What second order concepts tell us about the continuum

The continuum is that object that serves as the foundation of mathematical analysis - the differential and integral calculus. A well-known unsolved problem arising from the work of Cantor is that of the size of the continuum. The reason why this problem has gone unsolved in the C20th is due to the insistence on using first-order set theory. Melampus explores the phenomenological and empirical foundations of the classical concept of the continuum, constructs a model of the continuum, and derives the continuum hypothesis from the second-order Axiom of Completeness.

On the Continuum

This is the original work of Melampus, and goes into more detail than the above.

Contents

1. The problem of the continuum

2. The potential and actual infinite

3. The phenomenological and arithmetical continuum

4. Axiom of Completeness

5. Partitions and lattices

6. The potentially infinite division of the unit interval

7. The actually infinite skeleton of the continuum and the one-point compactification

8. The Derived set of the one-point compactification

9. The algebraic numbers

10. Decomposition of the skeleton

11. Lowering the floor and notional atoms

12. Cohen forcing and Cohen reals

13. Amoeba reals

14. Forcing and generic sets

15. Transcendental numbers

16. Mahler’s classification of the transcendental numbers

17. Posets and forcings in general

18. The Derived set and the continuum hypothesis

Contents

1. The problem of the continuum

2. The potential and actual infinite

3. The phenomenological and arithmetical continuum

4. Axiom of Completeness

5. Partitions and lattices

6. The potentially infinite division of the unit interval

7. The actually infinite skeleton of the continuum and the one-point compactification

8. The Derived set of the one-point compactification

9. The algebraic numbers

10. Decomposition of the skeleton

11. Lowering the floor and notional atoms

12. Cohen forcing and Cohen reals

13. Amoeba reals

14. Forcing and generic sets

15. Transcendental numbers

16. Mahler’s classification of the transcendental numbers

17. Posets and forcings in general

18. The Derived set and the continuum hypothesis

Beyond the Axiom of Completeness

Supplement to On the Continuum exploring alternative structures to the continuum other than that provided by the Axiom of Completeness.

Contents

1. The countable chain condition and the continuum

2 Fine-tuning the structure of the continuum

3. Motivation for Change

Contents

1. The countable chain condition and the continuum

2 Fine-tuning the structure of the continuum

3. Motivation for Change