# PAI Private Arithmetics Institute

## Natural Numbers, Recursion, Geometry, Symmetry, Axioms, Logic, Metalanguage and more

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..

- 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