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The critical number of a variable in a function

Published online by Cambridge University Press:  12 March 2014

Gaisi Takeuti
Gaisi Takeuti*
Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801, E-mail: takeuti@symcom.math.uiuc.edu
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Let L0 be a language on ℕ consisting of 0, 1, +, ∸, ·, ⌊½a⌋, ∣a∣, #, ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b), and μx ≤ ∣s∣t(x). Here μx ≤ ∣s∣t(x) is the smallest number x ≤ ∣s∣ satisfying t(x) > 0 and 0 if there exist no such x and we stipulate that if s and t(a) are terms in Lo, then μx ≤ ∣s∣t(x) is also a term in Lo. The defining axioms of functions ∧(a,b), ∨(a,b), ¬(a), ≤ (a,b) are as follows:

Let L a language on ℕ with only predicate constant = and L0 ⊆ L. Let f (b, a1,…,am) be a function for ℕm+1 into ℕ. We say “f is weakly expressed by terms t1(b, a1,…, am),…, tr(b, a1,…, am) in L” if for every b, a1,…, am ∈ ℕ, f (b, a1,…,am) is equal to one of ti(b, a1, …, am). The critical number of b in f with respect to L is the minimum number n such that whenever f(b, a1,…, an) is weakly expressed by terms t1(b, a1, …, an),…, the number of occurrences of b in some ti(b, a1,…, an) is at least n.

As is defined in [1], a function f is defined by a limited iteration from g and h with respect to L iff the following holds: Let τ be defined as

with the condition ; and is defined by

where and are terms in L. We say “f is defined by a short limited iteration from g and h” if is defined by

where τ, s, t are the same as above satisfying the condition .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 59 , Issue 4 , December 1994 , pp. 1228 - 1244
Copyright
Copyright © Association for Symbolic Logic 1994

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References

REFERENCES

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