Why PAI Private Arithmetics Institute ?
A critical analysis of the connection between logics and mathematics shows that the foundations of mathematics often are not treated thoroughly. It is the aim of PAI to distribute a method and precise languages that adhere to the following principles in so-called Bavaria notation :
Clear distinction of language levels
- supralanguage (English)
- metalanguage (Mencish)
- object-language (Funcish, obeying the principle of context-independence)
Clear distinction of systems, that are called
- abstract calcules (purely axiomatic)
- concrete calcules (at most recursive)
Clear distinction of logics with respect to types
- first order (e.g. natural numbers, rational numbers, radical numbers i.e. with roots)
- higher order (e.g. real numbers)
It is always the question what a system is talking about, in other words, what is the ontological basis to start with. The requirement for precision is such that a computer can decide if a certain step of reasoning is in accordance with the rules - if it fulfills the Calculation Criterion of Truth .
On the other hand it is tried to stay as close to normal mathematical language based on predicate logic as possible. Set theory is not used; on the contrary, axiomatic set theory can be expressed in FUME, the language system of Funcish and Mencish..
As of November 2019 there are eight publications available via pdf-download:
- Geometries of O , relating to axiomatic planar geometry
- Representation of processive functions in Robinson arithmetic ? , relating to recursive functions
- Church's thesis is questioned by new calculation paradigm , relating to recursive functions
- The Snark, a counterexample for Church's thesis ? , relating to recursive functions
- Programming primitive functions and beyond , relating to recursive functions
- Confusing conventional notation , relating to axiomatic set theory
- A flaw in Separation Axioms ? , relating to axiomatic set theory
- Number Theory beyond Frege , relating to predicate logic