7 research outputs found
Librationist cum classical set theories
A librationist set theoretic system \Pfund, which is inter alia geared to
deal with set theoretic paradoxes in new ways, is developed. It descends from
work in a semantic setting, for truth, initiated by by Kripke, Herzberger and
Gupta. \Pfund \ extends the author's contribution in Librationist closures of
the paradoxes in Logic and Logical Philosophy 21(4), 323-361, 2012. It is shown
that \Pfund \ provides an interpretation of a set theory published by D. Scott
in More on the axiom of extensionality, in Bar-Hillel et alia, Essays on the
foundations of mathematics, North-Holland Publishing Company, 1961, 115-131.
Given this, \Pfund \ also obtains an interpretation of ZFC vi results of von
Neumann on regularity in 1929, and G\"odel on the Axiom og Choice in 1938.
However, \Pfund \ offers alternative ways to include choice and regularity by
means of principles which are informative, and natural. \Pfund \ retains the
idea, of Bj{\o}rdal 2012, that the set theoretic universe is countable. But the
set within which ZF is interpreted "believes" that there are sets which are not
countable. The situation can be resolved much as by Skolem, though one need not
suggest that the notion of 'set' is imprecice: for the bijection from the set
of finite von Neumann ordinals to the full universe is itself not a member of a
classical set theory
Librationist cum classical theories of sets
We give an interpretation of classical set theory in librationist set theory
Friedman-reflexivity
In the present paper, we explore an idea of Harvey Friedman to obtain a coordinate-free presentation of consistency. For some range of theories, Friedman's idea delivers actual consistency statements (modulo provable equivalence). For a wider range, it delivers consistency-like statements. We say that a sentence C is an interpreter of a finitely axiomatised A over U iff it is the weakest statement C over U, with respect to U-provability, such that U+C interprets A. A theory U is Friedman-reflexive iff every finitely axiomatised A has an interpreter over U. Friedman shows that Peano Arithmetic, PA, is Friedman-reflexive. We study the question which theories are Friedman-reflexive. We show that a very weak theory, Peano Corto, is Friedman-reflexive. We do not get the usual consistency statements here, but bounded, cut-free, or Herbrand consistency statements. We illustrate that Peano Corto as a base theory has additional desirable properties. We prove a characterisation theorem for the Friedman-reflexivity of sequential theories. We provide an example of a Friedman-reflexive sequential theory that substantially differs from the paradigm cases of Peano Arithmetic and Peano Corto. Interpreters over a Friedman-reflexive U can be used to define a provability-like notion for any finitely axiomatised A that interprets U. We explore what modal logics this idea gives rise to. We call such logics interpreter logics. We show that, generally, these logics satisfy the Löb Conditions, aka K4. We provide conditions for when interpreter logics extend S4, K45, and Löb's Logic. We show that, if either U or A is sequential, then the condition for extending Löb's Logic is fulfilled. Moreover, if our base theory U is sequential and if, in addition, its interpreters can be effectively found, we prove Solovay's Theorem. This holds even if the provability-like operator is not necessarily representable by a predicate of Gödel numbers. At the end of the paper, we briefly discuss how successful the coordinate-free approach is