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A boundedness theorem in ID1(W)

Published online by Cambridge University Press:  12 March 2014

Gerhard Jäger
Gerhard Jäger*
Affiliation:
Mathematik, Eth-Zentrum, Zürich, Switzerland
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Extract

In this paper we prove a boundedness theorem in the theory ID1(W). This answers a question asked by Feferman, for example in [3]. The background is the following.

Let A[X, x] be an X-positive formula arithmetic in X. The theory ID1(PA) is an extension of Peano arithmetic PA by the following axioms:

for arbitrary formulas F; PA is a constant for the least fixed point of A[X, x]. Set-theoretically, PA can be defined by recursion on the ordinals as follows:

where is the first nonrecursive ordinal.

Now let a ≺ b be the arithmetic relation which expresses that the recursive tree coded by a is a proper subtree of the tree coded by b, and define

The least fixed point of Tree[X, x] is the set PTree of all well-founded recursive trees. We write W or Wα for PTree or , respectively. Since W is complete we have for all α < . If we define for each element a ∈ W its inductive norm ∣a∣ by ∣a∣≔ min{ξ: a ∈ Wξ}, then we have = {∣a∣: a ∈ W} and the elements of W can be used as codes for the ordinals less than .

Assume that B[X, x] is an X-positive formula arithmetic in X with the only free variables X and x, and assume that QB is a relation that satisfies

If we define

then we obviously have PB = IB.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 51 , Issue 4 , December 1986 , pp. 942 - 947
Copyright
Copyright © Association for Symbolic Logic 1986

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References

REFERENCES

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