Generalized measurement on size of set

TL;DR

This paper introduces a generalized measurement framework for set size based on an expanded concept of cover, unifying dimension, measure, and cardinality, and offering new insights into infinity and set properties.

Contribution

It proposes a novel approach to measuring set size using pairs of covers and functions, extending traditional concepts to unify dimension, measure, and cardinality.

Findings

The size measure satisfies outer measure properties in general cases.

In the case of graduation 1, it satisfies measure properties.

Rewrites the Continuum Hypothesis using a dimension-based approach.

Abstract

We generalize the measurement using an expanded concept of cover, in order to provide a new approach to size of set other than cardinality. The generalized measurement has application backgrounds such as a generalized problem in dimension reduction, and has reasons from the existence of the minimum of both the positive size and the positive graduation, i.e., both the minimum is the size of the set ${0}$. The minimum of positive graduation in actual measurement provides the possibility that an object cannot be partitioned arbitrarily, e.g., an interval $[0, 1]$ cannot be partitioned by arbitrarily infinite times to keep compatible with the minimum of positive size. For the measurement on size of set, it can be assumed that this minimum is the size of ${0}$, in symbols $|{0}|$ or graduation 1. For a set $S$, we generalize any graduation as the size of a set $C_i$ where $\exists x \in S (x \in C_i)$, and $|S|$ is represented by a pair, in symbols $(C, N(C))$, where ${C} = \cup {C_i}$ and $N(C)$ is a set function on $C_i$, with $C_i$ independent of the order $i$ and $N(C)$ reflecting the quantity of $C_i$. This pair is a generalized form of box-counting dimension. The yielded size satisfies the properties of outer measure in general cases, and satisfies the properties of measure in the case of graduation 1; while in the reverse view, measure is a size using the graduation of size of an interval. As for cardinality, the yielded size is a one-to-one correspondence where only addition is allowable, a weak form of cardinality, and rewrites Continuum Hypothesis using dimension as $\omega \dot |{0,1}| = 1$. In the reverse view, cardinality of a set is a size in the graduation of the set. The generalized measurement provides a unified approach to dimension, measure, cardinality and hence infinity.