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 ...
... 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
... 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 ...
... 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: 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. ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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 ...
SECTION B Subsets
... We can use this definition to prove two sets, A and B, are equal, A B , by showing A B and B A . This means that sets A and B are equal if and only if A is a subset of B and B is a subset of A. This is similar to proving 2 numbers are equal. To show that numbers a and b are equal we show a b ...
... We can use this definition to prove two sets, A and B, are equal, A B , by showing A B and B A . This means that sets A and B are equal if and only if A is a subset of B and B is a subset of A. This is similar to proving 2 numbers are equal. To show that numbers a and b are equal we show a b ...
CS1231 - Lecture 09
... – When you think of the number of elements in the set, you tend to think of ‘1,2,3,…’, and if you can’t give a number, you just say ‘infinite’. – DO NOT BRING this restricted idea of ‘number’ in here. – A Cardinal Number is a descriptive numerical object, generalized to include ALL SORTS OF SETS – F ...
... – When you think of the number of elements in the set, you tend to think of ‘1,2,3,…’, and if you can’t give a number, you just say ‘infinite’. – DO NOT BRING this restricted idea of ‘number’ in here. – A Cardinal Number is a descriptive numerical object, generalized to include ALL SORTS OF SETS – F ...
God, the Devil, and Gödel
... a mind: that the mind can outstrip any machine in deductive process. We must take care here, for there are some trivial ways in which the thesis could be false, and Lucas is well aware of many of these and does not intend it in these ways. For example, it is clear that present digital computers can ...
... a mind: that the mind can outstrip any machine in deductive process. We must take care here, for there are some trivial ways in which the thesis could be false, and Lucas is well aware of many of these and does not intend it in these ways. For example, it is clear that present digital computers can ...
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 ...
... 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
... 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 ...
... 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
... (see [2]), whose proclaimed objective was to establish the consistency of mathematics (analysis and set theory) by using finitary tools. This problem was regarded by the representatives of Hilbert’s school as the central problem of mathematical logic. However, it follows from Gödel’s second theorem ...
... (see [2]), whose proclaimed objective was to establish the consistency of mathematics (analysis and set theory) by using finitary tools. This problem was regarded by the representatives of Hilbert’s school as the central problem of mathematical logic. However, it follows from Gödel’s second theorem ...
HKT Chapters 1 3
... The order ≤ is well-defined on ≡-equivalence classes; that is, if a ≤ b, a ≡ a , and b ≡ b , then a ≤ b . It therefore makes sense to define [a] ≤ [b] if a ≤ b. The resulting order on ≡-classes is a partial order. A strict partial order is a binary relation < that is irreflexive and transitive. Any ...
... The order ≤ is well-defined on ≡-equivalence classes; that is, if a ≤ b, a ≡ a , and b ≡ b , then a ≤ b . It therefore makes sense to define [a] ≤ [b] if a ≤ b. The resulting order on ≡-classes is a partial order. A strict partial order is a binary relation < that is irreflexive and transitive. Any ...
Advanced Logic —
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... It is the most general method of repetitive computation. Introduces functional programming style. ...
... It is the most general method of repetitive computation. Introduces functional programming style. ...
Supplemental Reading (Kunen)
... reader to a text on the subject, such as [Enderton 19723, [Kleene 19521, or [Shoenfield 19671, for a more detailed treatment. We shall give a precise definition of the formal language, as this is easy to do and is necessary for stating the axioms of ZFC. We shall only hint at the rules of formal ded ...
... reader to a text on the subject, such as [Enderton 19723, [Kleene 19521, or [Shoenfield 19671, for a more detailed treatment. We shall give a precise definition of the formal language, as this is easy to do and is necessary for stating the axioms of ZFC. We shall only hint at the rules of formal ded ...
Recurrent points and hyperarithmetic sets
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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
... 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 ...
... 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 ...
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 ...
... 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
... 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 ...
... 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 ...