Taking the Pirahã Seriously

We propose first order formal theories Γn, with n ≥ 1, and Γ = ⋃ n≥1 Γn, which can be roughly described as follows: each one of these theories axiomatizes a bounded universe, with a greatest element, modelled on Sergeyev’s so-called grossone; each such theory is consistent if predicative arithmetic I∆0 + Ω1 is; inside each such theory one can represent (in a weak, precisely specified, sense) the partial computable functions, and thus develop computability theory; each such theory is undecidable; the consistency of Γn implies the consistency of Γn∪{ConΓn}, where ConΓn “asserts” the consistency of Γn (this however does not conflict with Gödel’s Second Incompleteness Theorem); if n > 1, then there is a precise way in which we can say that Γn proves that each set has cardinality bigger than every proper subset, although two sets have the same cardinality if and only if they are bijective; if n > 2, inside Γn there is a precise sense in which we can talk about integers, rational numbers, and real numbers; in particular, we can develop some measure theory; we can show that every series converges, and is invariant under any rearrangement of its terms (at least, those series and those rearrangements we are allowed to talk about); we also give a basic example, showing that even transcendental functions can be approximated up to infinitesimals in our theories: this example seems to provide a general method to replace a significant part of the mathematics of the continuum by discrete mathematics.