Arithmetic as a theory modulo
... applied to any axiomatic theory. Deduction modulo [2, 3] is one attempt, among others, towards this aim. In deduction modulo, a theory is not a set of axioms but a set of axioms combined with a set of rewrite rules. For instance, the axiom ∀x x + 0 = x can be replaced by the rewrite rule x + 0 −→ x. ...
... applied to any axiomatic theory. Deduction modulo [2, 3] is one attempt, among others, towards this aim. In deduction modulo, a theory is not a set of axioms but a set of axioms combined with a set of rewrite rules. For instance, the axiom ∀x x + 0 = x can be replaced by the rewrite rule x + 0 −→ x. ...
Section 2.6 Cantor`s Theorem and the ZFC Axioms
... theorem. We assume we can match every real number in (0,1) with a realvalued function on ( 0,1) . We then construct a “rogue” function not on the list, which contradicts the our assumption that such a correspondence exists. Need for Axioms in Set Theory The reader should not entertain the belief tha ...
... theorem. We assume we can match every real number in (0,1) with a realvalued function on ( 0,1) . We then construct a “rogue” function not on the list, which contradicts the our assumption that such a correspondence exists. Need for Axioms in Set Theory The reader should not entertain the belief tha ...
Math 117: The Completeness Axiom
... Note. Theorem 12.1 in the book is stated only for prime natural numbers. However, the proof can be adapted to work for all natural numbers that are not a perfect squares by using a little bit of number theory (like prime factorizations). Notice the proof given in the book is also a proof by contradi ...
... Note. Theorem 12.1 in the book is stated only for prime natural numbers. However, the proof can be adapted to work for all natural numbers that are not a perfect squares by using a little bit of number theory (like prime factorizations). Notice the proof given in the book is also a proof by contradi ...
MS-Word version
... weaks arithmetics are undecidability of the field of rational numbers, MatiyasevichDavis-Robinson-Putnam theorem, solving Hilbert’s tenth problem, Erdös-Woods conjecture, Buss arithmetic and study of the real exponential field. The proposed project is constituted of nine teams from university of Par ...
... weaks arithmetics are undecidability of the field of rational numbers, MatiyasevichDavis-Robinson-Putnam theorem, solving Hilbert’s tenth problem, Erdös-Woods conjecture, Buss arithmetic and study of the real exponential field. The proposed project is constituted of nine teams from university of Par ...
Lecture 9. Model theory. Consistency, independence, completeness
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
Infinite natural numbers: an unwanted phenomenon, or a useful
... From Gödel 1st incompleteness theorem, published in 1931, we know that PA is incomplete. From that (and from the completeness theorem published also by Gödel in 1930 but perhaps known to Skolem even before 1930) it is clear that PA has models that differ in validity of some sentences. If two model ...
... From Gödel 1st incompleteness theorem, published in 1931, we know that PA is incomplete. From that (and from the completeness theorem published also by Gödel in 1930 but perhaps known to Skolem even before 1930) it is clear that PA has models that differ in validity of some sentences. If two model ...
Square roots
... There are many ways to see this fact, already accessible to us after the first lecture! (In case you are wondering why this might be interesting, it is rumoured that Hippassus of Metapontum was killed over his discovery of this fact, so this was a very astonishing and abrasive discovery at one time. ...
... There are many ways to see this fact, already accessible to us after the first lecture! (In case you are wondering why this might be interesting, it is rumoured that Hippassus of Metapontum was killed over his discovery of this fact, so this was a very astonishing and abrasive discovery at one time. ...
Study Guide Unit Test2 with Sample Problems
... Let x and y be two odd numbers. By the definition of odd numbers, there is some p integer such that x = 2p -1, and there is some q integer such that y = 2q – 1. x + y = 2p – 1 + 2q – 1 = 2p + 2q – 2 = 2(p + q – 1) Let s = p + q – 1 Then, x + y = 2s x + y has the representation of an even number, the ...
... Let x and y be two odd numbers. By the definition of odd numbers, there is some p integer such that x = 2p -1, and there is some q integer such that y = 2q – 1. x + y = 2p – 1 + 2q – 1 = 2p + 2q – 2 = 2(p + q – 1) Let s = p + q – 1 Then, x + y = 2s x + y has the representation of an even number, the ...
KRIPKE-PLATEK SET THEORY AND THE ANTI
... In order to obtain a lower bound for the strength of KPA it will be shown that the fragment ∆12 -CA0 of second order arithmetic can be interpreted in KPA. In addition to the axioms of ACA0 (cf. [11], I.3 ), ∆12 -CA0 has the axiom scheme of ∆12 Comprehension, i.e., ∀n(φ(n) ↔ ψ(n)) ← ∃X ∀n (n ∈ X ↔ φ( ...
... In order to obtain a lower bound for the strength of KPA it will be shown that the fragment ∆12 -CA0 of second order arithmetic can be interpreted in KPA. In addition to the axioms of ACA0 (cf. [11], I.3 ), ∆12 -CA0 has the axiom scheme of ∆12 Comprehension, i.e., ∀n(φ(n) ↔ ψ(n)) ← ∃X ∀n (n ∈ X ↔ φ( ...
Number Set
... • A + 1 = A . That is, adding 1 to any number gives the successor of that number. • A + B = (A + B) . That is, adding the successor of any number to any number gives the successor of the sum. By repeated induction on B we get A + B = the Bth successor of A. We also define a multiplication operati ...
... • A + 1 = A . That is, adding 1 to any number gives the successor of that number. • A + B = (A + B) . That is, adding the successor of any number to any number gives the successor of the sum. By repeated induction on B we get A + B = the Bth successor of A. We also define a multiplication operati ...
Relating Infinite Set Theory to Other Branches of Mathematics
... Emil Post’s early work. In 1921, ten years before Gödel’s incompleteness theorem, Post proved, without publishing the work, that any sound formal system contains true unprovable statements. Nevertheless, Stillwell’s description of Post as a “neglected figure” seems to me an overstatement. Aside from ...
... Emil Post’s early work. In 1921, ten years before Gödel’s incompleteness theorem, Post proved, without publishing the work, that any sound formal system contains true unprovable statements. Nevertheless, Stillwell’s description of Post as a “neglected figure” seems to me an overstatement. Aside from ...
PDF
... inference rule modus ponens preserves truth. Since theorems are deduced from axioms and by applications of modus ponens, they are tautologies as a result. Using truth tables, one easily verifies that every axiom is true (under any valuation). For example, if the axiom is of the form A → (B → A), the ...
... inference rule modus ponens preserves truth. Since theorems are deduced from axioms and by applications of modus ponens, they are tautologies as a result. Using truth tables, one easily verifies that every axiom is true (under any valuation). For example, if the axiom is of the form A → (B → A), the ...
Sub-Birkhoff
... The sequences in the (axiom)-clause correspond to the sequence of variables occurring in the axiom. Two remarks on notation used in the definition: ≡[L] expresses that corresponding components are identical (as terms) except for one index where they are related by L. The notation s(~t) expresses tha ...
... The sequences in the (axiom)-clause correspond to the sequence of variables occurring in the axiom. Two remarks on notation used in the definition: ≡[L] expresses that corresponding components are identical (as terms) except for one index where they are related by L. The notation s(~t) expresses tha ...
1 The Principle of Mathematical Induction
... using the PMI. The work involved in prove that (1) holds for Z(n) is called the base case. The work is to prove that Z(1) is true. Exercise 6. For each of the following statements, suppose you want to prove them true for all natural numbers using the PMI. Write the proof from the beginning until you ...
... using the PMI. The work involved in prove that (1) holds for Z(n) is called the base case. The work is to prove that Z(1) is true. Exercise 6. For each of the following statements, suppose you want to prove them true for all natural numbers using the PMI. Write the proof from the beginning until you ...
ppt
... Russell’s Doctrine “I wish to propose for the reader's favourable consideration a doctrine which may, I fear, appear wildly paradoxical and subversive. The doctrine in question is this: that it is undesirable to believe a proposition when there is no ground whatever for supposing it true.” (Russell ...
... Russell’s Doctrine “I wish to propose for the reader's favourable consideration a doctrine which may, I fear, appear wildly paradoxical and subversive. The doctrine in question is this: that it is undesirable to believe a proposition when there is no ground whatever for supposing it true.” (Russell ...
Document
... can be arranged into a list that contains all the members of the list. Then we assume that a value can be composed from the members of the list creating a new member in the set that is not in the list. Only if this impossible value exists can we prove that the list does not exist and, by extension t ...
... can be arranged into a list that contains all the members of the list. Then we assume that a value can be composed from the members of the list creating a new member in the set that is not in the list. Only if this impossible value exists can we prove that the list does not exist and, by extension t ...
Russell`s logicism
... talking about the number 3, and number in general, as properties or characteristics. Now he is moving from this to talking about the number 3, and number in general, as sets. The next question is, what sets are they? Russell says: “Reurning now to the definition of number, it is clear that number i ...
... talking about the number 3, and number in general, as properties or characteristics. Now he is moving from this to talking about the number 3, and number in general, as sets. The next question is, what sets are they? Russell says: “Reurning now to the definition of number, it is clear that number i ...
Quiz 1 - NISER
... 1. Let A = N × N, where N is the set of natural number. Define a relation ∼ on A by (a, b) ∼ (c, d) ⇔ b − a = d − c. (a) Show that ∼ is an equivalence relation on A. (b) Find the elements of the equivalence class of (1, 2). (c) Think A = N × N as the points in the first quadrant of the Cartesian pla ...
... 1. Let A = N × N, where N is the set of natural number. Define a relation ∼ on A by (a, b) ∼ (c, d) ⇔ b − a = d − c. (a) Show that ∼ is an equivalence relation on A. (b) Find the elements of the equivalence class of (1, 2). (c) Think A = N × N as the points in the first quadrant of the Cartesian pla ...
First-order logic;
... Representation: Understand the relationships between different representations of the same information or idea. I ...
... Representation: Understand the relationships between different representations of the same information or idea. I ...
MAS110 Problems for Chapter 2: Summation and Induction
... a = b = 1. So P (1) is true. Inductive step. Assume that P (k) is true for some k ≥ 1. Thus if the maximum of two positive integers is k then those two numbers are the same. Now let a, b be two positive integers such that max{a, b} = k + 1. Then max{a − 1, b − 1} = k. Hence by the inductive hypothes ...
... a = b = 1. So P (1) is true. Inductive step. Assume that P (k) is true for some k ≥ 1. Thus if the maximum of two positive integers is k then those two numbers are the same. Now let a, b be two positive integers such that max{a, b} = k + 1. Then max{a − 1, b − 1} = k. Hence by the inductive hypothes ...
(1) Find all prime numbers smaller than 100. (2) Give a proof by
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
... (1) Find all prime numbers smaller than 100. (2) Give a proof by induction (instead of a proof by contradiction given in class) that any natural number > 1 has a unique (up to order) factorization as a product of primes. (3) Give a proof by induction that if a ≡ b( mod m) then an ≡ bn ( mod m) for a ...
The Non-Euclidean Revolution Material Axiomatic Systems and the
... Primitive terms: person, collection -- these are used in their everyday senses. Definitions: The Turtle Club is a collection of one or more persons. A person in the Turtle Club is called a Turtle. The Committees are certain collections of one or more Turtles. A Turtle in a Committee is called a memb ...
... Primitive terms: person, collection -- these are used in their everyday senses. Definitions: The Turtle Club is a collection of one or more persons. A person in the Turtle Club is called a Turtle. The Committees are certain collections of one or more Turtles. A Turtle in a Committee is called a memb ...
Lecture 7. Model theory. Consistency, independence, completeness
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
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.