Hilbert’s Machine and the Axiom of Infinity

Hilbert’s machine is a supertask machine inspired by Hilbert’s Hotel whose functioning leads to a contradiction that compromises the Axiom of Infinity. 1. Hilbert’s Machine In the following conceptual discussion we will make use of a theoretical device that will be referred to as Hilbert’s machine, composed of the following elements (see Figure 1): (1) An infinite tape similar to those of Turing machines which is divided in two infinite parts, the left and the right side: (a) The right side is divided into an ω-ordered sequence of adjacent cells 〈ci〉i∈N 1 which are indexed from left to right as c1, c2, c3, . . . . These cells will be referred to as right cells. (b) The left side is also divided into an ω-ordered sequence of adjacent cells 〈c′i〉i∈N indexed now from right to left as c′1, c ′ 2, c ′ 3, . . . , being c ′ 1 adjacent to c1. These cells will be referred to as left cells. (2) An ω-ordered sequence of rings 〈ri〉i∈N being each ring ri initially placed on the right cell ci and permanently bound to its successor ri+1 by means of a rigid rod of the appropriate length. The rings ri will be termed Hilbert’s rings and the sequence 〈ri〉i∈N Hilbert’s chain. (3) A multidisplacement mechanism which moves simultaneously all Hilbert’s rings one cell to the left, so that the ring placed on ck, k>1 is placed on ck−1, the one placed on c1 is placed on c ′ 1, and the one placed on c ′ k, k≥1 is placed on c ′ k+1. This simultaneous displacement of all Hilbert’s rings one cell to the left will be termed multidisplacement. Multidisplacements are the only actions performed by Hilbert’s machine. Figure 1. Hilbert’s machine just before performing the second multidisplacement. The functioning of Hilbert’s machine is always subjected to the following Hilbert’s restriction: the machine will perform a multidisplacement if, and only if, the multidisplacement 1As usual, N stands for the set of natural numbers {1, 2, 3, . . . }