TITLE:
Countability of Infinite Paths in the Infinity Tree: Proof of the Continuum Hypothesis in a Non-Cantorian Infinity Theory
AUTHORS:
Philip C. Jackson
KEYWORDS:
Infinity, Countable, Uncountable, Diagonalization, Real Numbers, Infinity Tree, Continuum Hypothesis, Turing Machine, Infinite Time Turing Machine, Non-Cantorian
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.15 No.1,
January
27,
2025
ABSTRACT: A previous paper showed that the real numbers between 0 and 1 could be represented by an infinite tree structure, called the ‘infinity tree’, which contains only a countably infinite number of nodes and arcs. This paper discusses how a finite-state Turing machine could, in a countably infinite number of state transitions, write all the infinite paths in the infinity tree to a countably infinite tape. Hence it is argued that the real numbers in the interval [0, 1] are countably infinite in a non-Cantorian theory of infinity based on Turing machines using countably infinite space and time. In this theory, Cantor’s Continuum Hypothesis can also be proved. And in this theory, it follows that the power set of the natural numbers P(ℕ) is countably infinite, which contradicts the claim of Cantor’s Theorem for the natural numbers. However, this paper does not claim there is an error in Cantor’s arguments that [0, 1] is uncountably infinite. Rather, this paper considers the situation as a paradox, resulting from different choices about how to represent and count the continuum of real numbers.