Infinite numbers: what are they and what are they good for?
... Theorem (Kirby and Paris 1982). Goodstein’s theorem cannot be proved (or disproved) in Peano arithmetic. This is a theorem of mathematical logic, proved using mathematical models of mathematical reasoning (under the assumption that standard set theory is consistent). We conclude that Peano’s axioms ...
... Theorem (Kirby and Paris 1982). Goodstein’s theorem cannot be proved (or disproved) in Peano arithmetic. This is a theorem of mathematical logic, proved using mathematical models of mathematical reasoning (under the assumption that standard set theory is consistent). We conclude that Peano’s axioms ...
Supplemental Reading (Kunen)
... set theory they use. It is generally understood which principles are correct beyond any doubt, and which are subject to question. For example, it is generally agreed that the Continuum Hypothesis (CH) is not a basic principle, but rather an open conjecture, and we are all able, without the benefit o ...
... set theory they use. It is generally understood which principles are correct beyond any doubt, and which are subject to question. For example, it is generally agreed that the Continuum Hypothesis (CH) is not a basic principle, but rather an open conjecture, and we are all able, without the benefit o ...
1 Chapter III Set Theory as a Theory of First Order Predicate Logic
... are generated by different sets of Urelements there is one that is special. This is the hierachy which results when we start with nothing, so to speak, i.e. when we begin with the empty set. It may not be immediately obvious that this will get us anything at all, but only a little reflection shows t ...
... are generated by different sets of Urelements there is one that is special. This is the hierachy which results when we start with nothing, so to speak, i.e. when we begin with the empty set. It may not be immediately obvious that this will get us anything at all, but only a little reflection shows t ...
Prolog arithmetic
... Each operator has a precedence value associated with it. Precedence values are used to decide which operator is carried out first. In Prolog, multiplication and division have higher precedence values than addition and subtraction. ...
... Each operator has a precedence value associated with it. Precedence values are used to decide which operator is carried out first. In Prolog, multiplication and division have higher precedence values than addition and subtraction. ...
Infinity - Tom Davis
... Before we plunge into what it means to “count” an infinite number of objects, let’s take a quick review of what it means to count a finite number of objects. What does it mean when you say, “This set contains 7 objects”? The starting point is usually to begin by saying what it means for two sets to ...
... Before we plunge into what it means to “count” an infinite number of objects, let’s take a quick review of what it means to count a finite number of objects. What does it mean when you say, “This set contains 7 objects”? The starting point is usually to begin by saying what it means for two sets to ...
Difficulties of the set of natural numbers
... The notation ‘+’ in above definition represents the successor operator which can be applied to any set to obtain its successor. In set theory the first natural number 0 is defined with the empty set φ, then number 1 with the successor of 0 and so on. To make the expression more intuitively we usuall ...
... The notation ‘+’ in above definition represents the successor operator which can be applied to any set to obtain its successor. In set theory the first natural number 0 is defined with the empty set φ, then number 1 with the successor of 0 and so on. To make the expression more intuitively we usuall ...
Big Numbers
... Obviously there is no limit; whatever number you name, a larger one can be obtained simply by adding one. But what is interesting is just how large a number can be expressed in a fairly compact way. Using the standard arithmetic operations (addition, subtraction, multiplication, division and exponen ...
... Obviously there is no limit; whatever number you name, a larger one can be obtained simply by adding one. But what is interesting is just how large a number can be expressed in a fairly compact way. Using the standard arithmetic operations (addition, subtraction, multiplication, division and exponen ...
Revised Version 070216
... since we add odd to odd and even to even. Even though we will always have to deal with one ungrouped number, this sum is still even and a natural number. We can use the same technique used above to form a general view of what happens when we add the first n natural numbers. The general case is also ...
... since we add odd to odd and even to even. Even though we will always have to deal with one ungrouped number, this sum is still even and a natural number. We can use the same technique used above to form a general view of what happens when we add the first n natural numbers. The general case is also ...
HERE - University of Georgia
... since we add odd to odd and even to even. Even though we will always have to deal with one ungrouped number, this sum is still even and a natural number. We can use the same technique used above to form a general view of what happens when we add the first n natural numbers. The general case is also ...
... since we add odd to odd and even to even. Even though we will always have to deal with one ungrouped number, this sum is still even and a natural number. We can use the same technique used above to form a general view of what happens when we add the first n natural numbers. The general case is also ...
Big Numbers - Our Programs
... Obviously there is no limit; whatever number you name, a larger one can be obtained simply by adding one. But what is interesting is just how large a number can be expressed in a fairly compact way. Using the standard arithmetic operations (addition, subtraction, multiplication, division and exponen ...
... Obviously there is no limit; whatever number you name, a larger one can be obtained simply by adding one. But what is interesting is just how large a number can be expressed in a fairly compact way. Using the standard arithmetic operations (addition, subtraction, multiplication, division and exponen ...
Discussion
... using the both the commutative and associative laws of addition, could be to change the order and groupings of the numbers. In our example, the first grouping could be the largest number with the smallest number (i.e. 1 + 16), next grouping the second largest number with the second smallest number ( ...
... using the both the commutative and associative laws of addition, could be to change the order and groupings of the numbers. In our example, the first grouping could be the largest number with the smallest number (i.e. 1 + 16), next grouping the second largest number with the second smallest number ( ...
pdf format
... Definition The sum of two ordinals is defined using transfinite induction as: 1. α + 0 = α. 2. α + S(β) = S(α + β). 3. α + β = sup{α + γ : γ < β} for β a limit ordinal. Ordinal addition can also be characterized in terms of concatenation of well-orderings. Let (X, <) and (Y, <0 ) be two well-ordered ...
... Definition The sum of two ordinals is defined using transfinite induction as: 1. α + 0 = α. 2. α + S(β) = S(α + β). 3. α + β = sup{α + γ : γ < β} for β a limit ordinal. Ordinal addition can also be characterized in terms of concatenation of well-orderings. Let (X, <) and (Y, <0 ) be two well-ordered ...
Lecture 5 MATH1904 • Disjoint union If the sets A and B have no
... Cartesian product is the set A × B of all ordered pairs (x, y), where x is an element of A and y is an element of B. That is, A × B = { z | z = (x, y), x ∈ A and y ∈ B }. The size of A × B is |A| × |B|. This can ...
... Cartesian product is the set A × B of all ordered pairs (x, y), where x is an element of A and y is an element of B. That is, A × B = { z | z = (x, y), x ∈ A and y ∈ B }. The size of A × B is |A| × |B|. This can ...
The Art of Ordinal Analysis
... presented as atomic intuitionistic sequents (also called Horn clauses), yielding the completeness of Robinsons resolution method. Partial cut elimination also pays off in the case of fragments of PA and set theory with restricted induction schemes, be it induction on natural numbers or sets. This me ...
... presented as atomic intuitionistic sequents (also called Horn clauses), yielding the completeness of Robinsons resolution method. Partial cut elimination also pays off in the case of fragments of PA and set theory with restricted induction schemes, be it induction on natural numbers or sets. This me ...
Annals of Pure and Applied Logic Ordinal machines and admissible
... In this article we show that α -recursion does indeed correspond naturally to computations by (abstract) machines: recursive enumerability and recursiveness in admissible recursion theory is equivalent to enumerability and computability by certain Turing machines working on ordinals. This was proved ...
... In this article we show that α -recursion does indeed correspond naturally to computations by (abstract) machines: recursive enumerability and recursiveness in admissible recursion theory is equivalent to enumerability and computability by certain Turing machines working on ordinals. This was proved ...
A Finite Model Theorem for the Propositional µ-Calculus
... where σ is either µ or ν. CL(p) is finite, and is in fact no larger than the number of symbols of p [2]. ...
... where σ is either µ or ν. CL(p) is finite, and is in fact no larger than the number of symbols of p [2]. ...
Kripke Models of Transfinite Provability Logic
... Λ validates all frame conditions except for condition (ii). We shall only approximate it in that we require, for ζ < ξ, v <ξ w ⇒ ∃ v 0 <ζ w such that v 0 -p v. Here p will be a set of parameters and u0 -p u denotes that u0 is p-bisimilar to u. The parameters p can be adjusted depending on φ in order ...
... Λ validates all frame conditions except for condition (ii). We shall only approximate it in that we require, for ζ < ξ, v <ξ w ⇒ ∃ v 0 <ζ w such that v 0 -p v. Here p will be a set of parameters and u0 -p u denotes that u0 is p-bisimilar to u. The parameters p can be adjusted depending on φ in order ...
Peano and Heyting Arithmetic
... π(σ _ hni) for actual sequences σ and natural numbers n. But this isn’t enough to give the last two clauses, because HA can’t actually prove that the numerals n are the only numbers. So the last two clauses say that HA can actually prove that φ_ represents a well-defined function. (For instance, the ...
... π(σ _ hni) for actual sequences σ and natural numbers n. But this isn’t enough to give the last two clauses, because HA can’t actually prove that the numerals n are the only numbers. So the last two clauses say that HA can actually prove that φ_ represents a well-defined function. (For instance, the ...
Hierarchical Introspective Logics
... and to be able to deal with "number theory", "functional calculus", and "set theory". And we prefer the viewpoint of Zermelo, that sets can be based on ur-elements which are not themselves sets. Also we like to think of ordinal numbers as being basically a special type of mathematical objects such ...
... and to be able to deal with "number theory", "functional calculus", and "set theory". And we prefer the viewpoint of Zermelo, that sets can be based on ur-elements which are not themselves sets. Also we like to think of ordinal numbers as being basically a special type of mathematical objects such ...
Ordinal Arithmetic
... Hint: You can define a bijection by transfinite recursion. Say something like “if x is the least element for which f is not yet defined, define f (x) to be. . . ” ...
... Hint: You can define a bijection by transfinite recursion. Say something like “if x is the least element for which f is not yet defined, define f (x) to be. . . ” ...
UNIVERSITY OF LONDON BA EXAMINATION PHILOSOPHY
... (i) Prove that any infinite ordinal α is equinumerous to its successor α+ . (ii) Define the cardinal |A| of a set A. What is it for an ordinal to be a limit ordinal? Prove that every infinite cardinal is a limit ordinal. (iii) State the theorem that legitimates definition by transfinite recursion on ...
... (i) Prove that any infinite ordinal α is equinumerous to its successor α+ . (ii) Define the cardinal |A| of a set A. What is it for an ordinal to be a limit ordinal? Prove that every infinite cardinal is a limit ordinal. (iii) State the theorem that legitimates definition by transfinite recursion on ...
Transfinite Chomp
... Every Chomp position X has ordinal size, size(X) Decompose position into finite, overlapping sum of boxes S Each component box has each side length ωe, for non-negative integer e Discard any box contained within another to ...
... Every Chomp position X has ordinal size, size(X) Decompose position into finite, overlapping sum of boxes S Each component box has each side length ωe, for non-negative integer e Discard any box contained within another to ...
Well-foundedness of Countable Ordinals and the Hydra Game
... results in a smallest such ordinal, which must be 0. This paper will develop the necessary definitions and theorems to prove this in ZF; actually, this result will be proven in the weaker system ACA0 + W F (COrd), where W F (COrd) is the statement that every collection of countable well-orders is we ...
... results in a smallest such ordinal, which must be 0. This paper will develop the necessary definitions and theorems to prove this in ZF; actually, this result will be proven in the weaker system ACA0 + W F (COrd), where W F (COrd) is the statement that every collection of countable well-orders is we ...