Episodes in Model-Theoretic Xenology: Rationals as Positive Integers in R#

Meyer and Mortensen’s Alien Intruder Theorem includes the extraor- dinary observation that the rationals can be extended to a model of the relevant arithmetic R♯, thereby serving as integers themselves. Al- though the mysteriousness of this observation is acknowledged, little is done to explain why such rationals-as-integers exist or how they operate. In this paper, we show that Meyer and Mortensen’s models can be identified with a class of ultraproducts of finite models of R♯, providing insights into some of the more mysterious phenomena of the rational models

Episodes in Model-Theoretic Xenology: Rationals as Positive Integers in R#

Meyer and Mortensen’s Alien Intruder Theorem includes the extraor- dinary observation that the rationals can be extended to a model of the relevant arithmetic R♯, thereby serving as integers themselves. Al- though the mysteriousness of this observation is acknowledged, little is done to explain why such rationals-as-integers exist or how they operate. In this paper, we show that Meyer and Mortensen’s models can be identified with a class of ultraproducts of finite models of R♯, providing insights into some of the more mysterious phenomena of the rational models

Consistency and Decidability in Some Paraconsistent Arithmetics

The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in incon- sistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals

Consistency and Decidability in Some Paraconsistent Arithmetics

The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent models seem likely to differ in terms of complexity. In this paper, I begin to explore this terrain, working, particularly, in incon- sistent theories of arithmetic couched in three-valued paraconsistent logics which have strong (i.e. detaching) conditionals