×

Arithmetical independence results using higher recursion theory. (English) Zbl 1067.03074

Summary: We extend an independence result proved in our previous paper [Ann. Pure Appl. Logic 112, 27–41 (2001; Zbl 0986.03051)]. We show that, for all \(n\), there is a special set of \(\Pi_n\) sentences \(\{\phi_a\}_{a\in H}\) corresponding to elements of a linear ordering \((H,<_H)\) of order type \(\omega^{CK}_1(1+\eta)\). These sentences allow us to build completions \(\{T_a\}_{a\in H}\) of PA such that for \(a <_H b\), \(T_a\cap\Sigma_n\subset T_b\cap \Sigma_n\), with \(\varphi_a\in T_a\), \(\neg\varphi_a\in T_b\). Our method uses the Barwise-Kreisel Compactness Theorem.

MSC:

03H15 Nonstandard models of arithmetic
03C57 Computable structure theory, computable model theory
03C62 Models of arithmetic and set theory
03F30 First-order arithmetic and fragments

Citations:

Zbl 0986.03051
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Computable structures and the hyperarithmetical hierarchy 144 (2000) · Zbl 0960.03001
[2] DOI: 10.1016/S0168-0072(01)00095-1 · Zbl 0986.03051
[3] Notices of the American Mathematical Society 5 pp 679– (1958)
[4] Theory of recursive functions and effective computability (1967) · Zbl 0183.01401
[5] Models of Peano Arithmetic 15 (1991) · Zbl 0744.03037
[6] Large discrete parts of the E-tree 53 pp 980– (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.