Totalities of Infinite Sets

Based on the paper "Totalities of Infinite Sets" by William Johnston, here is a detailed technical summary covering the problem, methodology, key contributions, results, and significance.

1. Problem Statement

The paper addresses the limitations of Georg Cantor's traditional definition of set cardinality, which relies solely on the existence of a bijection between two sets. Under the Zermelo-Fraenkel (ZF) axiomatic system and the Continuum Hypothesis (CH), infinite real sets are categorized into only two primary infinite cardinalities: countable ($\aleph_0$) and uncountable ($\aleph_1$).

The author argues that this binary (or ternary, if distinguishing finite) classification is insufficient for distinguishing between different types of uncountable sets that share the same cardinality (e.g., the set of normal numbers vs. the set of irrationals, or the Cantor set vs. an interval). The goal is to create an enhancement to Cantor's cardinality that:

• Preserves the existing bijection-based cardinality framework.
• Provides a finer-grained comparison of arbitrary infinite sets.
• Allows for comparisons between sets of fundamentally different types (e.g., subsets of a metric space $X$ vs. subsets of its power set $P(X)$).

2. Methodology

The paper proposes four distinct constructions, collectively termed "Totalities," which refine the concept of set size. These constructions move beyond simple bijections by incorporating topological properties, measure theory, and iterative partitioning.

A. Complementary Cardinality (Definition 1)

The first construction modifies the notation of cardinality based on the cardinality of the set's complement within the real numbers ($\mathbb{R}$).

• Notation: Instead of just $|A| = \aleph_1$, the notation becomes $|A| = \aleph_1 \setminus |A^c|$.
• Examples:
  • If $A^c$ is countable, $|A| = \aleph_1 \setminus \aleph_0$.
  • If $A^c$ is finite ($n$), $|A| = \aleph_1 \setminus n$.
  • If $A^c$ is uncountable, $|A| = \aleph_1 \setminus \aleph_1$.
• Application: This distinguishes the set of normal numbers ($\mathbb{N}$) from the set of irrationals ($\mathbb{I}$). While both are uncountable, $\mathbb{I}$ has a countable complement (rationals), whereas $\mathbb{N}$ has an uncountable complement (non-normal numbers), yielding $|\mathbb{N}| = \aleph_1 \setminus \aleph_1$ and $|\mathbb{I}| = \aleph_1 \setminus \aleph_0$.

B. Step-by-Step Paradigm (Definition 2)

This method uses a recursive process to partition sets with equal cardinality into finer "totalities" ($\Omega_s$).

• Step 1: Distinguishes sets based on the countability of the set and its complement ($\Omega_0, \Omega_{1/2}, \Omega_1$).
• Step 2: Further partitions sets with $\Omega_{1/2}$ based on topological properties (e.g., whether a closed interval containing the set has an uncountable or countable complement).
• Mechanism: Each step increases the number of defined totalities, creating a hierarchy where sets with the same Cantor cardinality are separated by their topological "density" or "gaps."

C. Strategic Totality Pairs via Outer Measure (Theorem 1)

This construction aims to compare subsets of a metric space $X$ with subsets of its power set $P(X)$, addressing the fact that Cantor's theorem states $|P(X)| > |X|$.

• Method: It utilizes Metric Outer Measure.
  • For a metric space $(X, d)$, an outer measure $m$ is defined where the measure of an open ball is its diameter ($2r$).
  • For the power set $P(X)$, the Hausdorff metric is used to define a corresponding outer measure $\mu$.
• Totality Assignment:
  • Finite sets: Cardinality.
  • Countable infinite sets: $\Omega_{-1}$.
  • Uncountable sets: $\Omega_{m(A)}$ (for $A \subseteq X$) or $\Omega_{\mu(P)}$ (for $P \subseteq P(X)$).
• Result: This creates a "Strategic Totality Pair" where $X$ and $P(X)$ can have equal totalities (e.g., both $\Omega_\infty$) even though their Cantor cardinalities differ, allowing for a meaningful comparison of "size" across different set types.

D. Infinite-Step Process and Limit (Definition 3 & Theorem 2)

This construction generalizes the step-by-step approach to an infinite sequence of refinements based on Lebesgue measure.

• Mechanism: At step $n$, sets are partitioned based on the sum of lengths of open intervals covering them (or their complements).
• Notation: Totalities are denoted as $\Omega_{k/2^n}$, $\Omega_{M - k/2^n}$, and $\Omega_M$ (the "middle" infinity).
• Limit: The final totality is defined as the limit of the sequence: $|A|_\infty = \lim_{n \to \infty} |A|_n = \Omega_{s_\infty}$.
• Properties: This creates a dense ordering of totalities where between any two uncountable totalities, another exists.

3. Key Contributions

1. Terminological Shift: Introduces the term "Totality" to distinguish these refined measures from standard cardinality, emphasizing that they provide additional structural information beyond bijections.
2. Refinement of Uncountable Sets: Demonstrates that uncountable sets can be strictly ordered and compared based on the properties of their complements (Definition 1) and their topological covering properties (Definitions 2 & 4).
3. Cross-Type Comparison: Successfully constructs a framework (Theorem 1) to compare subsets of a space $X$ with subsets of $P(X)$ using metric outer measures, bypassing the strict inequality of Cantor's theorem to find "equal" sizes in a strategic sense.
4. Dense Ordering of Infinities: The limit process (Section 4) generates a continuum of infinite totalities, suggesting a dense hierarchy of infinities between $\aleph_0$ and the full continuum.

4. Results

• Differentiation of Normal Numbers: The paper successfully distinguishes the set of normal numbers from the set of irrationals. While both are uncountable, the normal numbers have a strictly "smaller" totality ($\aleph_1 \setminus \aleph_1$) compared to irrationals ($\aleph_1 \setminus \aleph_0$) because the set of non-normal numbers is uncountable.
• Equality of Totalities in Power Sets: For a metric space $X$ with infinite measure, the totalities of $X$ and $P(X)$ are shown to be equal ($\Omega_\infty$), despite $|P(X)| > |X|$ in Cantor's sense.
• Existence of Intermediate Totalities: Theorem 2 proves that in the limit construction, there is no "gap" between $\Omega_{-1}$ and $\Omega_0$, and between any two uncountable totalities, another totality exists.
• Monotonicity: All constructed totalities satisfy monotonicity (if $A \subseteq B$, then $|A| \leq |B|$).

5. Significance

• Mathematical Flexibility: The paper suggests that while the Continuum Hypothesis remains undecidable within the standard ZF axiomatic system, the definition of set size can be refined to resolve specific undecidables at the level of the refinement.
• Enhanced Intuition: It provides a "strong mathematical sense" of which infinite sets are "larger" or "smaller" in terms of their distribution and structure, rather than just their existence of a bijection.
• Application to Analysis: By linking totalities to metric outer measures and Hausdorff dimensions, the framework connects set theory with topology and measure theory, potentially offering new tools for analyzing fractals (like the Cantor set) and function spaces (like $L^2(\mathbb{R})$).
• Philosophical Implication: The conclusion posits that deductive mathematics can become "more muscular" by iteratively refining definitions to resolve undecidables, even if the underlying axiomatic system remains unchanged.

In summary, Johnston's work proposes a multi-layered approach to infinity that complements Cantor's theory, offering a richer vocabulary for describing the "size" and "structure" of infinite sets through topological and measure-theoretic lenses.