• Study Resource
  • Explore
    • Arts & Humanities
    • Business
    • Engineering & Technology
    • Foreign Language
    • History
    • Math
    • Science
    • Social Science

    Top subcategories

    • Advanced Math
    • Algebra
    • Basic Math
    • Calculus
    • Geometry
    • Linear Algebra
    • Pre-Algebra
    • Pre-Calculus
    • Statistics And Probability
    • Trigonometry
    • other →

    Top subcategories

    • Astronomy
    • Astrophysics
    • Biology
    • Chemistry
    • Earth Science
    • Environmental Science
    • Health Science
    • Physics
    • other →

    Top subcategories

    • Anthropology
    • Law
    • Political Science
    • Psychology
    • Sociology
    • other →

    Top subcategories

    • Accounting
    • Economics
    • Finance
    • Management
    • other →

    Top subcategories

    • Aerospace Engineering
    • Bioengineering
    • Chemical Engineering
    • Civil Engineering
    • Computer Science
    • Electrical Engineering
    • Industrial Engineering
    • Mechanical Engineering
    • Web Design
    • other →

    Top subcategories

    • Architecture
    • Communications
    • English
    • Gender Studies
    • Music
    • Performing Arts
    • Philosophy
    • Religious Studies
    • Writing
    • other →

    Top subcategories

    • Ancient History
    • European History
    • US History
    • World History
    • other →

    Top subcategories

    • Croatian
    • Czech
    • Finnish
    • Greek
    • Hindi
    • Japanese
    • Korean
    • Persian
    • Swedish
    • Turkish
    • other →
 
Profile Documents Logout
Upload
venn diagram review
venn diagram review

Bridge to Abstract Mathematics: Mathematical Proof and
Bridge to Abstract Mathematics: Mathematical Proof and

... techniques, including induction, indirect proof, specialization, division into cases, and counterexample, are also studied. Solved examples and exercises calling for the writing of proofs are selected from set theory, intermediate algebra, trigonometry, elementary calculus, matrix algebra, and eleme ...
KURT GÖDEL - National Academy of Sciences
KURT GÖDEL - National Academy of Sciences

HOW TO DEFINE A MEREOLOGICAL (COLLECTIVE) SET
HOW TO DEFINE A MEREOLOGICAL (COLLECTIVE) SET

... or what are its essential features. First, we can see that every single one of objects a, b, c and d is part of x. Second, whatever part of x we take (any its fragment), it overlaps one of the four objects in question. On the other hand we can notice as well that any object which is exterior to ever ...
Artificial Intelligence
Artificial Intelligence

Completeness or Incompleteness of Basic Mathematical Concepts
Completeness or Incompleteness of Basic Mathematical Concepts

... Whether I am right or wrong about Gödel’s use, I will always use “implied be the concept” in a non-epistemic sense.56 (4) The concept of the natural numbers is first-order complete: it determines truth values for all sentences of first-order arithmetic. That is, it implies each first-order sentence ...
week6-recursion
week6-recursion

... recursive call gets closer to a base case. • Recursion can always be used instead of a loop. (This is a mathematical fact.) In declarative programming languages, like Prolog, there are no loops. There is only recursion. • Recursion is elegant and sometimes very handy, but it is marginally less effic ...
Incompleteness
Incompleteness

... Note that the definition of a language is purely “syntactical,” that is, no meaning is in anyway assigned to these symbols of the language (e.g., there is nothing that says ` somehow really represents what we think of a addition on the natural numbers, whatever that means). For any first-order langu ...
Proof, Sets, and Logic - Boise State University
Proof, Sets, and Logic - Boise State University

... Some comments: I may want to do some forcing and some FrankelMostowski methods in type theory. It is useful to be able to do basic consistency and independence results in the basic foundational theory, and it may make it easier to follow the development in NFU or NF. Definitely introduce TNTU (and T ...
Proof, Sets, and Logic - Department of Mathematics
Proof, Sets, and Logic - Department of Mathematics

... Added material on forcing in type theory. Note that I have basic constructions for consistency and independence set up in the outline in to be in the first instance in type theory rather than in either untyped or stratified theories. My foundation for mathematics as far as possible is in TSTU here, ...
Comparing sizes of sets
Comparing sizes of sets

... Sets A and B are the same size if there is a bijection from A to B. (That was a definition!) For finite sets A, B, it is not difficult to verify that there is a bijection from A to B iff |A| = |B|. Let’s do it. . . Take arbitrary finite sets A and B. LR: Assume f : A → B is bijective. Then f is inje ...
slides - University of Edinburgh
slides - University of Edinburgh

... Now, the number is getting larger - what's going on? The point is that the DIFFERENCE between m and n is getting smaller. It's essential that something gets smaller - otherwise the recursion will never terminate. ...
Recursive call to factorial(1)
Recursive call to factorial(1)

... The loop computes each Fibonacci number by starting at 2 and working its way upward Clearly, the number of iterations is bounded above by n The amount of space required is constant ...
REVERSE MATHEMATICS, WELL-QUASI
REVERSE MATHEMATICS, WELL-QUASI

... A(Pf[ (Q)) and U(Pf[ (Q)), this is because if Q is a wqo, then Pf[ (Q) is also a wqo (the U(Pf[ (Q)) case also follows from Theorem 1.3). The U(P ] (Q)) case follows from Theorem 1.3 because for every A ∈ P(Q) there is a B ∈ Pf (Q) that is equivalent to A in the sense that A ≤]Q B and B ≤]Q A. Notic ...
Incompleteness in the finite domain
Incompleteness in the finite domain

... tures without any formal supporting evidence. There are, essentially, two reasons for stating some sentences as conjectures. First, we believe that some basic theorems of proof theory should also hold true with suitable bounds on the lengths of proofs. The prime example is the Second Incompleteness ...
Incompleteness in the finite domain
Incompleteness in the finite domain

... and bounded arithmetic seem to follow a general pattern. For example, as we noted above, polynomial time computations are associated with the theory S21 by a witnessing theorem. If we take S22 , which we believe is a stronger theory, then the corresponding function class is PNP ,2 which we believe i ...
Gödel`s Theorems
Gödel`s Theorems

... by the familiar rules. But what is the real content of the thought that the truthvalues of all basic arithmetic propositions are thereby ‘fixed’ ? Here’s one initially attractive way of giving non-metaphorical content to that thought. The idea is that we can specify a bundle of fundamental assumptio ...
Recursion
Recursion

Foundations of Computation - Department of Mathematics and
Foundations of Computation - Department of Mathematics and

... When these operators are used in expressions, in the absence of parentheses to indicate order of evaluation, we use the following precedence rules: The exclusive or operator, ⊕, has the same precedence as ∨. The conditional operator, →, has lower precedence than ∧, ∨, ¬, and ⊕, and is therefore eval ...
lecture notes in logic - UCLA Department of Mathematics
lecture notes in logic - UCLA Department of Mathematics

... “singleton operation”, i.e., for every x, x ∈ z =⇒ {x} ∈ z. (7) Choice: for every set x whose members are all non-empty and pairwise disjoint, there exists a set z which intersects each member of x in exactly one point, i.e., if y ∈ x, then there exists exactly one u such that u ∈ y and also u ∈ z. ...
Gödel Without (Too Many) Tears
Gödel Without (Too Many) Tears

... But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effective decidability. So let’s say: Defn. 1 A property P (defined over some domain of objects D) is effectively decidable iff there’s an algorithm (a finite set of instructions for a det ...
you can this version here
you can this version here

... But to explain what we mean here, we first need to take some steps towards pinning down the intuitive notion of effective decidability. So let’s say: Defn. 1. A property P (defined over some domain of objects D) is effectively decidable iff there’s an algorithm (a finite set of instructions for a de ...
MARTIN`S CONJECTURE, ARITHMETIC EQUIVALENCE, AND
MARTIN`S CONJECTURE, ARITHMETIC EQUIVALENCE, AND

... restricted to the class of uniformly Turing invariant functions. Theorem 1.2 (Slaman and Steel [25]). Part I of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem 1.3 (Steel [26]). Part II of Martin’s conjecture holds for all uniformly Turing invariant functions. Theorem ...
Document
Document

... that specifies at least one member of the set, and a recursive part that specifies how additional members of the set can be generated in terms of given members. • The prototypical example is ℕ, the set of natural numbers. It can be defined as follows: – Basis statement: 0  ℕ – Recursive part: if n ...
Document
Document

... that specifies at least one member of the set, and a recursive part that specifies how additional members of the set can be generated in terms of given members. • The prototypical example is ℕ, the set of natural numbers. It can be defined as follows: – Basis statement: 0  ℕ – Recursive part: if n ...
1 2 3 4 5 ... 17 >

Computability theory

Computability theory, also called recursion theory, is a branch of mathematical logic, of computer science, and of the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees.The basic questions addressed by recursion theory are ""What does it mean for a function on the natural numbers to be computable?"" and ""How can noncomputable functions be classified into a hierarchy based on their level of noncomputability?"". The answers to these questions have led to a rich theory that is still being actively researched. The field has since grown to include the study of generalized computability and definability. Invention of the central combinatorial object of recursion theory, namely the Universal Turing Machine, predates and predetermines the invention of modern computers. Historically, the study of algorithmically undecidable sets and functions was motivated by various problems in mathematics that turned to be undecidable; for example, word problem for groups and the like. There are several applications of the theory to other branches of mathematics that do not necessarily concentrate on undecidability. The early applications include the celebrated Higman's embedding theorem that provides a link between recursion theory and group theory, results of Michael O. Rabin and Anatoly Maltsev on algorithmic presentations of algebras, and the negative solution to Hilbert's Tenth Problem. The more recent applications include algorithmic randomness, results of Soare et al. who applied recursion-theoretic methods to solve a problem in algebraic geometry, and the very recent work of Slaman et al. on normal numbers that solves a problem in analytic number theory.Recursion theory overlaps with proof theory, effective descriptive set theory, model theory, and abstract algebra.Arguably, computational complexity theory is a child of recursion theory; both theories share the same technical tool, namely the Turing Machine. Recursion theorists in mathematical logic often study the theory of relative computability, reducibility notions and degree structures described in this article. This contrasts with the theory of subrecursive hierarchies, formal methods and formal languages that is common in the study of computability theory in computer science. There is a considerable overlap in knowledge and methods between these two research communities; however, no firm line can be drawn between them. For instance, parametrized complexity was invented by a complexity theorist Michael Fellows and a recursion theorist Rod Downey.
  • studyres.com © 2025
  • DMCA
  • Privacy
  • Terms
  • Report