Further Remarks on the Infinity Tree

A 2023 paper discussed how 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. A 2025 paper discussed 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, supporting an argument that the real numbers in the interval [0, 1] are countably infinite, in a non-Cantorian theory of infinity. However, it is not claimed that there is an error in Cantor’s arguments that [0, 1] is uncountably infinite. Rather, the situation is considered as a paradox, resulting from different choices about how to represent and count the continuum of real numbers. The present, short paper addresses some further questions that mathematicians have asked, and might ask, about this approach.