# Construction and Set Theory

## TL;DR

This paper presents a construction-based perspective on set theory, proposing a logarithmic-time search algorithm for set membership that relates to the Generalized Continuum Hypothesis under information minimization assumptions.

## Contribution

It introduces a novel construction approach to set theory and outlines a logarithmic-time membership search algorithm linked to the GCH.

## Key findings

- A logarithmic-time search algorithm for set membership.
- Equivalence of membership decision complexity to GCH under information minimization.
- Construction perspective offers new insights into set representation and decision processes.

## Abstract

This paper argues that mathematical objects are constructions and that constructions introduce a flexibility in the ways that mathematical objects are represented (as sets of binary sequences for example) and presented (in a particular order for example). The construction approach is then applied to searching for a mathematical object in a set, and a logarithm-time search algorithm outlined which applies to a set X of all binary sequences of length ordinal $\beta$ with a binary label appended to each sequence to indicate that sequence is a member of X or not. It follows that deciding membership of a set for a given binary sequence of length of binary sequence of cardinal length $\beta$ takes $\beta+1$ bits, which is shown to be equivalent to the Generalised Continuum Hypothesis on the assumption that information is minimised when a mathematical object is created.