Mathematical Logic with Special Reference to the Natural Numbers

Monotone inductive definitions occur frequently throughout mathematical logic. The set of formulas in a given language and the set of consequences of a given axiom system are examples of (monotone) inductively defined sets. The class of Borel subsets of the continuum can be given by a monotone inductive definition. Kleene's inductive definition of recursion in a higher type functional (see [6]) is fundamental to modern recursion theory; we make use of it in §2.Inductive definitions over the natural numbers have been studied extensively, beginning with Spector [11]. We list some of the results of that study in §1 for comparison with our new results on inductive definitions over the continuum. Note that for our purposes the continuum is identified with the Baire space ωω.It is possible to obtain simple inductive definitions over the continuum by introducing real parameters into inductive definitions over N—as in the definition of recursion in [5]. This is itself an interesting concept and is discussed further in [4]. These parametric inductive definitions, however, are in general weaker than the unrestricted set of inductive definitions, as is indicated below.In this paper we outline, for several classes of monotone inductive definitions over the continuum, solutions to the following characterization problems:(1) What is the class of sets which may be given by such inductive definitions ?(2) What is the class of ordinals which are the lengths of such inductive definitions ?These questions are made more precise below. Most of the results of this paper were announced in [2].

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Partition Theorems and Computability Theory

Bulletin of Symbolic Logic ◽

10.2178/bsl/1122038995 ◽

2005 ◽

Vol 11 (3) ◽

pp. 411-427 ◽

Cited By ~ 2

Author(s):

Joseph R. Mileti

Keyword(s):

Mathematical Logic ◽

Reverse Mathematics ◽

Computability Theory ◽

Natural Numbers ◽

Ramsey’S Theorem ◽

Finite Sequences ◽

Partition Theorems ◽

Combinatorial Principles ◽

Rich History ◽

König’S Lemma

The connections between mathematical logic and combinatorics have a rich history. This paper focuses on one aspect of this relationship: understanding the strength, measured using the tools of computability theory and reverse mathematics, of various partition theorems. To set the stage, recall two of the most fundamental combinatorial principles, König's Lemma and Ramsey's Theorem. We denote the set of natural numbers by ω and the set of finite sequences of natural numbers by ω<ω. We also identify each n ∈ ω with its set of predecessors, so n = {0, 1, 2, …, n − 1}.

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1936: Post, Turing and ‘a kind of miracle’ in mathematical logic

The Mathematical Gazette ◽

10.1017/s0025557200174170 ◽

2004 ◽

Vol 88 (511) ◽

pp. 2-15

Author(s):

G. T. Q. Hoare

Keyword(s):

Mathematical Logic ◽

Natural Number ◽

Natural Numbers ◽

Alan Turing

In the 1930s several mathematicians, principally Alonzo Church (1903-1995), Stephen Kleene (1909-1994), Emil Post (1897-1954) and Alan Turing (1912-1954), began investigating the notion of effective calculability. (A function from natural numbers to natural numbers is effectively calculable if there is some finite rule or mechanism which will calculate the value of the function for any natural number.) Central to this activity was the notion of recursiveness. Loosely, recursion is a process of defining a function by specifying each of its values in terms of previously defined values.

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