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College Geometry University of Memphis MATH 3581 Mathematical
... postulates include “Two points uniquely determine a line” and the Parallel Postulate. The first statement is true in the plane, but it is false on the surface of the sphere. The Parallel Postulate can be stated in more than one way, changing the structure of the geometry. The basic point is that Dif ...
... postulates include “Two points uniquely determine a line” and the Parallel Postulate. The first statement is true in the plane, but it is false on the surface of the sphere. The Parallel Postulate can be stated in more than one way, changing the structure of the geometry. The basic point is that Dif ...
4 slides/page
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...
Arithmetic Series Homework - Niskayuna Central Schools
... eight seats and increasing by two seats per row thereafter. How many seats are in the new black-box theatre? Show the calculations that lead to your answer. ...
... eight seats and increasing by two seats per row thereafter. How many seats are in the new black-box theatre? Show the calculations that lead to your answer. ...
CHAPTER 0: WELCOME TO MATHEMATICS A Preface of Logic
... allowed to deduce truth values once we accept assumptions or proved propositions into the system. This topic will be discussed once we have defined some more language in the chapter on logic. To summarize, the game we call mathematics is an internally consistent system within which we carefully stat ...
... allowed to deduce truth values once we accept assumptions or proved propositions into the system. This topic will be discussed once we have defined some more language in the chapter on logic. To summarize, the game we call mathematics is an internally consistent system within which we carefully stat ...
1. Counting (1) Let n be natural number. Prove that the product of n
... Ai “ tS P A | i P Su In other words, elements of Ai are subsets of rns with cardinality 2 and which has the natural number i as one of its elements. Clearly, Ai Ă A. (iii) Each element of Ai is a set of the form ti, ju. Now j can take any value from the set t1, 2, 3 . . . , nu ´ tiu. So, there are n ...
... Ai “ tS P A | i P Su In other words, elements of Ai are subsets of rns with cardinality 2 and which has the natural number i as one of its elements. Clearly, Ai Ă A. (iii) Each element of Ai is a set of the form ti, ju. Now j can take any value from the set t1, 2, 3 . . . , nu ´ tiu. So, there are n ...
Platonism in mathematics (1935) Paul Bernays
... At first sight, such disjunctions seem trivial, and we must be attentive in order to notice that an assumption slips in. But analysis is not content with this modest variety of platonism; it reflects it to a stronger degree with respect to the following notions: set of numbers, sequence of numbers, ...
... At first sight, such disjunctions seem trivial, and we must be attentive in order to notice that an assumption slips in. But analysis is not content with this modest variety of platonism; it reflects it to a stronger degree with respect to the following notions: set of numbers, sequence of numbers, ...
A simple proof of Parsons` theorem
... tautologies. In the sequel, we need a form of Herbrand’s theorem for ∃∀∃consequences of universal theories.6 The particular form in question has a quite elegant statement, and can be proved by a very simple compactness argument due to Jan Krajı́ček, Pavel Pudlák and Gaisi Takeuti in [KPT91]. Their ...
... tautologies. In the sequel, we need a form of Herbrand’s theorem for ∃∀∃consequences of universal theories.6 The particular form in question has a quite elegant statement, and can be proved by a very simple compactness argument due to Jan Krajı́ček, Pavel Pudlák and Gaisi Takeuti in [KPT91]. Their ...
Exercises: Sufficiently expressive/strong
... 2. In this exercise, take ‘theory’ to mean any set of sentences equipped with deductive rules, whether or not effectively axiomatizable: (a) If a theory is effectively decidable, must it be negation complete? (b) If a theory is effectively decidable, must it be effectively axiomatizable? (c) If a th ...
... 2. In this exercise, take ‘theory’ to mean any set of sentences equipped with deductive rules, whether or not effectively axiomatizable: (a) If a theory is effectively decidable, must it be negation complete? (b) If a theory is effectively decidable, must it be effectively axiomatizable? (c) If a th ...
Chapter 4, Mathematics
... ‘algorithm’. In cookery a reliable recipe is an algorithm for producing the soup, cake, stew or whatever it is that it tells us how to cook. A computer program, if it works, embodies some sort of algorithm. Only a Rationalist could have supposed there might be such a procedure for determining the tr ...
... ‘algorithm’. In cookery a reliable recipe is an algorithm for producing the soup, cake, stew or whatever it is that it tells us how to cook. A computer program, if it works, embodies some sort of algorithm. Only a Rationalist could have supposed there might be such a procedure for determining the tr ...
Proof Theory - Andrew.cmu.edu
... parts of ordinary mathematics, but weak enough, on the other hand, to be amenable to proof-theoretic analysis. He then suggested “calibrating” various mathematical theorems in terms of their axiomatic strength. Whereas in ordinary (meta)mathematics, one proves theorems from axioms, Friedman noticed ...
... parts of ordinary mathematics, but weak enough, on the other hand, to be amenable to proof-theoretic analysis. He then suggested “calibrating” various mathematical theorems in terms of their axiomatic strength. Whereas in ordinary (meta)mathematics, one proves theorems from axioms, Friedman noticed ...
A Brief Note on Proofs in Pure Mathematics
... to the above proof, we knew that our final result should be 2(n + 1) ≤ 2n+1 , so we did what we had to to get the correct right-hand side. ...
... to the above proof, we knew that our final result should be 2(n + 1) ≤ 2n+1 , so we did what we had to to get the correct right-hand side. ...
MathsReview
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
Lecture 3.1
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
Lecture 3.1
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
Lecture 3
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
... Bi-condition: p q [p if and only if q (p iff q)] Logical equivalence: p q [p is logically equivalent to q] ...
Axioms and Theorems
... impossible. Therefore the first 30 dominoes (wherever they are put) must cover 30 white squares and 30 black. This MUST leave two black squares uncovered. And since these can’t be together, they cannot be covered by one domino. Therefore it is impossible. ...
... impossible. Therefore the first 30 dominoes (wherever they are put) must cover 30 white squares and 30 black. This MUST leave two black squares uncovered. And since these can’t be together, they cannot be covered by one domino. Therefore it is impossible. ...
3.1 Definition of a Group
... This section contains the definitions of a binary operation, a group, an abelian group, and a finite group. These definitions provide the language you will be working with, and you simply must know this language. Try to learn it so well that you don’t have even a trace of an accent! Loosely, a group ...
... This section contains the definitions of a binary operation, a group, an abelian group, and a finite group. These definitions provide the language you will be working with, and you simply must know this language. Try to learn it so well that you don’t have even a trace of an accent! Loosely, a group ...
§0.1 Sets and Relations
... (d) The real number π is a set. Like wise, each real number x is a set. (e) Then, of course, the collection of all integers Z is a set. The collection of all rational numbers Q is a set. The collection of all real numbers R is a set. The collection of all complex numbers C is a set. (f) A function f ...
... (d) The real number π is a set. Like wise, each real number x is a set. (e) Then, of course, the collection of all integers Z is a set. The collection of all rational numbers Q is a set. The collection of all real numbers R is a set. The collection of all complex numbers C is a set. (f) A function f ...
The Anti-Foundation Axiom in Constructive Set Theories
... subsection, it is pivotal to observe that, unlike Aczel’s type of iterative sets, a strong system type is not required to have an inductive structure, i.e. there need not be an elimination rule for it. The strength of a type theory ML1 with the type constructors Π, Σ, +, I, N, N0 , N1 and one univer ...
... subsection, it is pivotal to observe that, unlike Aczel’s type of iterative sets, a strong system type is not required to have an inductive structure, i.e. there need not be an elimination rule for it. The strength of a type theory ML1 with the type constructors Π, Σ, +, I, N, N0 , N1 and one univer ...
Completeness Theorem for Continuous Functions and Product
... short, is considered as a minimal subsystem of ZF necessary for a good notion of computation. KP arises from ZF by omitting the Power Set Axiom and restricting Separation and Collection to ∆0 -formulas. An admissible set is a transitive set A such that (A, ∈) is a model of KP. The smallest example o ...
... short, is considered as a minimal subsystem of ZF necessary for a good notion of computation. KP arises from ZF by omitting the Power Set Axiom and restricting Separation and Collection to ∆0 -formulas. An admissible set is a transitive set A such that (A, ∈) is a model of KP. The smallest example o ...
1 Proof by Contradiction - Stony Brook Mathematics
... For a compact axiom set for N, we need only use these three axioms involving S, the successor function a 7→ a + 1 (we don’t write it as “+1” since the axioms don’t require or define addition or the constant 1). 1. There a natural number “0”. 2. S(a) = S(b) =⇒ a = b for all a, b. 3. S(a) 6= 0 for all ...
... For a compact axiom set for N, we need only use these three axioms involving S, the successor function a 7→ a + 1 (we don’t write it as “+1” since the axioms don’t require or define addition or the constant 1). 1. There a natural number “0”. 2. S(a) = S(b) =⇒ a = b for all a, b. 3. S(a) 6= 0 for all ...
Multi-Agent Only
... I First-order modal logic for multi-agent only-knowing I Faithfully generalizes intuitions of Levesque’s logic I Semantics not based on Kripke structures or canonical models, and thus avoids some problems of ...
... I First-order modal logic for multi-agent only-knowing I Faithfully generalizes intuitions of Levesque’s logic I Semantics not based on Kripke structures or canonical models, and thus avoids some problems of ...
Number systems. - Elad Aigner
... We cannot appeal to the WOP with S as S is a set of unordered pairs and not a subset of N. To be able to appeal to the WOP we consider the projection of S namely S 0 = {n ∈ Z+ : ∃m ∈ Z+ s.t. {m, n} ∈ S}. The set S 0 is non-empty (since S is non-empty) and S 0 ⊆ Z+ ⊆ N. By the WOP S 0 has a least ele ...
... We cannot appeal to the WOP with S as S is a set of unordered pairs and not a subset of N. To be able to appeal to the WOP we consider the projection of S namely S 0 = {n ∈ Z+ : ∃m ∈ Z+ s.t. {m, n} ∈ S}. The set S 0 is non-empty (since S is non-empty) and S 0 ⊆ Z+ ⊆ N. By the WOP S 0 has a least ele ...
CSE 1400 Applied Discrete Mathematics Proofs
... The non-logical axioms of number theory are: • Closure of addition: If n and m are natural numbers, then n + m is a natural number. • Closure of multiplication: If n and m are natural numbers, then n · m is a natural number. • Closure of exponentiation: If n and m are natural numbers, then nm is a n ...
... The non-logical axioms of number theory are: • Closure of addition: If n and m are natural numbers, then n + m is a natural number. • Closure of multiplication: If n and m are natural numbers, then n · m is a natural number. • Closure of exponentiation: If n and m are natural numbers, then nm is a n ...
SECOND-ORDER LOGIC, OR - University of Chicago Math
... I(A) is a subset of d × d. A variable-assignment s is a function from the variables of L1K(=) to d. Each formula is assigned a truth-value in the standard inductive way. An atomic formula P (t1 ...tn ) is assigned to ‘true’ if hd1 ...dn i ∈ I(P ), where d1 , . . . , dn are the evaluation of the term ...
... I(A) is a subset of d × d. A variable-assignment s is a function from the variables of L1K(=) to d. Each formula is assigned a truth-value in the standard inductive way. An atomic formula P (t1 ...tn ) is assigned to ‘true’ if hd1 ...dn i ∈ I(P ), where d1 , . . . , dn are the evaluation of the term ...
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.