On the Continuum
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ABSTRACT
In this paper it is argued that determination of the structure of the continuum has always been an empirical matter. This thesis is supported by a demonstration in second-order logic from the Axiom of Completeness and via the one-point Alexandroff compactification of the real line that the Mahler U, T transcendental reals are Cohen reals and the Mahler S reals are amoeba reals. An important consideration is the distinction between the potential and actual infinite, which is used extensively in the derivation. |
Flash introductions to On the Continuum
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Part 1: Lattices
This presentation covers the essentials of lattice theory required to understand the theorem presented as a whole. Note: - (1) There is an exact derivation between two lattices - one called a skeleton and the other an algebra. (2) The skeleton is a partition of the continuum. (3) To solve the problem of the continuum we only have to determine what its skeleton is. (4) The Cantor set is a representing set for the continuum, but it is not the continuum because it is not homeomorphic to it. (5) There is no solution in ZFC (Zermelo Frankel Set theory) to the problem of the continuum because the Cantor set in that theory is an ambiguous structure that stands for a collection of solutions. |
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Part 2: The potential and actual infinite
Describes an important distinction implicit in set theory between the potential and actual infinite. The Cantor set is an actually infinite lattice. It contains within it a potentially infinite lattice which is the ideal of all finite subsets of the least infinite ordinal. This ideal is denoted Fin. |
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Part 3: The arithmetical continuum
The arithmetical continuum is not to be found in experience. It is a construct of science. Any theory of real numbers derives its validity from empirical observation. |
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Part 4: Complementarity and extension
Our search for the skeleton of the continuum begins with finite partitions. Immediately, two extraordinary observations emerge. (1) The skeleton must be both a chain and an antichain. (2) First order set theory is inadequate to describe the continuum because it has only one primitive notion of set membership, whereas the continuum requires a second primitive - of extension. |
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Part 5: The potentially infinite skeleton
A potentially infinite skeleton is not adequate to describe the continuum because it is non-atomic and does not cover the unit interval, since it leaves out a neighbourhood of 1. There is a distinction between first and second order logic. The widely held belief that the anything that can be expressed in second-order logic may be translated into a first-order statement is false. The Completeness Axiom is irreducibly second-order and provides a categorical structure for the continuum. It is equivalent to the Heine-Borel theorem, which states that the unit interval is closed and bounded. |
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The Derived Set of the actually infinite skeleton
The skeleton of the continuum is an actually infinite partition generated by the one-point Alexandroff compactification, which is a consequence of the Completeness Axiom. The Derived Set of the actually infinite skeleton is homeomorphic to the continuum subject to the second-order Axiom of Completeness. The structure of the Derived Set demonstrates that it is split into two parts, which never meet. The boundary is of size continuum and contains all transcendental reals. |
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Part 7: The meagre-null decomposition
A standard theorem demonstrates that the continuum may be decomposed into two sets: A: Null B: Meagre, of measure 1 in the unit interval. In fact, the skeleton of the Derived set is a collection of half-open, half-closed intervals. This illustrates the distinction between the Derived Set and the Cantor set, because the Cantor set is a collection of nowhere dense isolated boundary points. There must be two sub-skeletons of the skeleton. Furthermore, there must be two kinds of transcendental real in the boundary: A: Boundary points of absolutely zero measure B: Extension points that are non-measurable. Note
Experts are invited to substitute the term "amoeba real" for "random real". |
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Part 7: Trees and Cohen forcing
Boundary points and extension points are generated by two distinct trees. The tree of boundaries is associated with a process of constructing transcendental real numbers that are called Cohen reals. The tree of intervals is associated with transcendental numbers called Amoeba reals. Important note
The flash presentation uses the term "random real" for "amoeba real" and indicates that this is a process of forcing. This is internally consistent with the definitions initially provided, but in the literature the terms "random real" and "random forcing" are not used in precisely this way, where random forcing is associated with a family of Borel codes and a poset. Here, what might be called Amoeba forcing is not forcing in that sense, since it comprises only what is left over from the interval once Cohen forcing has been performed. Subject to the Axiom of Completeness, the continuum has only one species of forcing, which is Cohen forcing from omega into 2. |
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Part 9: Generic forcing
There is a forcing language whose purpose is to construct transcendental reals as generic ultrafilters. These transcendental numbers are generated on completion and closure of the ultrafilters of Fin, the ideal of all finite subsets of omega. An important principle is that these ultrafilters are generated by inductive sequences. The assumption that a transcendental number can be constructed at any finite stage of iteration of such an inductive sequence leads to a contradiction. So an infinite sequence of embeddings of finite Boolean algebras must be completed by an actually infinite extension. Note
Experts are invited to substitute the term "amoeba real" for "random real". |
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Part 10: Transcendental numbers
Detailed analysis of the proof the existence of Liouville numbers shows that a Liouville number is a generic ultrafilter. (This specifically falsifies the Axiom of Constructibility.) Examination of the Mahler classification of real numbers demonstrates that: (1) The Mahler S numbers are amoeba reals. They are non-measurable. (It is a further claim that the collectively form a Suslin line.) (2) The Mahler U, T numbers are Cohen reals. They have absolutely zero measure. These results are confirmed by a standard theorem demonstrating that the measure of all S numbers is 1 in the unit interval. In turn these results confirm that the previous argument in favour of a distinction between boundary and extension points is correct. Note
Experts are invited to substitute the term "amoeba real" for "random real". |
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Proof of the continuum hypothesis
The proof that the Axiom of Completeness entails that the continuum is homeomorphic to the Derived Set and that it satisfies the Continuum Hyopthesis may be found in the last section of the text: On the Continuum.
Beyond the Completeness Axiom
There is also an extension paper entitled Beyond the Completeness Axiom. By introducing how alternative models of the continuum may be constructed this work demystifies the Problem of the Continuum, by showing that the arithmetical continuum is a construct of science. It shows that the Continuum Hypothesis represents a base model for the continuum, and invites us to consider what the benefits to science might be of developing alternative models.
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