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...the basis of which lies a certain synthesis of ideas in the theory of sets and algorithms (see [[#References|[2]]]). ...TD valign="top">[2]</TD> <TD valign="top"> J. Barwise, "Admissible sets and structures" , Springer (1975)</TD></TR></table>

$#C+1 = 7 : ~/encyclopedia/old_files/data/M065/M.0605390 Multiple recursion Please remove this comment and the {{TEX|auto}} line below,

...ained by primitive recursion from an $n$-place function $g(x_1,\dots,x_n)$ and an $(n+2)$-place function $h(x_1,\dots,x_n,y,z)$ if for all natural number and

...ly enumerable. Many sets playing an important role in recursive set theory and its applications are productive (e.g. the set of all Gödel numbers of gene ...lign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165</TD></TR></table>

A set is ''hypersimple'' if it is recursively enumerable and its complement is a [[hyperimmune set]].

...re of interest from the point of the view of the recursive analogue of the theory of [[cardinal number]]s. In [[recursive set theory]] and its applications one also uses certain special subclasses of the class of i

...t]]; in other words, a set $A$ is creative if it is recursively enumerable and if there exists a [[partial recursive function]] $\phi(x)$ such that, for a ...numerable sets. The concept of creativity generalizes to sequences of sets and other objects.

$#C+1 = 86 : ~/encyclopedia/old_files/data/R080/R.0800210 Recursion Please remove this comment and the {{TEX|auto}} line below,

Please remove this comment and the {{TEX|auto}} line below, ...umber of the operations of composition and [[Primitive recursion|primitive recursion]].

Please remove this comment and the {{TEX|auto}} line below, A method used in [[Recursion|recursion]] theory for the construction of sets with a simple structure (from the recursive po

...valign="top"> H. Rogers jr., "Theory of recursive functions and effective computability" , McGraw-Hill (1967) pp. 164–165</TD></TR> ...artin, "Bounded Queries in Recursion Theory", Progress in Computer Science and Applied Logic '''16'''. Springer (1999) {{ISBN|0817639667}}</TD></TR>

Please remove this comment and the {{TEX|auto}} line below, ...], defined as follows. One examines functions given on the natural numbers and with natural values. The functions are assumed to be partial, i.e. they are

..., by its table; the number of arguments of the function may depend on $a$) and numbers $b_1,\ldots,b_n$ such that $a\in A$ is equivalent to the truth of $ ...quivalence relation $\equiv_{tt}$, namely $A\equiv_{tt}B$ if $A\leq_{tt}B$ and $B\leq_{tt}A$. The equivalence classes for this relation are called truth-t

Please remove this comment and the {{TEX|auto}} line below, ...of the simpler classes. The most important hierarchies in descriptive set theory are defined as follows. If $ T $

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...duction (see below). It subsumes addition, multiplication, exponentiation, and all higher-order analogues of these operations. Because of this it grows to

where the $R_i$ and $R$ are atomic formulas (and $R$ could also be $x_p = x_q$ or $\texttt{false}$. (Strict) propositional H ...exttt{false}$. In the strict case $q = \texttt{false}$ is excluded. A Horn theory is a set of Horn clauses.

$#C+1 = 130 : ~/encyclopedia/old_files/data/R080/R.0800340 Recursive set theory Please remove this comment and the {{TEX|auto}} line below,

If the TeX and formula formatting is correct, please remove this message and the {{TEX|semi-auto}} category. ...ed his machines (cf. [[Turing machine]]) and showed that Turing computable and lambda definable are equivalent notions. These are arguments for the Church

Please remove this comment and the {{TEX|auto}} line below, ...m for the solution of a given infinite series of problems of a given type, and of finding such an algorithm if it exists.