**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