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Introduction to HyperReals
... Since b is finite there are real numbers s and t with s < b < t. Let A = { x | x is real and x < b }. A is non-empty since it contains s and is bounded above by t. Thus there is a real number r which is the least upper bound of A. We claim r b. Suppose not. Thus r b and Hence r-b is positive or ...
... Since b is finite there are real numbers s and t with s < b < t. Let A = { x | x is real and x < b }. A is non-empty since it contains s and is bounded above by t. Thus there is a real number r which is the least upper bound of A. We claim r b. Suppose not. Thus r b and Hence r-b is positive or ...
Gödel on Conceptual Realism and Mathematical Intuition
... strong enough for arithmetic are either inconsistent or incomplete. Language is subordinate to reality. Goldstein points out that Gödel is committed to the possibility of reaching out beyond our experiences to describe the world “out yonder.” Mathematics is a means of unveiling the features of objec ...
... strong enough for arithmetic are either inconsistent or incomplete. Language is subordinate to reality. Goldstein points out that Gödel is committed to the possibility of reaching out beyond our experiences to describe the world “out yonder.” Mathematics is a means of unveiling the features of objec ...
WhichQuantifiersLogical
... and quantifiers is to be established semantically in one way or another prior to their inferential role. Their meanings may be the primitives of our reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a ...
... and quantifiers is to be established semantically in one way or another prior to their inferential role. Their meanings may be the primitives of our reasoning in general“and”, “or”, “not”, “if…then”, “all”, “some”or they may be understood informally like “most”, “has the same number as”, etc. in a ...
LPSS MATHCOUNTS 2004–2005 Lecture 1: Arithmetic Series—4/6/04
... Suppose an auditorium consists of 40 rows of seats. The first row contains 10 seats, the second row 12 seats, the third row 14 seats, and so on. Each row contains two more seats than its predecessor. How many seats S are there in the auditorium? This is an arithmetic series: S = 10 ...
... Suppose an auditorium consists of 40 rows of seats. The first row contains 10 seats, the second row 12 seats, the third row 14 seats, and so on. Each row contains two more seats than its predecessor. How many seats S are there in the auditorium? This is an arithmetic series: S = 10 ...
CHAPTER 1 INTRODUCTION 1 Mathematical Paradoxes
... the theory which are not expressed in the axioms, require proof. For some time this degree of exactness in the construction of theories seemed sufficient. However, it turned out that the assumption of a consistent set of axioms does not prevent the occurrence of another kind of paradoxes, called sem ...
... the theory which are not expressed in the axioms, require proof. For some time this degree of exactness in the construction of theories seemed sufficient. However, it turned out that the assumption of a consistent set of axioms does not prevent the occurrence of another kind of paradoxes, called sem ...
Is the principle of contradiction a consequence of ? Jean
... from the first to the third through the second and that all are true is a procedure which still common in contemporary books of algebra as well as the details of the fonts: italic for the variable, no italics for the numbers, the parentheses, the exponent and the identity sign. It is also common to ...
... from the first to the third through the second and that all are true is a procedure which still common in contemporary books of algebra as well as the details of the fonts: italic for the variable, no italics for the numbers, the parentheses, the exponent and the identity sign. It is also common to ...
Induction handout and worksheet 1
... Solution. We will show by induction that any set of N horses consists of horses of the same color. The base case is easy. If we have a set with one horse, then all horses in the set are the same color. We assume as our induction hypothesis that any set of N horses consists of horses of the same col ...
... Solution. We will show by induction that any set of N horses consists of horses of the same color. The base case is easy. If we have a set with one horse, then all horses in the set are the same color. We assume as our induction hypothesis that any set of N horses consists of horses of the same col ...
APPENDIX B EXERCISES In Exercises 1–8, use the
... Annie continued, “Number the days next week like this: Monday=1, Tuesday=2, …, Friday=5. Let S(n) be the statement ‘There can’t be a quiz on day 5 − n .’ Now S(0) says ‘There can’t be a quiz on day 5.’ That’s Friday and if we haven’t had the quiz by Thursday, then we’ll know it’s on Friday and it wo ...
... Annie continued, “Number the days next week like this: Monday=1, Tuesday=2, …, Friday=5. Let S(n) be the statement ‘There can’t be a quiz on day 5 − n .’ Now S(0) says ‘There can’t be a quiz on day 5.’ That’s Friday and if we haven’t had the quiz by Thursday, then we’ll know it’s on Friday and it wo ...
SOME AXIOMS FOR CONSTRUCTIVE ANALYSIS Introduction
... logic, and intuitionistic arithmetic IA0 in a language with only the constants =, 0, 0 , +, ·, as subsystems obtained from the corresponding classical theories by weakening the law of double negation ¬¬A → A to ¬A → (A → B). To further clarify the relation between intuitionistic and classical mathem ...
... logic, and intuitionistic arithmetic IA0 in a language with only the constants =, 0, 0 , +, ·, as subsystems obtained from the corresponding classical theories by weakening the law of double negation ¬¬A → A to ¬A → (A → B). To further clarify the relation between intuitionistic and classical mathem ...
An Axiomatization of G'3
... We consider a logic simply as a set of formulas that, moreover, satisfies the following two properties: (i) is closed under modus ponens (i.e. if A and A → B are in the logic, then so is B) and (ii) is closed under substitution (i.e. if a formula A is in the logic, then any other formula obtained by ...
... We consider a logic simply as a set of formulas that, moreover, satisfies the following two properties: (i) is closed under modus ponens (i.e. if A and A → B are in the logic, then so is B) and (ii) is closed under substitution (i.e. if a formula A is in the logic, then any other formula obtained by ...
A. Formal systems, Proof calculi
... formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, then it proves all th ...
... formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algorithm, then the calculus is decidable. If the calculus is complete, then it proves all th ...
(pdf)
... functions. Functions are of course familiar to us from set theory, where they are defined as sets of ordered pairs, associating inputs to their outputs. Functions in the λ-calculus are different in that the process by which an input is transformed into an output is of primary concern. It is this pro ...
... functions. Functions are of course familiar to us from set theory, where they are defined as sets of ordered pairs, associating inputs to their outputs. Functions in the λ-calculus are different in that the process by which an input is transformed into an output is of primary concern. It is this pro ...
Modular Arithmetic
... We’re interested in the algebraic properties of mathematical structures—the formal, symbolic, structural properties of those systems. So far, we have at least three examples of mathematical structures— arithmetic, logic, and set theory. But there’s another useful structure we should talk about—the i ...
... We’re interested in the algebraic properties of mathematical structures—the formal, symbolic, structural properties of those systems. So far, we have at least three examples of mathematical structures— arithmetic, logic, and set theory. But there’s another useful structure we should talk about—the i ...
1 Chapter III Set Theory as a Theory of First Order Predicate Logic
... thev other, none of the new axioms that were proposed seem to qualify as unequivocally true. So, from a conceptual point of view the CH is na open question to this day For our present purposes the first question - What is the total range of infinite cardinalities? - is of more immediate importance. ...
... thev other, none of the new axioms that were proposed seem to qualify as unequivocally true. So, from a conceptual point of view the CH is na open question to this day For our present purposes the first question - What is the total range of infinite cardinalities? - is of more immediate importance. ...
Introduction to the Theory of Computation
... (ii) for any i ∈ N, if i ∈ A, then i + 1 ∈ A. Then N ⊆ A. Principle of Complete Induction: Let A be a set that satisfies the following properties: (*) for any i ∈ N, if ∀j < i, j ∈ A, then i ∈ A. Then N ⊆ A. Monday, January 11, 2010 ...
... (ii) for any i ∈ N, if i ∈ A, then i + 1 ∈ A. Then N ⊆ A. Principle of Complete Induction: Let A be a set that satisfies the following properties: (*) for any i ∈ N, if ∀j < i, j ∈ A, then i ∈ A. Then N ⊆ A. Monday, January 11, 2010 ...
Incompleteness - the UNC Department of Computer Science
... In general H(Tj,j) is true but not provable in L Paradox: How do we know that this formula is true when L does not know it? What logical system are we in when we think this way? We seem to have the ability to get outside of any logical system. Thus we must not reason using a fixed logical system. L ...
... In general H(Tj,j) is true but not provable in L Paradox: How do we know that this formula is true when L does not know it? What logical system are we in when we think this way? We seem to have the ability to get outside of any logical system. Thus we must not reason using a fixed logical system. L ...
PARADOX AND INTUITION
... mean that we at the same time establish any structure-preserving correspondence between the objects from the initial model and arithmetical objects. Hilary Putnam’s paper Models and Reality (Putnam 1980) has become one of the most frequently quoted works discussing problems connected with determinac ...
... mean that we at the same time establish any structure-preserving correspondence between the objects from the initial model and arithmetical objects. Hilary Putnam’s paper Models and Reality (Putnam 1980) has become one of the most frequently quoted works discussing problems connected with determinac ...
Euclidian Roles in Description Logics
... Figure 1: The (a) Euclidean closure, (b) grid, (c) role hierarchy and (d) coincidence enforcing As it happens with many role axioms of SRIQ [2], it comes to no surprise that Euclidian roles are a syntactic sugar for SRIQ. More precisely, an axiom Eucl(R) can be represented by the RIA R− R v R. This ...
... Figure 1: The (a) Euclidean closure, (b) grid, (c) role hierarchy and (d) coincidence enforcing As it happens with many role axioms of SRIQ [2], it comes to no surprise that Euclidian roles are a syntactic sugar for SRIQ. More precisely, an axiom Eucl(R) can be represented by the RIA R− R v R. This ...
ARITHMETIC TRANSLATIONS OF AXIOM SYSTEMS
... many elements plus classes of them. Such a model seems to make it clear that N is roughly as strong as a second-order predicate calculus founded on natural numbers. N may also be characterized as a system obtained from a well-developed system of Quine(9) by replacing all his elementhood axioms by a ...
... many elements plus classes of them. Such a model seems to make it clear that N is roughly as strong as a second-order predicate calculus founded on natural numbers. N may also be characterized as a system obtained from a well-developed system of Quine(9) by replacing all his elementhood axioms by a ...
Supplemental Reading (Kunen)
... reader to a text on the subject, such as [Enderton 19723, [Kleene 19521, or [Shoenfield 19671, for a more detailed treatment. We shall give a precise definition of the formal language, as this is easy to do and is necessary for stating the axioms of ZFC. We shall only hint at the rules of formal ded ...
... reader to a text on the subject, such as [Enderton 19723, [Kleene 19521, or [Shoenfield 19671, for a more detailed treatment. We shall give a precise definition of the formal language, as this is easy to do and is necessary for stating the axioms of ZFC. We shall only hint at the rules of formal ded ...
A course in Mathematical Logic
... x, y, . . . := sequences of indeterminate length. xn := sequence x1 , . . . , xn ; and similarly, for xk , xm , yn , yk , ym . z ∈ x := z is an element of the sequence x. x ∩ y = ∅ := sequences x and y do not have common elements. X ⊆f Y := X is a finite subset of Y . 1 := true 0 := false ...
... x, y, . . . := sequences of indeterminate length. xn := sequence x1 , . . . , xn ; and similarly, for xk , xm , yn , yk , ym . z ∈ x := z is an element of the sequence x. x ∩ y = ∅ := sequences x and y do not have common elements. X ⊆f Y := X is a finite subset of Y . 1 := true 0 := false ...
CPS130, Lecture 1: Introduction to Algorithms
... 5. The Integers and Induction The integers is the set I = {0, 1, , …, n, …}with the “usual” set of properties of integer arithmetic which are stated explicitly in the appendix, section 6. The set P = {1,2, …, n, …} is the set of positive integers (natural numbers – {0}). One property of I, ca ...
... 5. The Integers and Induction The integers is the set I = {0, 1, , …, n, …}with the “usual” set of properties of integer arithmetic which are stated explicitly in the appendix, section 6. The set P = {1,2, …, n, …} is the set of positive integers (natural numbers – {0}). One property of I, ca ...
Jordan Bradshaw, Virginia Walker, and Dylan Kane
... Take an arbitrary object a Suppose a is an F Since all Fs are Gs, a is a G Since all Gs are Hs, a is an H So if a is an F then a is an H But this argument works for any a So all Fs are Hs ...
... Take an arbitrary object a Suppose a is an F Since all Fs are Gs, a is a G Since all Gs are Hs, a is an H So if a is an F then a is an H But this argument works for any a So all Fs are Hs ...
Peano axioms
![](https://commons.wikimedia.org/wiki/Special:FilePath/Domino_effect_visualizing_exclusion_of_junk_term_by_induction_axiom.jpg?width=300)
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.