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Reverse mathematics - Wikipedia, the free encyclopedia For example, the ω-model consisting of (the usual natural numbers together with) the set of recursive sets of natural numbers is an ω-model of recursive comprehension (in fact, the smallest one) which is not a model of arithmetical comprehension. Recursive comprehension serves as our core system, so to state that a theorem is “equivalent” to recursive comprehension merely means that it is provable even in that weak system. We add to recursive comprehension a weak form of König's lemma, namely the statement that every infinite subtree of the full binary tree (the tree of all finite sequences of 0's and 1's) has an infinite path. en.wikipedia.org /wiki/Reverse_mathematics   (3382 words)

Transfinite induction - Wikipedia, the free encyclopedia If P(b) follows from the truth of P(a) for all a < b, then it is simply a special case to say that P(0) is true, since it is vacuously true that P(a) holds for all a < 0. Transfinite recursion is a notion closely related to transfinite induction, but whereas the latter is a method of proof, the former is a method of definition or construction. Relationship to AC There is a popular misconception that transfinite induction, or transfinite recursion, or both, require the axiom of choice. en.wikipedia.org /wiki/Transfinite_induction   (629 words)

Transfinite induction - Encyclopedia, History, Geography and Biography   (Site not responding. Last check: 2007-10-29) Transfinite induction is the proof technique of mathematical induction when applied to (large) well-ordered sets, for instance to sets of ordinals or cardinals, or even to the class of all ordinals. It may be regarded as one of three forms of mathematical induction. If you are trying to prove that a property P holds for all ordinals then you can apply transfinite induction: www.arikah.net /encyclopedia/Transfinite_recursion   (268 words)

paper available   (Site not responding. Last check: 2007-10-29) The objective is to support the formalization of particular recursive definitions for use in verification, semantics proofs and other computational reasoning. Recursive functions are defined by well-founded recursion and its derivatives, such as transfinite recursion. Recursive data structures are expressed by applying the Knaster-Tarski Theorem to a set that is closed under Cartesian product and disjoint sum. www.seas.upenn.edu /~sweirich/types/archive/1993/msg00131.html   (227 words)

Foundations of Mathematics By David Hilbert (1927) The recursion associated with the number-theoretic variable is "ordinary recursion", by means of which t function of t number-theoretic variable n is defined when we indicate what value it has for n = 0 and how the value for n' is obtained from that for n. The generalisation of ordinary recursion is transfinite recursion; it rests upon the general principle that the value of the function for a value of the variable is determined by means of the preceding values of the function. As soon as Cantor had discovered his first transfinite numbers, the numbers of the second number class as they are called, the question arose whether by means of this transfinite counting one could actually enumerate the elements of sets known in other contexts but not denumerable in the ordinary sense. members.fortunecity.com /iewulib3/libkj/hilbert.htm   (4234 words)

the theology company > synopsis > 30 the transfinite network   (Site not responding. Last check: 2007-10-29) This businesslike idea is here expanded, using Cantor's theory of transfinite cardinal and ordinal numbers, into an infinite abstract structure we call the transfinite network. Since the transfinite network is taken as a model of the whole universe, every o-machine within it has, in principle, the ability to consult all the other machines, and can thus tap the power of the whole network. This is the first step in a recursive process, since we can imagine that once each machine has learnt everything all the others have to teach it, this new generation of more powerful machines can begin another round of consultation. www.physicaltheology.com /Synopsis/s30Network.html   (799 words)

2.3 Ordinal Arithmetic Because of this, the theorem is often called the Recursion Theorem, and definitions that use the Recursion Theorem in the justification of their existence and uniqueness are called definitions by transfinite recursion or transfinite induction The definition of addition for ordinals is done using transfinite recursion, and the proof of the theorem below shows how much this definition relies on Theorem 2.18. Thus, by transfinite induction the theorem is true for all ordinals. www.rdegraaf.nl /mirror/www.u.arizona.edu/~miller/thesis/node8.html   (600 words)

Metamath Proof Explorer - mmtheorems33   (Site not responding. Last check: 2007-10-29) Principle of Transfinite Recursion, part 1 of 3. Principle of Transfinite Recursion, part 2 of 3. A recursive definition requires advanced metalogic to justify - in particular, eliminating a recursive definition is very difficult and often not even shown in textbooks. us.metamath.org /mpegif/mmtheorems33.html   (716 words)

Well-founded relation   (Site not responding. Last check: 2007-10-29) An important reason that well-founded relations are interesting is because a version of transfinite induction can be used on them: if (X, R) is a well-founded relation and P(x) is some property of elements of X and you want to show that P(x) holds for all elements of X, it suffices to show that: Then induction on S is the usual mathematical induction, and recursion on S gives primitive recursion. When the well-founded relation is the usual ordering on the class of all ordinal numbers, the technique is called transfinite induction. www.tocatch.info /en/Well-founded.htm   (729 words)

Steinhart Article Abstracts   (Site not responding. Last check: 2007-10-29) recursiveness; well-ordering principles; continuity at transfinite limits; minimality, and identification of n with the set of all numbers less than n). I analyze the complexity of physical systems into (1) universes founded on finite recursion; (2) universes founded on transfinite recursion; and (3) universes with non-recursive foundations. There are finite as well as transfinite algorithms and machines of any degree of complexity (e.g. www.wpunj.edu /cohss/philosophy/faculty/ESTEINHA/abstract.htm   (2423 words)

Practical Foundations of Mathematics   (Site not responding. Last check: 2007-10-29) Transfinite recursion Since ordinals are a special case of well founded relations, for any \ev Besides transfinite recursion, ordinals also provide a convenient way of performing constructions which require the axiom of choice. The induction scheme must be restricted to a smaller class of predicates, such as the Scott-continuous ones in Theorem 3.7.13. www.cs.man.ac.uk /~pt/Practical_Foundations/html/s67.html   (2057 words)

S2004 272Descr   (Site not responding. Last check: 2007-10-29) When such an operator is not ‘finitely based’, one has to keep iterating it ‘into the transfinite.’ Trying to make sense of these transfinite inductions was the original motivation for Cantor to develop his theory of infinite numbers. We can also justify definitions by transfinite recursion on ordinals. Hence, we can also use these names to carry transfinite recursion and induction. home.gwu.edu /~harizanv/S2004_272Descr.html   (282 words)

[No title] Many other functions of interest are also defined by transfinite recursions, and we would like to know that they are absolute as well. Then the class function $G$ defined in the proof of the transfinite recursion theorem is absolute between $M$ and $N$ and in both $M,N$ satisfies the recursive definition $G(x)=F(x, G \restriction \pred(x,R))$. Since both $M,N$ satisfy $\zf-$Power, the transfinite recursion theorem in $M$ and $N$ shows that in both classes $G$ is a function defined on $A$ and satisfies the recursive definition. www.math.unt.edu /~sjackson/6010f02/absolute.tex   (4677 words)

the theology company > synopsis > 16 Cantor   (Site not responding. Last check: 2007-10-29) Cantor showed that the step from natural to real numbers is not unique, but the first an endless series of steps to even bigger number spaces, which he called the transfinite numbers. This structure, called the transfinite numbers, we take to be a language big enough to begin talking about God. Our claim that the universe is divine relies on the fact that only the transfinite space is big enough to describe even the smallest aspects of our world. www.physicaltheology.com /Synopsis/s16Cantor.html   (785 words)

Readings in Logic We have selected from the wealth of topics available some of those which deal with the basic concepts of the subject, or are particularly important for applications to other parts of mathematics, or both. The work moves on through the development of transfinite cardinals to applicability of this powerful instrument to analysis. Recursively enumerable sets of positive integers and their decision problems [1944] nfocentrale.net /orcmid/readings/logic.htm   (4287 words)

The Recursion Theorem Since we are defining a function recursively, this theorem is sometimes called the recursion theorem. Each new ordinal t extends the function further, and when all the ordinals are taken together, the composite function maps all of s into r, and is compatible with g. This is sometimes called recursion with choice, because the axiom of choice ensures a well ordering on any set, including r. www.mathreference.com /set-zf,rect.html   (868 words)

Strong Normalization Theorem for a Constructive Arithmetic with Definition by Transfinite Recursion and Bar Induction, ... Strong Normalization Theorem for a Constructive Arithmetic with Definition by Transfinite Recursion and Bar Induction, Osamu Takaki Strong Normalization Theorem for a Constructive Arithmetic with Definition by Transfinite Recursion and Bar Induction We prove the strong normalization theorem for the natural deduction system for the constructive arithmetic TRDB (the system with Definition by Transfinite Recursion and Bar induction), which was introduced by Yasugi and Hayashi. projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.ndjfl/1039700743   (180 words)

sci.math: Re: Countably infinite Hausdorff topology? forced to face with the ugly details of "transfinite recursion". Recursion Theorem (and hence, it would be more accurate to AC (expedient) or to transfinite recursion (with all its "glorious" details sci.tech-archive.net /Archive/sci.math/2004-09/4013.html   (1059 words)

Cornell Math - MATH 781, Fall 2005   (Site not responding. Last check: 2007-10-29) This begins with the constructive ordinals, effective transfinite recursion and Pi-1-1 sets. Then an analysis of the hyperarithmetic hierarchy based on iterating the Turing jump into the transfinite. If this topic is not covered in 784 it will be the topic for 781. www.math.cornell.edu /~www/Courses/GradCourses/FA05/781.html   (221 words)

[No title] So with these definitions, we could restate the axiom of replacement as: whenever F is a class term and a is a set, then the class {y : y=F(x) for some x in a} is a set. Now, I will need the recursion theorem for ordinals (transfinite recursion), and I'm going to give a restatement of it in terms of class terms. Since this is transfinite induction, it remains to check the case for limit ordinals, but this is considerably easier, and I'm not going to do it unless anyone wants me to. br.endernet.org /~loner/settheory/evcummhier.txt   (1964 words)

Countable Ordinals =Yes, transfinite recursion applies to general well-ordered sets, =and this fact does not require AC. Then as recursion produces unique functions, the long recursion restricted to nu+1 is the same as the short recursion. Since type is a well-defined operator which yields a ==unique set fro a given argument, one may apply the axiom of ==Replacement to form the set S above. www.forum-one.org /new-5971498-4346.html   (6085 words)

Subsystems of Second Order Arithmetic The Transfinite Recursion Scheme consists of the universal closures of all formulas of the form Thus the Transfinite Recursion Scheme asserts the existence of sets defined by transfinite recursion along arbitrary well orderings of The letters ATR stand for arithmetical transfinite recursion. www.math.psu.edu /simpson/papers/article-l/node2.html   (1184 words)

Constructing Recursion Operators in Intuitionistic Type Theory - Paulson (ResearchIndex)   (Site not responding. Last check: 2007-10-29) To handle recursion schemes other than primitive recursion, a theory of well-founded relations is presented. Using primitive recursion over higher types, induction and recursion are formally derived for a large class of well-founded relations. 1 On definition trees of ordinal recursive functionals: reduct.. citeseer.lcs.mit.edu /paulson84constructing.html   (668 words)

One Hundred Years of Russell's Paradox - Abstracts The main feature of this embedding is an operator R on infinitary derivations (in omega-arithmetic) defined by (transfinite) recursion on well-founded trees. In (2), for each finite PA-derivation d certain reduced PA-derivations d[i] are defined by primitive recursion on the build up of d in such a way that the endsequent of d results from the endsequents of the d[i]'s by an inference of PAomega. This distinction is analogous to the arithmetic and analytic hierarchies familiar in recursion theory and descriptive set theory. www.lrz-muenchen.de /~russell01/papers.html   (8694 words)

ISABELLE-91 It can be instantiated to any reflexive/transitive relation (such as = and <->) for which congruence rules can be proved. It handles conditional rewrite rules by a recursive invocation of rewriting. Isabelle's Zermelo-Fraenkel set theory now derives a theory of functions, transfinite recursion, and several recursive data structures (including mutually recursive trees and forests). www.seas.upenn.edu /~sweirich/types/archive/1991/msg00050.html   (436 words)

Conservation Results From this it follows that the recursive well orderings of ``all ordinals are recursive'', i.e., ``every well ordering is isomorphic to a recursive well ordering''. sentence ``all ordinals are recursive'', which is provable in www.math.psu.edu /simpson/papers/symm/node2.html   (198 words)

M112,Philosophy 134, Winter 2004, Poster, Moschovakis   (Site not responding. Last check: 2007-10-29) Contents: Naive set theory, Cantor's basic theorems, the paradoxes; axiomatic set theory, relations and functions, cardinal numbers; the natural numbers, proof by induction and definition by recursion; well orderings, proof  by transfinite induction and definition by transfinite recursion; the axiom of choice, cardinal arithmetic; replacement, ordinal numbers and the cumulative hierarchy of  sets. Prerequisites: One of Mathematics 110A, 121,131A, or Philosophy 135, or consent of the instructor. Those interested in taking this class should talk with me after the fourth week of classes. www.math.ucla.edu /~ynm/m112.1.04w/poster.htm   (316 words)

iqexpand.com   (Site not responding. Last check: 2007-10-29) This is a very strong limitation the allowed proof means (for example, recursive comprehension is not sufficient to prove König's lemma as described in the next subsection). Association for Symbolic Logic Topic: Reverse Mathematics and Computability Theory Title: andquot;Why the Recursion Theorists Should Thank Meandquot; Speaker: Stephen... Reverse Mathematics - definition of Reverse Mathematics in... reverse_mathematics.iqexpand.com   (3753 words)

The Mathematics of Boolean Algebra The definition is by transfinite recursion, where α, β are ordinals and λ is a limit ordinal: The B-valued universe is the proper class V(B) which is the union of all of these Vs. Next, one defines by a rather complicated transfinite recursion over well-founded sets the value of a set-theoretic formula with elements of the Boolean valued universe assigned to its free variables plato.stanford.edu /entries/boolalg-math   (2064 words)

mbox: Abstracts from J. of Automated Reasoning, Vol. 15, No. 2 is to support the formalization of particular recursive definitions {\it Recursive functions} are defined by well-founded recursion variable-branching trees, and mutually recursive trees and forests. www-unix.mcs.anl.gov /qed/mail-archive/volume-3/0070.html   (305 words)

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