P - Department of Computer Science
... Computability: What are the fundamental capabilities and limitations of ...
... Computability: What are the fundamental capabilities and limitations of ...
Introduction to Discrete Structures Instructional Material
... visitors. It is a European party and it is customary to shake the hands of the house owner before entering their premises. Let the class observe how many handshakes the birthday celebrant did when inviting her visitors in her house. Then allow each visitor to shake the hands of the other visitors. A ...
... visitors. It is a European party and it is customary to shake the hands of the house owner before entering their premises. Let the class observe how many handshakes the birthday celebrant did when inviting her visitors in her house. Then allow each visitor to shake the hands of the other visitors. A ...
1-1
... determines which objects are in the set. • List – if the set is finite, a list of objects belonging to the set is often used to specify the set. Exercise : Write pseudo code to perform set operations for two sets specified by ordered lists. • For infinite or large finite sets, the following methods ...
... determines which objects are in the set. • List – if the set is finite, a list of objects belonging to the set is often used to specify the set. Exercise : Write pseudo code to perform set operations for two sets specified by ordered lists. • For infinite or large finite sets, the following methods ...
Grade 7/8 Math Circles Sets Sets
... However, there is an even more fundamental way to determine if these two sets are of the same size. We could simply pair one member of one set to a member of another set. If there are any remaining members that can’t be paired up, then the set with remaining elements has more elements than the othe ...
... However, there is an even more fundamental way to determine if these two sets are of the same size. We could simply pair one member of one set to a member of another set. If there are any remaining members that can’t be paired up, then the set with remaining elements has more elements than the othe ...
Section 5.3 notes
... reduce the exponent until it becomes zero. We give this procedure in Algorithm 2. ALGORITHM 2 A Recursive Algorithm for Computing an . procedure power(a: nonzero real number, n: nonnegative integer) if n = 0 then return 1 else return a· power(a, n − 1) {output is an } Mathematical induction, and its ...
... reduce the exponent until it becomes zero. We give this procedure in Algorithm 2. ALGORITHM 2 A Recursive Algorithm for Computing an . procedure power(a: nonzero real number, n: nonnegative integer) if n = 0 then return 1 else return a· power(a, n − 1) {output is an } Mathematical induction, and its ...
Truth and proof
... human beings, at least in principle. This playability of our "language games" is one of the most characteristic features of the thought of both Wittgenstein and Dummett.” (Hintikka 1996) ...
... human beings, at least in principle. This playability of our "language games" is one of the most characteristic features of the thought of both Wittgenstein and Dummett.” (Hintikka 1996) ...
How Big Is Infinity?
... Did you notice that we've been looking at correspondences between infinite sets? We're going to use the idea of a one-toone correspondence to compare the sizes of sets, infinite as well as finite. If a one-to-one correspondence exists between two sets S and T then we say that S and T have the same c ...
... Did you notice that we've been looking at correspondences between infinite sets? We're going to use the idea of a one-toone correspondence to compare the sizes of sets, infinite as well as finite. If a one-to-one correspondence exists between two sets S and T then we say that S and T have the same c ...
Godel incompleteness
... a Gödel sentence (G) — that the system cannot prove. Nonetheless, we can see that the Gödel sentence is true, thus we have a capacity that the formal system lacks. G also stands for a number — a Gödel number Gp — which results from the assignment of a code number to each sentence in the language of ...
... a Gödel sentence (G) — that the system cannot prove. Nonetheless, we can see that the Gödel sentence is true, thus we have a capacity that the formal system lacks. G also stands for a number — a Gödel number Gp — which results from the assignment of a code number to each sentence in the language of ...
chapter1
... A set is said to be countably infinite if it is equinumerous with , and countable if it is finite or countably infinite. A set that is not countable is uncountable. Several techniques are useful for showing a set A to be countably infinite. The most direct way is to exhibit a bijection between some ...
... A set is said to be countably infinite if it is equinumerous with , and countable if it is finite or countably infinite. A set that is not countable is uncountable. Several techniques are useful for showing a set A to be countably infinite. The most direct way is to exhibit a bijection between some ...
On the paradoxes of set theory
... Theorem B.--The series of ordirials up to and including a given ordinal number, say~, has ordinal number ~ 1. Theorem C.--Tho series of all ordinal numbers is well ordered and hence, has an ordinal number, say.fl. The Burali.-Forti paradox accerbs the incompatibility of the above three theorems. ...
... Theorem B.--The series of ordirials up to and including a given ordinal number, say~, has ordinal number ~ 1. Theorem C.--Tho series of all ordinal numbers is well ordered and hence, has an ordinal number, say.fl. The Burali.-Forti paradox accerbs the incompatibility of the above three theorems. ...
Comp 205: Comparative Programming Languages
... Another way of implementing this is to use the built-in function zip: zip :: [a] -> [b] -> [(a,b)] zip xs [] = [] zip [] ys = [] zip (x:xs) (y:ys) = (x,y) : zip xs ys Main> zip [1,2,3] ["Those", "are"] [(1,"Those"),(2,"are")] ...
... Another way of implementing this is to use the built-in function zip: zip :: [a] -> [b] -> [(a,b)] zip xs [] = [] zip [] ys = [] zip (x:xs) (y:ys) = (x,y) : zip xs ys Main> zip [1,2,3] ["Those", "are"] [(1,"Those"),(2,"are")] ...
INDEX SETS FOR n-DECIDABLE STRUCTURES CATEGORICAL
... Definition 1. A computable structure M is d-computably categorical (also called d-autostable) if, for every computable structure A isomorphic to M, there exists a d-computable isomorphism from M onto A. In case d = 0, we simply say that M is computably categorical. A computable structure M is relati ...
... Definition 1. A computable structure M is d-computably categorical (also called d-autostable) if, for every computable structure A isomorphic to M, there exists a d-computable isomorphism from M onto A. In case d = 0, we simply say that M is computably categorical. A computable structure M is relati ...
Section 2.4 Countable Sets
... S . This says that every infinite set contains a denumerable proper subset, which means the cardinality of a denumerable set can not be greater than the cardinality of any infinite set. Hence, ℵ0 it is the smallest transfinite number. Note: The symbol " ∞ " , the reader is well aware of from calculu ...
... S . This says that every infinite set contains a denumerable proper subset, which means the cardinality of a denumerable set can not be greater than the cardinality of any infinite set. Hence, ℵ0 it is the smallest transfinite number. Note: The symbol " ∞ " , the reader is well aware of from calculu ...
Slide 1
... Theorem: The Entscheidungsproblem is unsolvable. Proof: (Due to Turing) 1. If we could solve the problem of determining whether a given Turing machine ever prints the symbol 0, then we could solve the problem of determining whether a given Turing machine halts. 2. But we can’t solve the problem of d ...
... Theorem: The Entscheidungsproblem is unsolvable. Proof: (Due to Turing) 1. If we could solve the problem of determining whether a given Turing machine ever prints the symbol 0, then we could solve the problem of determining whether a given Turing machine halts. 2. But we can’t solve the problem of d ...
1 Enumerability - George Belic Philosophy
... ◦ What is an infinite set? One kind can be defined in terms of enumerability ◦ function = function of positive integers, leaving it open whether the function is total or partial (Dfn 1.1) Informal: A set S is enumerable iff S is a set that can be enumerated i.e. arranged in a single list with a firs ...
... ◦ What is an infinite set? One kind can be defined in terms of enumerability ◦ function = function of positive integers, leaving it open whether the function is total or partial (Dfn 1.1) Informal: A set S is enumerable iff S is a set that can be enumerated i.e. arranged in a single list with a firs ...
Section 2.2 Families of Sets
... ▪ sets are denoted capital letters A, B, C ,... ▪ families of sets are denoted by script letters , , , … Example 3 (Indexed Family) If An = {n + 1, n + 2, , 2n} for each natural number n ∈ N , then ...
... ▪ sets are denoted capital letters A, B, C ,... ▪ families of sets are denoted by script letters , , , … Example 3 (Indexed Family) If An = {n + 1, n + 2, , 2n} for each natural number n ∈ N , then ...
Sets (section 3.1 )
... There is a strange loop here which yields a contradiction: S cannot exist. For a similar reason, the set of all sets cannot exist. See Gödel Esher Bach[?] for a lot of similar fun stu with strange loops. That's been a major source of trouble and work a century ago, when mathematicians tried to de n ...
... There is a strange loop here which yields a contradiction: S cannot exist. For a similar reason, the set of all sets cannot exist. See Gödel Esher Bach[?] for a lot of similar fun stu with strange loops. That's been a major source of trouble and work a century ago, when mathematicians tried to de n ...
23-24-TuringMachinesHandout
... One way or two way infinite tape: we're about to show that we can simulate two way infinite with ours. Rewrite and move at the same time: just affects (linearly) the number of moves it takes to solve a problem. Turing Machine Extensions In fact, there are lots of extensions we can make to our ba ...
... One way or two way infinite tape: we're about to show that we can simulate two way infinite with ours. Rewrite and move at the same time: just affects (linearly) the number of moves it takes to solve a problem. Turing Machine Extensions In fact, there are lots of extensions we can make to our ba ...
Cardinality, countable and uncountable sets
... With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify their elements can be placed in pairwise correspondence; that is, that there is a bijection between them. It is then natural to generalize this to infi ...
... With the notion of bijection at hand, it is easy to formalize the idea that two finite sets have the same number of elements: we just need to verify their elements can be placed in pairwise correspondence; that is, that there is a bijection between them. It is then natural to generalize this to infi ...
A Problem Course in Mathematical Logic Volume II Computability
... abstract models of computation in the 1930’s, including recursive functions, λ-calculus, Turing machines, and grammars. Although these models are very different from each other in spirit and formal definition, it turned out that they were all essentially equivalent in what they could do. This sugges ...
... abstract models of computation in the 1930’s, including recursive functions, λ-calculus, Turing machines, and grammars. Although these models are very different from each other in spirit and formal definition, it turned out that they were all essentially equivalent in what they could do. This sugges ...
+ 1 sO - Department of Mathematics, CCNY
... standard deck of 52 cards? Consider various interpretations of this question, as in the preceding problem. 2.17. How many ways can three coins fall? four coins? n coins, where n is any positive integer? 2.18. Two cards are drawn one after the other from a standard deck of 52 card.s. In how many ways ...
... standard deck of 52 cards? Consider various interpretations of this question, as in the preceding problem. 2.17. How many ways can three coins fall? four coins? n coins, where n is any positive integer? 2.18. Two cards are drawn one after the other from a standard deck of 52 card.s. In how many ways ...
CHAP02 Axioms of Set Theory
... Mathematics was in danger of collapsing! A few mathematicians, those interested in the foundations of mathematics, tried to prop it up. Most mathematicians simply ignored the problem and just got on with their business. The rescue came with replacing the one axiom by a set of axioms that avoided the ...
... Mathematics was in danger of collapsing! A few mathematicians, those interested in the foundations of mathematics, tried to prop it up. Most mathematicians simply ignored the problem and just got on with their business. The rescue came with replacing the one axiom by a set of axioms that avoided the ...