Abstract
The main goal of this paper is to formulate a constructive analogue of Ackermann's observation about finite set theory and arithmetic. We will see that Heyting arithmetic is bi-interpretable with CZFfin, the finitary version of CZF. We also examine bi-interpretability between subtheories of finitary CZF and Heyting arithmetic based on the modification of Fleischmann's hierarchy of formulas, and the set of hereditarily finite sets over CZF, which turns out to be a model of CZFfin but not a model of finitary IZF.
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