## Truth definitions without exponentiation and the $$\Sigma _{1}$$ collection scheme.(English)Zbl 1245.03058

The paper presents three results related to the notoriously difficult open problem: Is the theory $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}+\lnot \mathrm{B}\Sigma_1$$ consistent? It is well known that if there is a model of $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}$$ with a universal $$\Sigma_1$$ formula, then there is also a model of $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}+\lnot \mathrm{B}\Sigma_1$$. The first result weakens the assumption a bit. It is shown that if there is a model of $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}$$ with cofinal $$\Sigma_1$$-definable elements and a $$\Sigma_1$$ truth definition for sentences, then $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}+\lnot \mathrm{B}\Sigma_1$$ is consistent. It is not know whether such models exist, but in the second part the authors show that there is a model of $$\mathrm{I}\Delta_0+\Omega_1+\lnot \mathrm{exp}$$ in which $$\Sigma_1$$ definable elements are cofinal and for which there is a $$\Sigma_2$$ truth definitions and $$\Pi_2$$ truth definition for $$\Sigma_1$$ sentences and for all $$n\geq 2$$ there is a $$\Sigma_n$$ truth definition for $$\Sigma_n$$ sentences. The last section presents a proof of an older, but previously unpublished result, that the assuption “Lessan’s bound for truth definitions is optimal” implies that $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}+\lnot \mathrm{B}\Sigma_1$$ is consistent. The paper concludes with some comments on why the problem of consistency of $$\mathrm{I}\Delta_0+\lnot \mathrm{exp}+\lnot \mathrm{B}\Sigma_1$$ is so hard.

### MSC:

 03C62 Models of arithmetic and set theory 03F30 First-order arithmetic and fragments
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### References:

 [1] DOI: 10.1016/S0168-0072(97)00026-2 · Zbl 0891.03032 [2] Notes on polynomially bounded arithmetic 61 pp 942– (1996) [3] Logic, methodology, and philosophy of science VIII (Moscow 1987) pp 143– (1989) [4] DOI: 10.4064/fm180-2-2 · Zbl 1053.03032 [5] Metamathematics of first-order arithmetic (1993) · Zbl 0781.03047 [6] DOI: 10.1016/0168-0072(94)00057-A · Zbl 0829.03035 [7] DOI: 10.1016/S0049-237X(08)72003-2
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