TITLE:
The Infinity Tree: Representing Infinities of Real Numbers with Countably Infinite Tree Structures
AUTHORS:
Philip C. Jackson
KEYWORDS:
Infinity, Countable, Diagonalization, Real Numbers, Tree Structure, Infinity Tree, Continuum Hypothesis
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.13 No.4,
April
28,
2023
ABSTRACT: This paper discusses how the infinite set of real numbers between 0 and 1 could be represented by a countably infinite tree structure which would avoid Cantor’s diagonalization argument that the set of real numbers is not countably infinite. Likewise, countably infinite tree structures could represent all real numbers, and all points in any number of dimensions in multi-dimensional spaces. The objective of this paper is not to overturn previous research based on Cantor’s argument, but to suggest that this situation may be treated as a definitional or axiomatic choice. This paper proposes a “non-Cantorian” branch of cardinality theory, representing all these infinities with countably infinite tree structures. This approach would be consistent with the Continuum Hypothesis.