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.

Montagna, F., Simi, G., & Sorbi, A. (2015). Taking the Pirahã seriously. COMMUNICATIONS IN NONLINEAR SCIENCE & NUMERICAL SIMULATION, 21(1-3), 52-69 [10.1016/j.cnsns.2014.06.052].

### Taking the Pirahã seriously

#### Abstract

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.
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Utilizza questo identificativo per citare o creare un link a questo documento: `http://hdl.handle.net/11365/979974`