What second-order concepts tell us about the continuum
What follows is the abstract of my work on the continuum problem. This work is extensive, and has been evolving over many years. Earlier versions of the material are also presented below, because they contain other ideas that could be of benefit. My most considered version of this work is the most recent essay.
- The classical concept of the arithmetical continuum gives rise to the second-order Axiom of Completeness. Since this axiom is a second-order principle an excursus on second-order logic investigates the difference between proof in first-order logic and demonstration by means of second-order concepts. Not all methods of inference cohere with an axiomatic system and the nature of the proof that may be expected is an exercise in philosophical mathematics.
- A phenomenological investigation into the origin of our concept of the continuum shows that the arithmetical continuum is not a priori, that it is the subject of an empirical enquiry and that the standard model of the continuum is an idealised empirical object. The Zeno paradoxes lead to the notion of the continuum as a complete aggregate of limits.
- The Axiom of Completeness embraces a distinction between the potential infinite and the actual infinite since it adds all real numbers as the limits of potentially infinite sequences.
- Ordinals are data types and the Cantor set is merely a representing set of the reals in set-theory that cannot be identified with the arithmetical continuum.
- The view that set theory is the foundation of mathematics is false. The identification of the actually infinite collection of all ordinals with the potentially infinite aggregate of all natural numbers is a paralogism. Set theory without the Axiom of Completeness has no existential statement to call into being real numbers. Set theory is not in fact founded upon the single primitive of set membership. The need to systematically distinguish potentially infinite aggregates from their actually infinite counterparts leads to the introduction of an operation of completeness as the operation of taking a potentially infinite collection and presuming it to form a completed totality. It is also demonstrated that infinite aggregates may have simultaneously different descriptions with different properties. The set of all rational numbers is simultaneously a potentially infinite aggregate subject to the natural order whose order-type upon completion is and a potentially infinite aggregate subject to the process of generation, whose order-type is . The properties of the actually infinite Cantor set and its potentially infinite subset, Fin, which is the set of all finite subsets of , lead to a model of the arithmetical continuum as a double tree structure. An axiom of indestructibility of extension that every proper part of an extended portion of space is extended is required. The lattice structure of the continuum and the relation of exponentiation and logarithm upon the lattice reveal the centrality of the skeleton of the continuum to its structure. A model of the continuum which identifies it as the Derived Set by exponentiation of the one-point compactification of the skeleton of the continuum is advanced together with its complete Boolean algebra representation by the Cantor set. The importance of the inexhaustible boundary of the Cantor set in this model is demonstrated.
- The model of the Derived Set is proven from the Axiom of Completeness in the form of the Heine-Borel Theorem.
- The null-meagre decomposition is consistent with the model and shows that there are both points of zero measure and infinitesimals in the continuum.
- The relationship between real numbers in second-order theory and real number generators called ultrafilters in first order set theory is investigated leading to an excursus on ultrafilters, generic sets and forcing arguments. Cohen forcing is described. Concerning the Mahler classification of transcendental numbers theorems are advanced that every Liouville number is a point of zero measure and the transcendental S numbers of the Mahler classification are infinitesimals.
- A forcing theorem shows that subject to the Axiom of Completeness there is only Cohen forcing in the model of the continuum and that all transcendental numbers of the Mahler classification correspond to ultrafilters produced by Cohen forcing. A theorem that the Completeness Axiom entails the Continuum Hypothesis follows from this, the limitation on the number of data-types and the relationship between the skeleton and the lattice derived by exponentiation from it.
- There is an excursus on the differences between C19th and C20th mathematics.
The wilful blindness of Gandhari - detail from The Temple of Karma Mara by Peter Fekete