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

Anton Freund
2020 arXiv   pre-print
In the context of reverse mathematics, effective 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

Benjamin Rin
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

MARIA HÄMEEN-ANTTILA
2019 Bulletin of Symbolic Logic
The idea that 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.  ...

### 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 .  ...  Also, in analogy with , 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

Harvey M. Friedman, Andre Scedrov
1986 Annals of Pure and Applied Logic
We consider intuitionistic number theory with 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

M. Machover
1961 Bulletin of the American Mathematical Society
To do this, one should possess 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

S. C. Kleene
1952 Bulletin of the American Mathematical Society
showing that there is no inconsistency in supposing that the number-theoretic 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]

Jacques Patarin
2012 Lecture Notes in Computer Science
We will see that for these "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

Wayne Richter
1967 Bulletin of the American Mathematical Society
This is born out 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  (see also Kleene ). 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.  ...

### TURING DEGREES AND THE ERSHOV HIERARCHY

FRANK STEPHAN, YUE YANG, LIANG YU
2009 Proceedings of the 10th Asian Logic Conference
An n-r.e. set can be 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.  ...

### 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).  ...

### 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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