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Infinity 1. Introduction
Infinity 1. Introduction

... many facts about the world in a single statement. Quantifiers were invented by Frege in 1879 and Peirce and Mitchell in 1883. Peirce and Mitchell devised the notation Σ i xi , meaning that xi is true for some value of i, and Π i xi , meaning that xi is true for all values of i, by a conscious analog ...
mathematical induction
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... immutable law: No disk may be placed on a smaller disk. In the beginning of the world all 64 disks formed the Tower of Brahma on one needle. Now, however, the process of transfer of the tower from one needle to another is in mid course. When the last disk is finally in place, once again forming the ...
Recursion - Damian Gordon
Recursion - Damian Gordon

... Recursion: Decimal to Binary Conversion MODULE DecToBin(N): IF N == 0 THEN RETURN ‘0’; ELSE RETURN DecToBin(N DIV 2) + String(N MOD 2); ENDIF; END. PROGRAM DecimalToBinary: READ A; PRINT DecToBin(A); END. ...
SOME AXIOMS FOR CONSTRUCTIVE ANALYSIS Introduction
SOME AXIOMS FOR CONSTRUCTIVE ANALYSIS Introduction

... functions, symbols for λ-abstraction and function application, and a finite list of mathematical constants. When needed, additional constants could be added to the list, as for the precise formal treatment of recursive functionals in [11]. Troelstra [17] formalized intuitionistic arithmetic HA in a ...
Proof Pearl: Defining Functions over Finite Sets
Proof Pearl: Defining Functions over Finite Sets

... We pay particular attention to the algebraic properties required for iterating a function over a set. It turns out that there are two distinct cases: commutative monoids with a unit (useful for defining summation) and ordered structures (useful for defining minimum). Both require distinct fold functio ...
Complexity of Recursive Normal Default Logic 1. Introduction
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... propositional logic nonmonotonic formalisms, the basic results are found in [BF91, MT91, Got92, Van89]. In case of formalisms admitting variables and, more generally, infinite recursive propositional nonmonotonic formalisms, a number of results has been found. These include basic complexity results ...
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God, the Devil, and Gödel
God, the Devil, and Gödel

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Sets, Infinity, and Mappings - University of Southern California
Sets, Infinity, and Mappings - University of Southern California

... The chain of sets given in (2) shows there are infinitely many levels of infinity. That is, there are infinitely many different infinite cardinalities. Of course, this iterative method can only demonstrate existence of a countably infinite number of levels of infinity. This leads one to wonder if th ...
REVERSE MATHEMATICS AND RECURSIVE GRAPH THEORY
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... Now we will consider theorems on the existence of Hamilton paths. A path through a graph G is called a (one way) Hamilton path if it uses every vertex of G exactly once. There is no arithmetical analog of the characterization “pre-Eulerian” for graphs containing Hamilton paths. Consequently, all the ...
Gödel incompleteness theorems and the limits of their applicability. I
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HKT Chapters 1 3
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... P ROOF. Suppose IND is true, but for a reductio, suppose that LNP is not. Then there is some subset B ⊆ N such that: (1) B is not empty; and (2) B has no least element. Let A = N\B. Clearly 0 ∈ / B (for then 0 would be the least element), so 0 ∈ A. Moreover, since B has no least element, this just m ...
How complicated is the set of stable models of a recursive logic
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... We can rephrase the basic problem that we wish to study as: “What types of objects are described by a logic program?”. The first result in this direction is the classical result implicit in ([Smullyan, 1961]) and, more recently, reproved by Andreka and Nemeti ([Andreka and Nemeti, 1978]). This can b ...
Internal Inconsistency and the Reform of Naïve Set Comprehension
Internal Inconsistency and the Reform of Naïve Set Comprehension

... Set theory has long been instrumental to our ideas of formal logic and mathematics, yet no totally satisfying responses have been given to the following concerns: (1) What is at the root of the logical antinomies? (2) What set descriptions (or their predicative functions) must be excluded from the r ...
ppt - University of Birmingham
ppt - University of Birmingham

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Supplemental Reading (Kunen)
Supplemental Reading (Kunen)

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Recurrent points and hyperarithmetic sets
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... 1·3 REMARK The above use of AC could be reduced to an application of DC by working in L[a, f ] and appealing to Shoenfield’s absoluteness theorem. We may use the following lemma since in a metric space second countability and separability are equivalent conditions. 1·4 LEMMA (AC) In a second countab ...
Chapter 12 - Arms-A
Chapter 12 - Arms-A

... test a positive numeric quantity that is decreased on each recursion, and to provide a stopping case for some small value. In the example write_vertical, the parameter value is the quantity mentioned above, and the "small value" is 10. General Form of a Recursive Function Definition The general outl ...
EXHAUSTIBLE SETS IN HIGHER-TYPE
EXHAUSTIBLE SETS IN HIGHER-TYPE

... formulated by giving the algorithm the type (D → B) → B, where D is a domain, K ⊆ D, and B is the domain of booleans. The main question investigated in this work is what kinds of infinite sets are exhaustible. Clearly, finite sets of computable elements are exhaustible. What may be rather unclear is ...
EXHAUSTIBLE SETS IN HIGHER
EXHAUSTIBLE SETS IN HIGHER

... formulated by giving the algorithm the type (D → B) → B, where D is a domain, K ⊆ D, and B is the domain of booleans. The main question investigated in this work is what kinds of infinite sets are exhaustible. Clearly, finite sets of computable elements are exhaustible. What may be rather unclear is ...
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Enumerations in computable structure theory
Enumerations in computable structure theory

... is Σ0α (A) if ϕ(x) is computable Σα , and Π0α (A) if ϕ(x) is computable Πα . Moreover, this holds with all imaginable uniformity, over structures and formulas. It is easy to see that if A has a formally c.e. Scott family, then it is relatively computably categorical, so it is computably categorical ...
connections to higher type Recursion Theory, Proof-Theory
connections to higher type Recursion Theory, Proof-Theory

... the first is to look inside EN, the other is to extend EN in order to to get Cartesian Closure without loosing the simplicity of this category. Scott and Ershov suggested a way to stay inside EN. As we want the HPEF to satisfy property (4) , this is also what we are looking for. Observe that Scott's ...
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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.
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