Annals of Pure and Applied Logic Ordinal machines and admissible
... A standard Turing machine is based on the set ω = {0, 1, . . .} of natural numbers: it acts on a Turing tape of length ω within a discrete time axis which is also indexed by ω. In [5], the first author defined ordinal Turing machines by replacing the set ω of natural numbers by the class Ord of ordi ...
... A standard Turing machine is based on the set ω = {0, 1, . . .} of natural numbers: it acts on a Turing tape of length ω within a discrete time axis which is also indexed by ω. In [5], the first author defined ordinal Turing machines by replacing the set ω of natural numbers by the class Ord of ordi ...
The Art of Ordinal Analysis
... proof, Gentzen used his sequent calculus and employed the technique of cut elimination. As this is a tool of utmost importance in proof theory and ordinal analysis, a rough outline of the underlying ideas will be discussed next. The most common logical calculi are Hilbert-style systems. They are spe ...
... proof, Gentzen used his sequent calculus and employed the technique of cut elimination. As this is a tool of utmost importance in proof theory and ordinal analysis, a rough outline of the underlying ideas will be discussed next. The most common logical calculi are Hilbert-style systems. They are spe ...
[Write on board:
... (1) Show you a different proof that sqrt(2) is irrational which has the virtue that, if you turn it around, gives you a procedure for generating rational numbers that get closer and closer to sqrt(2). (2) Discuss the holes in the rationals, and state an axiom capturing the idea that “there are no ho ...
... (1) Show you a different proof that sqrt(2) is irrational which has the virtue that, if you turn it around, gives you a procedure for generating rational numbers that get closer and closer to sqrt(2). (2) Discuss the holes in the rationals, and state an axiom capturing the idea that “there are no ho ...
Document
... (1) Show you a different proof that sqrt(2) is irrational which has the virtue that, if you turn it around, gives you a procedure for generating rational numbers that get closer and closer to sqrt(2). (2) Discuss the holes in the rationals, and state an axiom capturing the idea that “there are no ho ...
... (1) Show you a different proof that sqrt(2) is irrational which has the virtue that, if you turn it around, gives you a procedure for generating rational numbers that get closer and closer to sqrt(2). (2) Discuss the holes in the rationals, and state an axiom capturing the idea that “there are no ho ...
Rational Numbers
... You have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However, integers are closed under addition, subtraction and multiplication but not under division. (iii) Rational numbers p Recall that a number which can be written in the form q , ...
... You have seen that whole numbers are closed under addition and multiplication but not under subtraction and division. However, integers are closed under addition, subtraction and multiplication but not under division. (iii) Rational numbers p Recall that a number which can be written in the form q , ...
Text (PDF format)
... a) Show that S takes its maximum value over all orderings of the two sequences when both sequences are sorted (so that the elements in each sequence are in nondecreasing order). b) Show that S takes its minimum value over all orderings of the two sequences when one sequence is sorted into nondecreas ...
... a) Show that S takes its maximum value over all orderings of the two sequences when both sequences are sorted (so that the elements in each sequence are in nondecreasing order). b) Show that S takes its minimum value over all orderings of the two sequences when one sequence is sorted into nondecreas ...
pdf format
... Theorem 12 If α and β are ordinals, then either α = β or α < β or β < α. Definition ON is the class of all ordinals. Definition A class C is a collection of sets which can be defined as follows: for some first-order formula ϕ(x, y1, . . ., yk ) and some fixed sets y1 , . . . , yk , C = {x : ϕ(x, y1 ...
... Theorem 12 If α and β are ordinals, then either α = β or α < β or β < α. Definition ON is the class of all ordinals. Definition A class C is a collection of sets which can be defined as follows: for some first-order formula ϕ(x, y1, . . ., yk ) and some fixed sets y1 , . . . , yk , C = {x : ϕ(x, y1 ...
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, can be realized as subfields of the surreals. It has also been shown (in Von Neumann–Bernays–Gödel set theory) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the axiom of global choice, this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all transfinite ordinal numbers; the arithmetic on them is given by the natural operations.In 1907 Hahn introduced Hahn series as a generalization of formal power series, and Hausdorff introduced certain ordered sets called ηα-sets for ordinals α and asked if it was possible to find a compatible ordered group or field structure. In 1962 Alling used a modified form of Hahn series to construct such ordered fields associated to certain ordinals α, and taking α to be the class of all ordinals in his construction gives a class that is an ordered field isomorphic to the surreal numbers.Research on the go endgame by John Horton Conway led to a simpler definition and construction of the surreal numbers. Conway's construction was introduced in Donald Knuth's 1974 book Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. In his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuth's term, and used surreals for analyzing games in his 1976 book On Numbers and Games.