Abstract
Akama et al. [1] introduced a hierarchical classification of first-order formulas for a hierarchical prenex normal form theorem in semi-classical arithmetic. In this paper, we give a justification for the hierarchical classification in a general context of first-order theories. To this end, we first formalize the standard transformation procedure for prenex normalization. Then we show that the classes \(\textrm{E}_k\) and \(\textrm{U}_k\) introduced in [1] are exactly the classes induced by \(\Sigma _k\) and \(\Pi _k\) respectively via the transformation procedure in any first-order theory.
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Acknowledgements
The authors thank Ulrich Kohlenbach for pointing them out that the definition of \(\textrm{P}_0\) in [1] is different from that in its preprint version, to which the authors referred in the previous version of this paper. They also thank Danko Ilik for providing some information about related works. The first author was supported by JSPS KAKENHI Grant Numbers JP19J01239, JP20K14354 and JP23K03205, and the second author by JP19K14586 and JP23K03200.
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Fujiwara, M., Kurahashi, T. Prenex normalization and the hierarchical classification of formulas. Arch. Math. Logic 63, 391–403 (2024). https://doi.org/10.1007/s00153-023-00899-x
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DOI: https://doi.org/10.1007/s00153-023-00899-x