PROOFS BY INDUCTION AND CONTRADICTION, AND WELL
... PROOFS BY INDUCTION AND CONTRADICTION, AND WELL-ORDERING OF N ...
... PROOFS BY INDUCTION AND CONTRADICTION, AND WELL-ORDERING OF N ...
5_1 Math 1 Notes Fall 2010
... The Associative Property for the set of Natural numbers under addition states that… For any natural numbers A, B, and C it is always true that (A + B) + C yields the same sum as A + (B + C). Or, (A + B) + C = A + (B + C) In other words, ...
... The Associative Property for the set of Natural numbers under addition states that… For any natural numbers A, B, and C it is always true that (A + B) + C yields the same sum as A + (B + C). Or, (A + B) + C = A + (B + C) In other words, ...
Compactness Theorem for First-Order Logic
... Let G be any set of formulas of first-order logic. Then G is satisfiable if every finite subset of G is satisfiable. ...
... Let G be any set of formulas of first-order logic. Then G is satisfiable if every finite subset of G is satisfiable. ...
PDF
... Call a wff of FO(Σ) quasi-atom if it is either atomic, or of the form ∀xA, where A is a wff of FO(Σ). Let Γ be the set of all quasi-atoms of FO(Σ). Proposition 1. Every wff of FO(Σ) can be uniquely built up from Γ using only logical connectives → and ¬. Proof. Induction on the complexity of wff. For ...
... Call a wff of FO(Σ) quasi-atom if it is either atomic, or of the form ∀xA, where A is a wff of FO(Σ). Let Γ be the set of all quasi-atoms of FO(Σ). Proposition 1. Every wff of FO(Σ) can be uniquely built up from Γ using only logical connectives → and ¬. Proof. Induction on the complexity of wff. For ...
REVERSE MATHEMATICS Contents 1. Introduction 1 2. Second
... 3. Arithmetical Formulas For each subsystem, induction is restricted to a certain level of arithmetical formula; there are limitations to what ϕ(n) can be used in the induction scheme. Different types of arithmetical formula are denoted by Σ0n , Π0n , and ∆0n . The 0 superscript indicates that quant ...
... 3. Arithmetical Formulas For each subsystem, induction is restricted to a certain level of arithmetical formula; there are limitations to what ϕ(n) can be used in the induction scheme. Different types of arithmetical formula are denoted by Σ0n , Π0n , and ∆0n . The 0 superscript indicates that quant ...
Solutions to assignment 6 File
... smallest positive integer which is divisible by both m and n. (a) Prove that gcd(m, n) · lcm(m, n) = mn. (b) Prove that gcd(m, n) ≤ lcm(m, n), with equality if and only if m = n. With an unfamiliar question, it is a good idea to do an example first to check what goes on. If m = 12 and n = 18, then g ...
... smallest positive integer which is divisible by both m and n. (a) Prove that gcd(m, n) · lcm(m, n) = mn. (b) Prove that gcd(m, n) ≤ lcm(m, n), with equality if and only if m = n. With an unfamiliar question, it is a good idea to do an example first to check what goes on. If m = 12 and n = 18, then g ...
Assignment 2: Proofs
... (A3) There is no line that contains every point. (A4) Any two lines intersect in at least one point. Prove that: (a) Any two lines intersect in exactly one point. (b) There is at least one point. ...
... (A3) There is no line that contains every point. (A4) Any two lines intersect in at least one point. Prove that: (a) Any two lines intersect in exactly one point. (b) There is at least one point. ...
Lecture 10: A Digression on Absoluteness
... non-well-founded models! Note, however, that any infinitely descending chain in such a model is not represented by an element in the universe. Consider also the naturals with addition, multiplication, successor, and zero, ordered by <, which is a well-founded relation. The preceding theorem shows th ...
... non-well-founded models! Note, however, that any infinitely descending chain in such a model is not represented by an element in the universe. Consider also the naturals with addition, multiplication, successor, and zero, ordered by <, which is a well-founded relation. The preceding theorem shows th ...
An Example of Induction: Fibonacci Numbers
... This short document is an example of an induction proof. Our goal is to rigorously prove something we observed experimentally in class, that every fifth Fibonacci number is a multiple of 5. As usual in mathematics, we have to start by carefully defining the objects we are studying. Definition. The s ...
... This short document is an example of an induction proof. Our goal is to rigorously prove something we observed experimentally in class, that every fifth Fibonacci number is a multiple of 5. As usual in mathematics, we have to start by carefully defining the objects we are studying. Definition. The s ...
If T is a consistent theory in the language of arithmetic, we say a set
... This is, in fact, just the formal calculation exhibited in section 6.1. Obviously this method is perfectly general, and whenever a + b = c we can prove a + b = c. Then also, again as in section 6.1, the recursion equations (Q5) and (Q6) for multiplication can be used to prove 2 · 3 = 6 and more gene ...
... This is, in fact, just the formal calculation exhibited in section 6.1. Obviously this method is perfectly general, and whenever a + b = c we can prove a + b = c. Then also, again as in section 6.1, the recursion equations (Q5) and (Q6) for multiplication can be used to prove 2 · 3 = 6 and more gene ...
natural numbers
... •There is a natural number 0. •Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1. •There is no natural number whose successor is 0. •Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b). •If a property is possessed by 0 and also ...
... •There is a natural number 0. •Every natural number a has a natural number successor, denoted by S(a). Intuitively, S(a) is a+1. •There is no natural number whose successor is 0. •Distinct natural numbers have distinct successors: if a ≠ b, then S(a) ≠ S(b). •If a property is possessed by 0 and also ...
An Introduction to Löb`s Theorem in MIRI Research
... So Gödel found that we can write undecidable statements about properties of natural numbers, and furthermore showed that adding new axioms won’t fix it, since you can repeat the process with the new rule system for whether a statement is a theorem. (There’s only one loophole, and it’s not a very ex ...
... So Gödel found that we can write undecidable statements about properties of natural numbers, and furthermore showed that adding new axioms won’t fix it, since you can repeat the process with the new rule system for whether a statement is a theorem. (There’s only one loophole, and it’s not a very ex ...
Mathematical Induction
... Basis: The sum of the first 0 natural numbers is indeed 0. Inductive step: Assume the sum of the first k natural numbers is k(k-1)/2 (inductive hypothesis). We want to show that then the same is true for k+1 instead of k, that is, the sum of the first k+1 natural numbers is (k+1)((k+1)-1)/2, i.e. it ...
... Basis: The sum of the first 0 natural numbers is indeed 0. Inductive step: Assume the sum of the first k natural numbers is k(k-1)/2 (inductive hypothesis). We want to show that then the same is true for k+1 instead of k, that is, the sum of the first k+1 natural numbers is (k+1)((k+1)-1)/2, i.e. it ...
Real Numbers - Will Rosenbaum
... number n larger than y. We will use the Archimedean property of R frequently in proofs of limits in the sequel. 1.4 Completeness From the perspective of calculus and analysis, the completeness property of the real numbers is what makes everything work. Unfortunately, it is also the most technical to ...
... number n larger than y. We will use the Archimedean property of R frequently in proofs of limits in the sequel. 1.4 Completeness From the perspective of calculus and analysis, the completeness property of the real numbers is what makes everything work. Unfortunately, it is also the most technical to ...
THE HITCHHIKER`S GUIDE TO THE INCOMPLETENESS
... some worlds. However, that is not true in our world of common sense, or in another viewpoint, we really do not care about those worlds where 1 + 1 = 3. We shall start writing down some basic sentences which are commonly accepted as true sentences and only discuss the worlds where those sentences are ...
... some worlds. However, that is not true in our world of common sense, or in another viewpoint, we really do not care about those worlds where 1 + 1 = 3. We shall start writing down some basic sentences which are commonly accepted as true sentences and only discuss the worlds where those sentences are ...
course notes - Theory and Logic Group
... Proof. Suppose that such a Γ exists and let I Γ. We have M ( I iff M is finite. Consider ∆ t I u Y tLn | n ¥ 1u. Let ∆0 be a finite subset of ∆, then ∆0 t I u Y tLn | 1 ¤ n ¤ mu for some m and every structure of size m 1 is a model of ∆0 . So by the compactness theorem ∆ would have a model whi ...
... Proof. Suppose that such a Γ exists and let I Γ. We have M ( I iff M is finite. Consider ∆ t I u Y tLn | n ¥ 1u. Let ∆0 be a finite subset of ∆, then ∆0 t I u Y tLn | 1 ¤ n ¤ mu for some m and every structure of size m 1 is a model of ∆0 . So by the compactness theorem ∆ would have a model whi ...
The Axiom of Choice
... to the axiom of choice, although they do not appear to be the same at all. I think this is partly why the axiom of choice shows up so often. Let me mention a couple of these which have to do with order relations. One statement that’s equivalent to the axiom of choice is the well-ordering theorem. It ...
... to the axiom of choice, although they do not appear to be the same at all. I think this is partly why the axiom of choice shows up so often. Let me mention a couple of these which have to do with order relations. One statement that’s equivalent to the axiom of choice is the well-ordering theorem. It ...
Strict Predicativity 3
... count as predicative; that is, are there mathematical theories that are predicative in the more usual sense but are not reducible to theories that are strictly predicative? (2) Assuming that there is such a difference, is there a body of arithmetic that is strictly predicative, and what are its limi ...
... count as predicative; that is, are there mathematical theories that are predicative in the more usual sense but are not reducible to theories that are strictly predicative? (2) Assuming that there is such a difference, is there a body of arithmetic that is strictly predicative, and what are its limi ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of the standard algebraic specification for arithmetic, designed to rule out finite models. The theoretical objective of adding the in ...
... Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of the standard algebraic specification for arithmetic, designed to rule out finite models. The theoretical objective of adding the in ...
Handout 14
... provable is a tautology. Thus, the formal system of propositional logic is not only sound (i.e. generates only valid formulas) but also generates all of them. Theorem 5.2 (completeness of propositional logic). Let T be a set of formulas and A a formula. Then T (A ...
... provable is a tautology. Thus, the formal system of propositional logic is not only sound (i.e. generates only valid formulas) but also generates all of them. Theorem 5.2 (completeness of propositional logic). Let T be a set of formulas and A a formula. Then T (A ...
Document
... The real numbers are an example of an ordered field. But there are other examples as well. The rational numbers ...
... The real numbers are an example of an ordered field. But there are other examples as well. The rational numbers ...
Real Analysis Lecture 2
... (i) Come up with a set of axioms for Q. (ii) Construct candidate Q from some previously defined thing. (Note: Q is usually constructed from the integers, while N is constructed directly from set theory; set theory is a basic axiomatic foundation of modern mathematics and is not constructed out of an ...
... (i) Come up with a set of axioms for Q. (ii) Construct candidate Q from some previously defined thing. (Note: Q is usually constructed from the integers, while N is constructed directly from set theory; set theory is a basic axiomatic foundation of modern mathematics and is not constructed out of an ...
MATH 532, 736I: MODERN GEOMETRY
... Part I. Each of the following is something or part of something you were asked to memorize. In the proofs you give below, I will assume you are using the axioms as you state them in your answers to (1) and (2). The first 2 problems in this section are worth 7 points each, and the second 2 problems a ...
... Part I. Each of the following is something or part of something you were asked to memorize. In the proofs you give below, I will assume you are using the axioms as you state them in your answers to (1) and (2). The first 2 problems in this section are worth 7 points each, and the second 2 problems a ...
An un-rigorous introduction to the incompleteness theorems
... We assign this formula a Gödel number by multiplying the Gödel number for the first digit of the formula (0) by 2, the Gödel number for the second digit of the formula (+) by 3, the Gödel number of the third digit (1) by 5, and so on, multiplying the Gödel number for the nth digit by the nth pr ...
... We assign this formula a Gödel number by multiplying the Gödel number for the first digit of the formula (0) by 2, the Gödel number for the second digit of the formula (+) by 3, the Gödel number of the third digit (1) by 5, and so on, multiplying the Gödel number for the nth digit by the nth pr ...
Peano axioms
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are a set of axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete.The need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, and in 1889 Peano published a more precisely formulated version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method (Latin: Arithmetices principia, nova methodo exposita).The Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set ""number"". The next four are general statements about equality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the ""underlying logic"". The next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema.