a basis for a mathematical theory of computation
... irrelevant to the statement being proved. 5. A way of defining new data spaces in terms of given base spaces and of defining functions on the new spaces in terms of functions on the base spaces. Lack of such a formalism is one of the main weaknesses of ALGOL but the business data processing language ...
... irrelevant to the statement being proved. 5. A way of defining new data spaces in terms of given base spaces and of defining functions on the new spaces in terms of functions on the base spaces. Lack of such a formalism is one of the main weaknesses of ALGOL but the business data processing language ...
Recursion
... 1st passes N-1 and 2nd passes N-2 …check If we assume Fib(N-1) and Fib(N-2) give us correct values of the N-1st and N-2 numbers in the series, and the assignment statement adds them together …this is the definition of the Nth Fibonacci number, so we know the function works for N > 2. Since the f ...
... 1st passes N-1 and 2nd passes N-2 …check If we assume Fib(N-1) and Fib(N-2) give us correct values of the N-1st and N-2 numbers in the series, and the assignment statement adds them together …this is the definition of the Nth Fibonacci number, so we know the function works for N > 2. Since the f ...
compactness slides
... A subset S of U is closed under F if x, y ∈ S ⇒ f (x, y ) ∈ S, g(x) ∈ S A set C ⊆ U is inductive if 1. B ⊆ C 2. C is closed under f , g Let C ∗ be the intersection of all inductive sets. Fact: C ∗ is inductive and it is the smallest inductive set. We call C ∗ the set generated from B by the functio ...
... A subset S of U is closed under F if x, y ∈ S ⇒ f (x, y ) ∈ S, g(x) ∈ S A set C ⊆ U is inductive if 1. B ⊆ C 2. C is closed under f , g Let C ∗ be the intersection of all inductive sets. Fact: C ∗ is inductive and it is the smallest inductive set. We call C ∗ the set generated from B by the functio ...
Set Theory - UVic Math
... the definition, this is the same as the logical implication x ∈ ∅ ⇒ x ∈ A which, in turn, is the same as the implication (x ∈ ∅) → (x ∈ A) being a tautology. The implication has only the truth value “true” because its hypothesis, x ∈ ∅, is false for any x. A different way to say it is that every ele ...
... the definition, this is the same as the logical implication x ∈ ∅ ⇒ x ∈ A which, in turn, is the same as the implication (x ∈ ∅) → (x ∈ A) being a tautology. The implication has only the truth value “true” because its hypothesis, x ∈ ∅, is false for any x. A different way to say it is that every ele ...
A preprint version is available here in pdf.
... of the ground model. Another consists of the sets represented in its theory Th(M ). These constructions generalise to the expanded structure (M, ω) and we can look at the set X of sets coded or represented in (M, ω). This will be a major topic for discussion in this paper, starting at Section 3. Par ...
... of the ground model. Another consists of the sets represented in its theory Th(M ). These constructions generalise to the expanded structure (M, ω) and we can look at the set X of sets coded or represented in (M, ω). This will be a major topic for discussion in this paper, starting at Section 3. Par ...
(pdf)
... a point x ∈ S 2 \ D. Two points are in the same orbit if and only if there exists a rotation which transports the first to the second. These orbits partition S 2 \ D since they are equivalence classes under the relation x ∼ y ⇐⇒ y ∈ F x. Using the Axiom of Choice, we will choose one member from each ...
... a point x ∈ S 2 \ D. Two points are in the same orbit if and only if there exists a rotation which transports the first to the second. These orbits partition S 2 \ D since they are equivalence classes under the relation x ∼ y ⇐⇒ y ∈ F x. Using the Axiom of Choice, we will choose one member from each ...
Document
... positive integers. The second row contains all the fractions with denominator equal to 2. The third row contains all the fractions with denominator equal to 3, etc. ...
... positive integers. The second row contains all the fractions with denominator equal to 2. The third row contains all the fractions with denominator equal to 3, etc. ...
A computably stable structure with no Scott family of finitary formulas
... Our construction is closely related to the work of Cholak, Goncharov, Khoussainov and Shore [3] and Hirschfeldt, Khoussainov and Shore [7]. In these papers, the authors give examples of computably categorical structures which do not remain computably categorical when a constant is added to the lang ...
... Our construction is closely related to the work of Cholak, Goncharov, Khoussainov and Shore [3] and Hirschfeldt, Khoussainov and Shore [7]. In these papers, the authors give examples of computably categorical structures which do not remain computably categorical when a constant is added to the lang ...
Section 7.5: Cardinality
... one-to-one correspondence between X and Y , then clearly X and Y have the same size. This motivates the following definition. Definition 1.1. Let X and Y be sets. We say that X has the same cardinality as Y if and only if there is a one-to-one correspondence from X to Y (when considering general set ...
... one-to-one correspondence between X and Y , then clearly X and Y have the same size. This motivates the following definition. Definition 1.1. Let X and Y be sets. We say that X has the same cardinality as Y if and only if there is a one-to-one correspondence from X to Y (when considering general set ...
1. Sets, relations and functions. 1.1 Set theory. We assume the
... (ii.) If A is a partition of X and no member of A is empty then r = ∪{A × A : A ∈ A} is an equivalence relation on X and X/r = A. Proof. We leave this as Exercise 1.4. for the reader. Definition. Suppose X is a set, r is a relation on X and A ⊂ X. We say a member u of X is an upper bound for A if (a ...
... (ii.) If A is a partition of X and no member of A is empty then r = ∪{A × A : A ∈ A} is an equivalence relation on X and X/r = A. Proof. We leave this as Exercise 1.4. for the reader. Definition. Suppose X is a set, r is a relation on X and A ⊂ X. We say a member u of X is an upper bound for A if (a ...
D • Recursively Palindromic Partitions
... is not palindromic while the second is. If a partition containing m integers is palindromic, its left half is the first floor(m/2) integers and its right half is the last floor(m/2) integers (which must be the reverse of the left half. (floor(x) is the greatest integer less than or equal to x.) A pa ...
... is not palindromic while the second is. If a partition containing m integers is palindromic, its left half is the first floor(m/2) integers and its right half is the last floor(m/2) integers (which must be the reverse of the left half. (floor(x) is the greatest integer less than or equal to x.) A pa ...
Computability and Incompleteness
... ization of pornography, “it may be hard to define precisely, but I know it when I see it.” Why, then, is such a definition desirable? In 1900 the great mathematician David Hilbert addressed the international congress of mathematicians in Paris, and presented a list of 23 problems that he hoped would ...
... ization of pornography, “it may be hard to define precisely, but I know it when I see it.” Why, then, is such a definition desirable? In 1900 the great mathematician David Hilbert addressed the international congress of mathematicians in Paris, and presented a list of 23 problems that he hoped would ...
Implementable Set Theory and Consistency of ZFC
... Another philosophical note is in place, when we are saying that we ”make with an axiom” and denote this as an implication A =⇒ B. In common mathematics, the implication =⇒ just means what is defined by a truth table in propositional logic. But there is another form of mathematics, called constructiv ...
... Another philosophical note is in place, when we are saying that we ”make with an axiom” and denote this as an implication A =⇒ B. In common mathematics, the implication =⇒ just means what is defined by a truth table in propositional logic. But there is another form of mathematics, called constructiv ...
3463: Mathematical Logic
... (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there is a unique longest sequence σ0 , σ1 , . . . such that σ0 = q0 x, the initial configuration on input x, and for j = 1, 2, . . . if σj is defined then σj−1 ⊢M σj . This longest sequence is bounded if and ...
... (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there is a unique longest sequence σ0 , σ1 , . . . such that σ0 = q0 x, the initial configuration on input x, and for j = 1, 2, . . . if σj is defined then σj−1 ⊢M σj . This longest sequence is bounded if and ...
1. Sets, relations and functions. 1.1. Set theory. We assume the
... is the set A characterized by the property that x ∈ A if and only if x = ai for some i = 1, . . . , n. In particular, for any object a we let singleton a equal {a} and note that x is a member of singleton a if and only if x = a. We let ∅ = {x : x 6= x} and call this set the empty set because it has ...
... is the set A characterized by the property that x ∈ A if and only if x = ai for some i = 1, . . . , n. In particular, for any object a we let singleton a equal {a} and note that x is a member of singleton a if and only if x = a. We let ∅ = {x : x 6= x} and call this set the empty set because it has ...
PowerPoint file for CSL 02, Edinburgh, UK
... The converse of arrows in the hierarchy of semi-classical principles are conjectured not to be derivable in HA. If the scheme S0n–DNE is not derivable from the scheme P0n–LEM, then the conjecture is proved for the n-level. The conjecture have been solved for n=1, 2 levels, which include all of the L ...
... The converse of arrows in the hierarchy of semi-classical principles are conjectured not to be derivable in HA. If the scheme S0n–DNE is not derivable from the scheme P0n–LEM, then the conjecture is proved for the n-level. The conjecture have been solved for n=1, 2 levels, which include all of the L ...
Hierarchical Introspective Logics
... uniquely describing them. (This was the "invariance" problem in the terminology of his paper.) Although there seems to be essentially no confusion about the naming or proper description of finite ordinal levels, beyond that there are serious difficulties that are classically known. Thus mathematical ...
... uniquely describing them. (This was the "invariance" problem in the terminology of his paper.) Although there seems to be essentially no confusion about the naming or proper description of finite ordinal levels, beyond that there are serious difficulties that are classically known. Thus mathematical ...
Recursion
... How many students total are directly behind you in your "column" of the classroom? – You have poor vision, so you can see only the people right next to you. So you can't just look back and count. – But you are allowed to ask questions of the person next to you. ...
... How many students total are directly behind you in your "column" of the classroom? – You have poor vision, so you can see only the people right next to you. So you can't just look back and count. – But you are allowed to ask questions of the person next to you. ...
CS311H: Discrete Mathematics Cardinality of Infinite Sets and
... Since such a sequence is a bijective function from Z+ to A, writing A as a sequence a1 , a2 , a3 , . . . establishes one-to-one ...
... Since such a sequence is a bijective function from Z+ to A, writing A as a sequence a1 , a2 , a3 , . . . establishes one-to-one ...
Theory of Computation
... Example. one example of an infinite language can be the constructed by allows Σ = {a, b, c}. Our language L then consists of all 1, 2, 3, 4.... letter words, and this generates an infinite number of words even though the length of the words has to remain finite. This is denoted Σ∗ . We can also talk ...
... Example. one example of an infinite language can be the constructed by allows Σ = {a, b, c}. Our language L then consists of all 1, 2, 3, 4.... letter words, and this generates an infinite number of words even though the length of the words has to remain finite. This is denoted Σ∗ . We can also talk ...
New York Journal of Mathematics Normality preserving operations for
... on a set of indices of density zero. In [4] he proved that the function σqz is b-normality preserving. It was shown in [2] that C. Aistleitner’s result does not generalize to at least one notion of normality for some of the Cantor series expansions, which we will be investigating in this paper. Ther ...
... on a set of indices of density zero. In [4] he proved that the function σqz is b-normality preserving. It was shown in [2] that C. Aistleitner’s result does not generalize to at least one notion of normality for some of the Cantor series expansions, which we will be investigating in this paper. Ther ...
CARLOS AUGUSTO DI PRISCO The notion of infinite appears in
... The notion of infinite appears in mathematics in many different ways. The notion of limit or endless processes of approximations have been considered since ancient times, but it was in the decade of 1870 that the systematic study of infinite collections as completed totalities was initiated by Georg ...
... The notion of infinite appears in mathematics in many different ways. The notion of limit or endless processes of approximations have been considered since ancient times, but it was in the decade of 1870 that the systematic study of infinite collections as completed totalities was initiated by Georg ...
Strict Predicativity 3
... Of course a ⊕ x is just a ∪{x}, but I have in mind a theory in which ⊕ is primitive. Then {x} is obviously ∅ ⊕ x. Union can be introduced by primitive recursion. ...
... Of course a ⊕ x is just a ∪{x}, but I have in mind a theory in which ⊕ is primitive. Then {x} is obviously ∅ ⊕ x. Union can be introduced by primitive recursion. ...