The Origin of Proof Theory and its Evolution

... First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithmetic is a first-order theory commonly formalized in ...

... First-order logic has sufficient expressive power for the formalization of virtually all of mathematics. A first-order theory consists of a set of axioms (usually finite or recursively enumerable) and the statements deducible from them. Peano arithmetic is a first-order theory commonly formalized in ...

Relating Infinite Set Theory to Other Branches of Mathematics

... of Goldstein, the other a graph-theoretic algorithm of Kirby and Paris—whose termination proofs depend on showing that the state corresponds to a descending sequence of infinite ordinals. I had never seen this, and would not have thought it possible. Leaving infinite sets for a while, Stillwell turn ...

... of Goldstein, the other a graph-theoretic algorithm of Kirby and Paris—whose termination proofs depend on showing that the state corresponds to a descending sequence of infinite ordinals. I had never seen this, and would not have thought it possible. Leaving infinite sets for a while, Stillwell turn ...

HW 12

... a. Provide a definitional axiom for A \ B (use a 2-place function symbol diff(x,y)) b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A B) \ C 5. Any set without any elements is an empty set a. Provide a definitional axiom that defines a 1-place predicate Empty(x ...

... a. Provide a definitional axiom for A \ B (use a 2-place function symbol diff(x,y)) b. Construct a formal proof that shows that for any sets A, B, and C: A (B \ C) = (A B) \ C 5. Any set without any elements is an empty set a. Provide a definitional axiom that defines a 1-place predicate Empty(x ...

INTEGER PARTITIONS: EXERCISE SHEET 6 (APRIL 27 AND MAY 4)

... Exercise 2: Refinement of Schur’s theorem The goal of this exercise is to prove the following refinement of Schur’s theorem due to Gleissberg. Theorem 1 (Gleissberg). Let C(m, n) denote the number of partitions of n into m distinct parts congruent to 1 or 2 modulo 3. Let D(m, n) denote the number of ...

... Exercise 2: Refinement of Schur’s theorem The goal of this exercise is to prove the following refinement of Schur’s theorem due to Gleissberg. Theorem 1 (Gleissberg). Let C(m, n) denote the number of partitions of n into m distinct parts congruent to 1 or 2 modulo 3. Let D(m, n) denote the number of ...

Math Vocabulary 3-1 - Clinton Public Schools

... factors- Numbers that are multiplied to give a product. Example: 3 x 8 = 24 factor x factor = product product- The answer to a multiplication problem. Example: 3 x 8 = 24 factor x factor = product dividend - The number to be divided. Example: 24/8=3 dividend / divisor = quotient divisor – The number ...

... factors- Numbers that are multiplied to give a product. Example: 3 x 8 = 24 factor x factor = product product- The answer to a multiplication problem. Example: 3 x 8 = 24 factor x factor = product dividend - The number to be divided. Example: 24/8=3 dividend / divisor = quotient divisor – The number ...

IPC-2221 3.4 Parts List

... All mechanical parts appearing on the assembly pictorial shall be assigned an item number which shall match the item number assigned on the parts list. Electrical components, such as capacitors, resistors, fuses, IC's, transistors, etc., shall be assigned reference designators, (Ex. C5, CR2, F1, R15 ...

... All mechanical parts appearing on the assembly pictorial shall be assigned an item number which shall match the item number assigned on the parts list. Electrical components, such as capacitors, resistors, fuses, IC's, transistors, etc., shall be assigned reference designators, (Ex. C5, CR2, F1, R15 ...

Math Vocabulary 3-1 - Clinton Public School District

... 10. numerator – The number above the fraction bar in a fraction; it tells how many equal parts. 11. denominator – The number below the fraction bar in a fraction; it tells the total number of equal parts. 12. unit fraction – A fraction with a numerator of 1. Example: ½ 13. benchmark fractions – Comm ...

... 10. numerator – The number above the fraction bar in a fraction; it tells how many equal parts. 11. denominator – The number below the fraction bar in a fraction; it tells the total number of equal parts. 12. unit fraction – A fraction with a numerator of 1. Example: ½ 13. benchmark fractions – Comm ...

A Short Glossary of Metaphysics

... by enduring. Contrast perdurantism. entity (from Latin ens, object) Object of any kind that exists. Often used more widely than ‘thing’ (Latin res). Entities comprise any object taken to exist, not just individual things, e.g. universals, sets, states of affairs. In Meinongian philosophy, entities a ...

... by enduring. Contrast perdurantism. entity (from Latin ens, object) Object of any kind that exists. Often used more widely than ‘thing’ (Latin res). Entities comprise any object taken to exist, not just individual things, e.g. universals, sets, states of affairs. In Meinongian philosophy, entities a ...

On a Symposium on the Foundations of Mathematics (1971) Paul

... axioms, as was done simultaneously by A.A. Fraenkel and Thoralf Skolem. The systems presented by W.V. Quine mediated, as it were, between type theory and the systems of axiomatic set theory. It turned out, however, by the results of Kurt Gödel and Skolem, that all such strictly formal frameworks fo ...

... axioms, as was done simultaneously by A.A. Fraenkel and Thoralf Skolem. The systems presented by W.V. Quine mediated, as it were, between type theory and the systems of axiomatic set theory. It turned out, however, by the results of Kurt Gödel and Skolem, that all such strictly formal frameworks fo ...

IntroToLogic - Department of Computer Science

... Introduced his formal language for making logical inferences in 1864. His work was entitled An Investigation of the Laws of Thought, on which are founded Mathematical Theories of Logic and Probabilities His system was a precursor to the fully developed propositional logic. ...

... Introduced his formal language for making logical inferences in 1864. His work was entitled An Investigation of the Laws of Thought, on which are founded Mathematical Theories of Logic and Probabilities His system was a precursor to the fully developed propositional logic. ...

COMPOSITION, IDENTITY, AND EMERGENCE

... has been criticized in [Bohn, 2012]3 and more recently in [Sider, 2014]. Using the formal resources of plural logic and extensional mereology Sider argues that CAI conﬂates extensionally identical pluralities. He labels this principle Collapse Principle (CP). CP in turns undermines McDaniel’s argume ...

... has been criticized in [Bohn, 2012]3 and more recently in [Sider, 2014]. Using the formal resources of plural logic and extensional mereology Sider argues that CAI conﬂates extensionally identical pluralities. He labels this principle Collapse Principle (CP). CP in turns undermines McDaniel’s argume ...

HOW TO DEFINE A MEREOLOGICAL (COLLECTIVE) SET

... the so-called Russell’s paradox this is very prone to inconsistencies and must be limited in one way or another. The set theoretical axiom of comprehension allows us to construct sets this way but only within sets whose existence were already proved from other axioms. In contemporary mathematics Fre ...

... the so-called Russell’s paradox this is very prone to inconsistencies and must be limited in one way or another. The set theoretical axiom of comprehension allows us to construct sets this way but only within sets whose existence were already proved from other axioms. In contemporary mathematics Fre ...

Multiply Fractions

... Marie and Matt are sharing a large Pizza. Marie cuts the pizza in half. Matt knows he can’t eat his half of the pizza, so he cuts it into fourths and eats one piece. What is the fraction of the whole pizza that Matt did not eat? Write and equation and solve for this question: ...

... Marie and Matt are sharing a large Pizza. Marie cuts the pizza in half. Matt knows he can’t eat his half of the pizza, so he cuts it into fourths and eats one piece. What is the fraction of the whole pizza that Matt did not eat? Write and equation and solve for this question: ...

Sets with dependent elements: Elaborating on Castoriadis` notion of

... boolean algebra of sets, i.e., they are inseparable. Perhaps more interesting is the case where the relation / is not symmetric, i.e., when we drop existence of complements. In this case one can ...

... boolean algebra of sets, i.e., they are inseparable. Perhaps more interesting is the case where the relation / is not symmetric, i.e., when we drop existence of complements. In this case one can ...

On the paradoxes of set theory

... Definition 1.—— A set M is called ordered if there exists a rule which tells us that for each two distinct elements in M which one precedes the other. Definition 2,~.— A~ ordered set is said to be well-ordered if every non-empty subset has a first element. Definition 3.——The “ordinal type” is a symb ...

... Definition 1.—— A set M is called ordered if there exists a rule which tells us that for each two distinct elements in M which one precedes the other. Definition 2,~.— A~ ordered set is said to be well-ordered if every non-empty subset has a first element. Definition 3.——The “ordinal type” is a symb ...

The Anti-Foundation Axiom in Constructive Set Theories

... The paper investigates the strength of the anti-foundation axiom on the basis of various systems of constructive set theories. ...

... The paper investigates the strength of the anti-foundation axiom on the basis of various systems of constructive set theories. ...

Implementable Set Theory and Consistency of ZFC

... being. The reason is that logic and predicates have not been done yet as an implementable theory, at least not by this author. It is noted, though, that if theorems are true in an implementable form of mathematics, then they are also true in a constructive mathematical sense. If the reverse is also ...

... being. The reason is that logic and predicates have not been done yet as an implementable theory, at least not by this author. It is noted, though, that if theorems are true in an implementable form of mathematics, then they are also true in a constructive mathematical sense. If the reverse is also ...

MATH 310 CLASS NOTES 1: AXIOMS OF SET THEORY Intuitively

... The barber shaves all those who do not shave themselves. The question is: What about the barber himself? If he shaves himself then he does not shave himself. But if he does not shave himself he shaves himself. To overcome Russell’s paradox, an axiomatic set theory was proposed in the 1920s, known as ...

... The barber shaves all those who do not shave themselves. The question is: What about the barber himself? If he shaves himself then he does not shave himself. But if he does not shave himself he shaves himself. To overcome Russell’s paradox, an axiomatic set theory was proposed in the 1920s, known as ...

Logic and Categories As Tools For Building Theories

... ‘pairing’ A C - B offers a decomposition of C into components in A and B, at the level of arrows rather than elements. The fact that pairs are uniquely determined by their components is expressed in arrow-theoretic terms by the universal property of the product; the fact that for every candidate p ...

... ‘pairing’ A C - B offers a decomposition of C into components in A and B, at the level of arrows rather than elements. The fact that pairs are uniquely determined by their components is expressed in arrow-theoretic terms by the universal property of the product; the fact that for every candidate p ...

1. Sets, relations and functions. 1.1. Set theory. We assume the

... is the set A characterized by the property that x ∈ A if and only if x = ai for some i = 1, . . . , n. In particular, for any object a we let singleton a equal {a} and note that x is a member of singleton a if and only if x = a. We let ∅ = {x : x 6= x} and call this set the empty set because it has ...

... is the set A characterized by the property that x ∈ A if and only if x = ai for some i = 1, . . . , n. In particular, for any object a we let singleton a equal {a} and note that x is a member of singleton a if and only if x = a. We let ∅ = {x : x 6= x} and call this set the empty set because it has ...

(pdf)

... a context, or an environment. We can bind a variable within the syntax of the λcalculus itself by prefixing λx to a term with x free, but we could also bind variables on the metalinguistic level by specifying a context containing bindings for the free variables and interpreting our term in that cont ...

... a context, or an environment. We can bind a variable within the syntax of the λcalculus itself by prefixing λx to a term with x free, but we could also bind variables on the metalinguistic level by specifying a context containing bindings for the free variables and interpreting our term in that cont ...

Canad. Math. Bull. Vol. 24 (2), 1981 INDEPENDENT SETS OF

... There are three possibilities for M: (i) M

... There are three possibilities for M: (i) M

1. Sets, relations and functions. 1.1 Set theory. We assume the

... (ii.) If A is a partition of X and no member of A is empty then r = ∪{A × A : A ∈ A} is an equivalence relation on X and X/r = A. Proof. We leave this as Exercise 1.4. for the reader. Definition. Suppose X is a set, r is a relation on X and A ⊂ X. We say a member u of X is an upper bound for A if (a ...

... (ii.) If A is a partition of X and no member of A is empty then r = ∪{A × A : A ∈ A} is an equivalence relation on X and X/r = A. Proof. We leave this as Exercise 1.4. for the reader. Definition. Suppose X is a set, r is a relation on X and A ⊂ X. We say a member u of X is an upper bound for A if (a ...

Russell`s logicism

... A first step in seeing how Russell aimed to show that mathematical truths are disguised versions of logical ones is to see what he thought numbers to be. Russell thinks that the first step is getting clear on the ‘grammar’ of numbers: “Many philosophers, when attempting to define number, are really ...

... A first step in seeing how Russell aimed to show that mathematical truths are disguised versions of logical ones is to see what he thought numbers to be. Russell thinks that the first step is getting clear on the ‘grammar’ of numbers: “Many philosophers, when attempting to define number, are really ...