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What is effective transfinite recursion in reverse mathematics?
[article]

2020
*
arXiv
*
pre-print

In the context of reverse mathematics, effective

arXiv:2006.08953v1
fatcat:bnbcuaqsmrbvhjqwzxgicdyici
*transfinite**recursion*refers to a principle that allows us to construct sequences of sets*by**recursion*along arbitrary well orders, provided that each set ... is Δ^0_1-*definable*relative to the previous stages of the*recursion*. ... Hence the hierarchy of*functions*F k with k ∈ N can be constructed*by*the principle of effective (*transfinite*)*recursion*, as specified in Definition 2. ...##
###
Transfinite recursion and computation in the iterative conception of set

2014
*
Synthese
*

*Transfinite*

*recursion*is an essential component of set theory. ... This is significant because, while the iterative conception of set has been widely recognized as insufficient to establish Replacement and

*recursion*, its supplementation

*by*considerations pertaining to ... This results in a

*function*whose existence and well-

*definedness*are guaranteed

*by*the theorem of

*transfinite*

*recursion*(TR). ...

##
###
NOMINALISTIC ORDINALS, RECURSION ON HIGHER TYPES, AND FINITISM

2019
*
Bulletin of Symbolic Logic
*

The idea that

doi:10.1017/bsl.2018.91
fatcat:aysfqpegergxfirhqrzl2c52a4
*recursion*on higher types could be used to simulate the limit-building in*transfinite**recursion*seems to originate from Bernays. ... In 1936, Gerhard Gentzen published a proof of consistency for Peano Arithmetic using*transfinite*induction up to ε 0 , which was considered a finitistically acceptable procedure*by*both Gentzen and Paul ... types*defined**by*the*functions*f 1 , f 2 , . . . f n . ...##
###
Page 198 of Mathematical Reviews Vol. 5, Issue 8
[page]

1944
*
Mathematical Reviews
*

Exponentiation of ordinals is then

*defined**by**recursion*, and the*recursive*definitions of addition and multiplication, which have previously been*defined*directly, are given. ... The present installment deals with the arithmetic of the*transfinite*ordinals, proof*by**transfinite*induction and definition*by**transfinite**recursion*. ...##
###
Functional interpretation and inductive definitions
[article]

2009
*
arXiv
*
pre-print

*defined*using

*transfinite*

*recursion*on well-founded trees. ... Extending Gödel's Dialectica interpretation, we provide a

*functional*interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type

*functionals*... Now, given ϕ, g, and h as in the statement of the lemma, we

*define*a

*function*k(α, x, n)

*by*primitive

*recursion*on n. ...

##
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Page 577 of American Mathematical Society. Bulletin of the American Mathematical Society Vol. 67, Issue 6
[page]

1961
*
American Mathematical Society. Bulletin of the American Mathematical Society
*

1961] THE THEORY OF

*TRANSFINITE**RECURSION*577 We can now*define*deduction (in tree form) in analogy with [3]. ... Also, in analogy with [3], we can*define*the notion the*function*$ is wa-*recursive*in the sequence of*functions*Po, Wi, -- +>, We °°. @,-*recursiveness*is transitive in the sense of THEOREM 1. ...##
###
Page 1357 of Mathematical Reviews Vol. , Issue 83d
[page]

1983
*
Mathematical Reviews
*

<n

*by*means of primitive*recursion*or*transfinite**recursion*over <. ... Section 3 deals with definitions*by*(*transfinite*)*recursion*. ...##
###
Intuitionistically provable recursive well-orderings

1986
*
Annals of Pure and Applied Logic
*

We consider intuitionistic number theory with

doi:10.1016/0168-0072(86)90004-7
fatcat:bqkogcphgng2fl4d3h3rfcafhe
*recursive*infinitary rules (HA*). Any primitive*recursive*binary relation for which*transfinite*induction schema is provable is in fact well founded. ... In fact, it suffices to consider*transfinite*induction with respect to one particular number-theoretic property. ... Primitive*recursive**defining*equations. ...##
###
The theory of transfinite recursion

1961
*
Bulletin of the American Mathematical Society
*

To do this, one should possess

doi:10.1090/s0002-9904-1961-10696-4
fatcat:ctomxliq3rbr3ajkzqcxwnbucu
*transfinite*analogues of the theory of*recursive**functions*and of arithmetization. ...*By*ordinals we shall mean ordinals a , a*function*letter (f.l.) "ƒ*" for each £ <co a . NUMERALS. 0 followed*by*a sequence of length j8 of strokes is the numeral for /3. We denote it*by*(3. ... To do this, one should possess*transfinite*analogues of the theory of*recursive**functions*and of arithmetization. ...##
###
Book Review: Rekursive Funktionen

1952
*
Bulletin of the American Mathematical Society
*

showing that there is no inconsistency in supposing that the number-theoretic

doi:10.1090/s0002-9904-1952-09607-5
fatcat:i5lhotxxvrekdhzvziwyitsnbq
*functions*are all*definable**by*use of forms of*recursion*associated with the*transfinite*ordinals of Cantor's second number ... For Hubert's proposal it was necessary to show that higher forms of*recursion*do give new*functions*; and the first demonstration of the existence of a*function**definable**by*a double*recursion*but not*by*...##
###
Some Transfinite Generalisations of Gödel's Incompleteness Theorem
[chapter]

2012
*
Lecture Notes in Computer Science
*

We will see that for these "

doi:10.1007/978-3-642-27654-5_14
fatcat:ktvsfri5uzhojilfsoytefjcpa
*transfinite*devices" almost all Gödel's limitations results have relatively simple generalisations. ... In this paper, we will study what happens when we consider more powerful computing devices: these "*transfinite*devices" will be able to perform α classical computations and to use α bits of memory, where ... (Choose for F the set of*recursive*semi-*functions**defined*on I α .) 4. If an α-software will always return values in a given subset B. (Choose for F the set of semi-*functions*whose output is in B.) ...##
###
Constructive transfinite number classes

1967
*
Bulletin of the American Mathematical Society
*

This is born out

doi:10.1090/s0002-9904-1967-11710-5
fatcat:efn43ij5o5fqtjhnkbs75q7rl4
*by*the characterization of the ordinals of (F, | |) given below. E\ is the type-2 representing*functional*of the predicate \a. ... (V/3)(Bx)[a(J3(x)) = Q] introduced*by*Tugué [12] (see also Kleene [4]). Let cof 1 be the smallest ordinal which is not the order type of any well-ordering*recursive*in JSi. ... Then using the techniques of [ç], the*recursion*theorem, and a proof*by**transfinite*induction, we obtain: THEOREM 1. For l£v£t<\F\, (1) F v ^ 0 F \ (2) |F,|==J?, (3) Ft gif* ÛiF. ...##
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TURING DEGREES AND THE ERSHOV HIERARCHY

2009
*
Proceedings of the 10th Asian Logic Conference
*

An n-r.e. set can be

doi:10.1142/9789814293020_0012
fatcat:njwekmmovbervkrtwydsgtizve
*defined*as the symmetric difference of n*recursively*enumerable sets. The classes of these sets form a natural hierarchy which became a well-studied topic in*recursion*theory. ... In this paper, we survey the early work*by*Ershov and others on this hierarchy and present the most fundamental results. We also provide some pointers to concurrent work in the field. ... This can be proven*by*using the class of all sets Note that this class is even uniformly*recursively*enumerable. ...##
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Page 1309 of Mathematical Reviews Vol. , Issue 96c
[page]

1996
*
Mathematical Reviews
*

A

*function*/ is said to be B-exotic*recursive*over C if it is computed in terms of a term t*by*<,-dr if possible, and is given a value*by*a C-*function*otherwise. ... The well-known non-primitive*recursive**function*g is (w” + 1)-DR, where g is*defined*as follows: fo(x) =2*, fn4i(0) = fn(l), fnsile +1) = fn(fnsilx)), g(n) = fn(n). ...##
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Page 4 of Mathematical Reviews Vol. , Issue 89M
[page]

1989
*
Mathematical Reviews
*

In the first one, using the method of

*transfinite**recursion*on the ordinals, we*define*fuzzy sets of all ranks and the class F of all fuzzy sets. ... Summary: “In addition to the basic system the primitive*recursive*arithmetic in the first class Ag1 consists of the following primitive*recursive**function*definition Ap and primitive*recursive*uniqueness ...
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