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.


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


Zbl 0986.03051
Full Text: DOI


[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)
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