Lecture notes from 5860
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
We can only see a short distance ahead, but we can see plenty
... that logic is to computer science what mathematics is to physics, and vice versa. Logic serves as the language and foundation for computer science as mathematics does for physics. Conversely, computer science supplies major problems and new arenas for logical analysis as physics does for mathematics ...
... that logic is to computer science what mathematics is to physics, and vice versa. Logic serves as the language and foundation for computer science as mathematics does for physics. Conversely, computer science supplies major problems and new arenas for logical analysis as physics does for mathematics ...
Real-time computability of real numbers by chemical
... case of deterministic CRNs, Stansifer has reportedly proven [5] that this model is Turing universal, i.e., that every algorithm can be simulated by a deterministic CRN. (Note: The title of [17] seems to make this assertion, but the paper only exhibits a way to use deterministic CRNs to simulate fini ...
... case of deterministic CRNs, Stansifer has reportedly proven [5] that this model is Turing universal, i.e., that every algorithm can be simulated by a deterministic CRN. (Note: The title of [17] seems to make this assertion, but the paper only exhibits a way to use deterministic CRNs to simulate fini ...
Variations on a Montagovian Theme
... object. The subject is the person who knows or believes; the object is that which is known or believed. But what kind of object is this? Two answers have been popular in the more systematic branches of epistemology and philosophy of mind. The first identifies objects of attitudes with something like ...
... object. The subject is the person who knows or believes; the object is that which is known or believed. But what kind of object is this? Two answers have been popular in the more systematic branches of epistemology and philosophy of mind. The first identifies objects of attitudes with something like ...
Discrete Mathematics, Chapter 5: Induction and Recursion
... The basis step specifies an initial collection of elements. The recursive step gives the rules for forming new elements in the set from those already known to be in the set. Sometimes the recursive definition has an exclusion rule, which specifies that the set contains nothing other than those eleme ...
... The basis step specifies an initial collection of elements. The recursive step gives the rules for forming new elements in the set from those already known to be in the set. Sometimes the recursive definition has an exclusion rule, which specifies that the set contains nothing other than those eleme ...
Slide 1
... ● There is always a next configuration of M and thus a next row in the tiling iff M does not halt. ● T is in TILES iff there is always a next row. ● So if it were possible to semidecide whether T is in TILES it would be possible to semidecide whether M fails to halt on . But H is not in SD. So ne ...
... ● There is always a next configuration of M and thus a next row in the tiling iff M does not halt. ● T is in TILES iff there is always a next row. ● So if it were possible to semidecide whether T is in TILES it would be possible to semidecide whether M fails to halt on . But H is not in SD. So ne ...
Difficulties in Factoring a Number: Prime Numbers
... captured by “mapping reduction”, e.g. ATM and ATM seem to be reducible to one another (a solution to either one could be used to solve the other by simply reversing the answer). However, ATM is not mapping reducible to ATM because it is not Turingrecognizable (find a solution to map each unaccepta ...
... captured by “mapping reduction”, e.g. ATM and ATM seem to be reducible to one another (a solution to either one could be used to solve the other by simply reversing the answer). However, ATM is not mapping reducible to ATM because it is not Turingrecognizable (find a solution to map each unaccepta ...
Document
... The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from P Q and ~ ...
... The significance of Russell's paradox can be seen once it is realized that, using classical logic, all sentences follow from a contradiction. For example, assuming both P and ~P, any arbitrary proposition, Q, can be proved as follows: from P we obtain P Q by the rule of Addition; then from P Q and ~ ...
Gödel on Conceptual Realism and Mathematical Intuition
... our knowledge, we may discover that there are actually two different concepts involved. Buzaglo adds that whenever there are several forced expansions, one of them is more natural than the others. On Gödel’s reading, the uniqueness of an expansion can be explained only if we assume that the outcome ...
... our knowledge, we may discover that there are actually two different concepts involved. Buzaglo adds that whenever there are several forced expansions, one of them is more natural than the others. On Gödel’s reading, the uniqueness of an expansion can be explained only if we assume that the outcome ...
Appendix A Sets, Relations and Functions
... The elements of a set are also called its members. To indicate that a is an element of a set A we write a ∈ A. To deny that a is an element of a set A we write a ∈ / A. The symbol ∈ is the symbol for membership. The elements of a set can be anything: words, colours, people, numbers. The elements of ...
... The elements of a set are also called its members. To indicate that a is an element of a set A we write a ∈ A. To deny that a is an element of a set A we write a ∈ / A. The symbol ∈ is the symbol for membership. The elements of a set can be anything: words, colours, people, numbers. The elements of ...
LOWNESS NOTIONS, MEASURE AND DOMINATION
... a set X with the infinite string given by its characteristic function. For X ⊆ ω and s ∈ ω, X[s] denotes the string hX(0), X(1), . . . , X(s − 1)i. For Y ⊆ 2<ω , [Y ] denotes the open class in 2ω of all X such that ∃σ ∈ Y (σ v X). If Z ⊆ 2ω , then Z c = 2ω \ Z. Finally, if M is any machine (viewed a ...
... a set X with the infinite string given by its characteristic function. For X ⊆ ω and s ∈ ω, X[s] denotes the string hX(0), X(1), . . . , X(s − 1)i. For Y ⊆ 2<ω , [Y ] denotes the open class in 2ω of all X such that ∃σ ∈ Y (σ v X). If Z ⊆ 2ω , then Z c = 2ω \ Z. Finally, if M is any machine (viewed a ...
Primitive Recursive Arithmetic and its Role in the Foundations of
... preferable to accept the notion of function as sui generis, to interpret A → B simply as the domain of functions from A to B, and to understand equations between objects of such a type to mean equality in the usual sense of extensional equality of functions. What makes T constructive is not that it ...
... preferable to accept the notion of function as sui generis, to interpret A → B simply as the domain of functions from A to B, and to understand equations between objects of such a type to mean equality in the usual sense of extensional equality of functions. What makes T constructive is not that it ...
NONSTANDARD MODELS IN RECURSION THEORY
... one hand, measuring the inductive strengths of recursion-theoretic theorems was a program with foundational flavour grounded in the overall objective of reverse mathematics. On the other hand, a number of important notions and techniques it uses have their origins in α-recursion theory. For instance ...
... one hand, measuring the inductive strengths of recursion-theoretic theorems was a program with foundational flavour grounded in the overall objective of reverse mathematics. On the other hand, a number of important notions and techniques it uses have their origins in α-recursion theory. For instance ...
Sets, Logic, Computation
... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...
... the relations that make up a first-order structure are described— characterized—by the sentences that are true in them. This in particular leads us to a discussion of the axiomatic method, in which sentences of first-order languages are used to characterize certain kinds of structures. Proof theory ...
Mathematical Structures for Reachability Sets and Relations Summary
... Hopcroft and Pansiot (1979) proved that the reachability sets of any VASS of dimension 2 are effectively definable in Presburger arithmetic, whereas there exist VASS of dimension 3 with reachability sets that are not definable in this logic. Hence, characterizing VASS reachability sets by formulæ in ...
... Hopcroft and Pansiot (1979) proved that the reachability sets of any VASS of dimension 2 are effectively definable in Presburger arithmetic, whereas there exist VASS of dimension 3 with reachability sets that are not definable in this logic. Hence, characterizing VASS reachability sets by formulæ in ...
On fuzzy semi-preopen sets and fuzzy semi
... studied in fuzzy topological spaces. In [7], the notions of fuzzy semi-preopen sets, fuzzy semi-precontinuous mappings and fuzzy semi-preopen mappings etc. are given according to the sense of Chang-Goguen spaces. By standard terminology in [5], these are corresponding with the following I-topologica ...
... studied in fuzzy topological spaces. In [7], the notions of fuzzy semi-preopen sets, fuzzy semi-precontinuous mappings and fuzzy semi-preopen mappings etc. are given according to the sense of Chang-Goguen spaces. By standard terminology in [5], these are corresponding with the following I-topologica ...
Section 2.2 Subsets
... • Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number. • Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number א0 read “aleph-null”. • Equal sets: Set A is equal t ...
... • Finite set: Set A is a finite set if n(A) = 0 ( that is, A is the empty set) or n(A) is a natural number. • Infinite set: A set whose cardinality is not 0 or a natural number. The set of natural numbers is assigned the infinite cardinal number א0 read “aleph-null”. • Equal sets: Set A is equal t ...
Chapter 5 Cardinality of sets
... As an aside, the vertical bars, | · |, are used throughout mathematics to denote some measure of size. For example, the absolute value of a real number measures its size in terms of how far it is from zero on the number line. According to the definition, set has cardinality n when there is a sequenc ...
... As an aside, the vertical bars, | · |, are used throughout mathematics to denote some measure of size. For example, the absolute value of a real number measures its size in terms of how far it is from zero on the number line. According to the definition, set has cardinality n when there is a sequenc ...
Chapter 1 Number Sets and Properties
... that x is greater than or equal to 0 or less than or equal to 5.” • It is important to note that this example lacks precision as it does not specify what sort of numbers the set elements should be (integers, rationals, etc.)? ...
... that x is greater than or equal to 0 or less than or equal to 5.” • It is important to note that this example lacks precision as it does not specify what sort of numbers the set elements should be (integers, rationals, etc.)? ...
BENIGN COST FUNCTIONS AND LOWNESS PROPERTIES 1
... and classes of degrees, most of which are known to be contained in the K-trivial degrees but are not yet known to be distinct from the ideal of Ktrivial degrees? Here we mostly think of classes derived from algorithmic randomess, such as the collection of degrees which are bounded by an incomplete r ...
... and classes of degrees, most of which are known to be contained in the K-trivial degrees but are not yet known to be distinct from the ideal of Ktrivial degrees? Here we mostly think of classes derived from algorithmic randomess, such as the collection of degrees which are bounded by an incomplete r ...
