Taking the Pirahã seriously

We propose first order formal theories $\Gamma_{n}$, with $n \ge 1$, and $\Gamma= \bigcup_{n\ge 1} \Gamma_{n}$, 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 $I\Delta_{0}+\Omega_{1}$ 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 $\Gamma_{n}$ implies the consistency of $\Gamma_{n}\cup \{\Con_{\Gamma_{n}}\}$, where $\Con_{\Gamma_{n}}$ ``asserts'' the consistency of $\Gamma_{n}$ (this however does not conflict with G\"odel's Second Incompleteness Theorem); if $n>1$, then there is a precise way in which we can say that $\Gamma_{n}$ 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 $\Gamma_{n}$ 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.