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