# The Problem of Foundations. An Adventurous Navigation from Sets to Entities From Gödel to Thomas Aquinas

INTRODUCTION

I - THE NBG THEORY AND ITS SET-LIKE INTERPRETATION
1. Primitive Notions of the Abstract Set Theory
1.1. Classes - Sets - Membership Relation
1.2. Analogy of Being and Problem of the Universal Class
2. First derived Notions and First Group of Axioms
2.1. Equality and Other Derived Relations
2.2. Derived Relations of Inclusion
2.3. Univalent Relations
3. Second Group of Axioms
4. Third Group of Axioms
5. Other Axioms

I I - ONTOLOGICAL INTERPRETATION AND ENLARGEMENT OF NBG THEORY
1. Ontological Interpretation of the Primitive Notions of NBG Theory
1.1. Ens – Entity – Relation “is in”
1.2. Categories
2. Derived Relations and First Group of Axioms
2.1. Relation of Equality
2.2. More Derived Relations
2.3. Derived Relations of “Inclusion” - Act and Potency
2.4. Empty Ens or Empty Entity?2.5. Univalent Relations
3. Second Group of Axioms
4. Third Group of Axioms
4.1. Axiom of Replacement
5. More Axioms

I I I - THE EIGHT WAYS TO SAY TO BE IN ACCORDING TO THOMAS AQUINAS
1. Analysis of a Text by Thomas Aquinas
1.1. The Eight Ways to say to be in
2. A Formalization of the Eight Ways to say to be in
2.1. The Eighth Way to say to be in
2.2. The Ways to say to be in according to which the Part is in the Whole
2.3. The Ways to say to be in according to which the Whole is in the Part

IV - THE QUANTIFIERS: UNIVERSALITY AND EXISTENCE
1. Analogical Character of Quantifiers
1.1. Universal Quantifier
1.2. Existential Quantifier
2. Logical Existence and Real Existence
2.1. The Axiomatic System as Ens/Entity
2.2. Universal Axiomatic System

V - FORM AND MATTER
1. The Extensionalist Reductionism
2. Irreducible Principles in AT
2.1. Form and Matter as Irreducible Principles
2.2. Material Entities and Elements

VI - THE THEORY OF CAUSALITY
1. Material and Formal Causality
2. Efficient Causality and Final Causality
3. Causal Ordering and Causal Chains
3.1. Univalent Causal Chains
3.2. Analogous Causal Chains and Levels of Causality

VII - COLLECTING ALL
1. Primitive Notions of FO and First Definitions
1.1. Logical Symbols and Ontological Notions

2. Logical Axioms and Inference Rules
3. Proper Axioms
3.1. First Group of Axioms
3.2. Second Group of Axioms
3.3. Third Group of Axioms
3.4. More Axioms
4. Applications and Developments
4.1. The Quantifiers as Entities
4.2. The Axiomatic Systems as Entities
5. Form and Matter
5.1. Definitions of Different Types of Form 5.2. Different Types and Ways to conceive Matter
6. Causality Relations and Causes
6.1. Causal Chains

CONCLUSION
BIBLIOGRAPHY
INDEX OF DEFI NITIONS, AXIOMS AND THEOREMS