**Category:**

Research Papers

**Date Published:**

July 30, 2023

**Keywords:**

actual infinity, potential infinity, Dedekind definition, axiom of infinity, Hilbert's machine, inconsistency of w-order, inconsistency of the actual infinity, inconsistency of actual infinite sets, theorem of the finite universe

**Abstract:**

Once made the necessary distinction between the actual infinity and the potential infinity, this paper ref{p3:finite versus infinite} of the series proves that the infinity involved in the Axiom of Infinity can only be the actual infinity, and that w-ordered collections are inconsistent, which in turn implies the inconsistency of the Axiom of Infinity itself, and the inconsistency of any infinite set when considered as a complete totality. It is also shown here (and the results will be much used in subsequent discussions) that every set is either discrete or can be ordered discretely (where being discrete means having a first and a last element and that every element (except the first) has an immediate predecessor, and an immediate successor (except the last). Obviously, the infinitist mathematics of modern physics (rarely put to the test) will be seriously affected by the inconsistency of the actual infinity (fortunately experimental physics can only be finitist and discrete). The consequences of this conclusion will be deduced in this and the following articles of the series. Here, one of such physical consequences will be demonstrated: in a consistent reality only a finite number of universes (if more than one) could exist, each with a finite number of physical objects.

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