so it cannot be ...
Many-Valued Models
... interesting general problem and has received attention from several different areas. In this tutorial we present an elementary but general approach on small finite models, showing their relevance and reviewing some elementary methods and techniques on their uses. There are many significant names in ...
... interesting general problem and has received attention from several different areas. In this tutorial we present an elementary but general approach on small finite models, showing their relevance and reviewing some elementary methods and techniques on their uses. There are many significant names in ...
The Development of Mathematical Logic from Russell to Tarski
... derived theorems from the basic propositions (definitions and axioms) but also to isolate the primitive notions, from which all the others are defined. An ideal to strive for is that of a system whose primitive ideas are irreducible, i.e., such that none of the primitive ideas can be defined by mean ...
... derived theorems from the basic propositions (definitions and axioms) but also to isolate the primitive notions, from which all the others are defined. An ideal to strive for is that of a system whose primitive ideas are irreducible, i.e., such that none of the primitive ideas can be defined by mean ...
Proofs in theories
... many-sorted Predicate logic—the notions of sort, language, term, proposition—, the notions of free and bound variables, alphabetic equivalence and substitution, and the basis of computability theory—the expression of computable functions as in arithmetic, in the language of rewrite rules and in the ...
... many-sorted Predicate logic—the notions of sort, language, term, proposition—, the notions of free and bound variables, alphabetic equivalence and substitution, and the basis of computability theory—the expression of computable functions as in arithmetic, in the language of rewrite rules and in the ...
Section 2.2: Introduction to the Logic of Quantified Statements
... If P (x) is false for all x ∈ D, then the universal conditional statement is true. In such circumstances, we say it is true by default or vacuously true. ...
... If P (x) is false for all x ∈ D, then the universal conditional statement is true. In such circumstances, we say it is true by default or vacuously true. ...
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.