31 Infinity One by One

31 Infinity One by One Chapter of the book Infinity Put to the Test by Antonio León available HERE Abstract.–Infinitist orthodoxy assumes the existence, as a complete totality, of the set of natural numbers, each of whose elements is defined by adding one unity to its immediate predecessor. The result of adding infinitely many units to the first natural number is not an infinite number but an infinitude of finite natural numbers. By means of the Unary System of Numeration, this chapter analyzes the consequences of such an assumption. Keywords: unary numeral system, unary table of natural numbers, actual infinity, potential infinity. The unary numeral system P570 A numeral is not a number but the symbol we use to represent a number. Thus, the numeral “5” is the symbol for the number 5 in the usual decimal numeral system. Perhaps the most primitive way to represent numbers [1] is what we now call the unary numeral system (UNS). As its name suggests, only one numeral is needed to represent any natural number. Here we will use the numeral “1”. The successive natural numbers will then be written as: 1, 11, 111, 1111, 11111, 111111, . P571 Although, for obvious reasons, the UNS is not the most appropriate for complex arithmetic calculations, it is the system that best represents the essential nature of the natural numbers: each natural number is exactly one unit greater than its immediate predecessor, and then the unary expression of each natural number has exactly one numeral more than the unary expression of its immediate predecessor. In addition, the UNS suggests a recursive arithmetic definition of the natural numbers: starting from the first of them, the number 1, add one unit to define the next one. P572 The result of defining the successive natural numbers (all of them finite) by adding one unit to the first natural number, and then to the successive numbers resulting from each of the infinitely many successive additions, is not an infinite number but an infinitude of finite numbers, each one unit greater than its immediate predecessor. In conformity with the hypothesis of the actual infinity, all these infinitely many finite natural numbers exist as a complete totality. Or in terms of the UNS, according to the infinitist orthodoxy it is possible to define infinitely many finite 1 2 Infinity One by One strings of 1s, each with one numeral 1 more than its immediate predecessor, without ever reaching a string with infinitely many 1s. On this belief is axiomatically founded the infinitist paradise. The Axiom of Infinity say, basically, the same: ∃N(∅∈ N ∧∀x ∈ N(x ∪ {x}∈ N)) (Chapter 4). P573 Let us put to the test the above hypothesis on the existence of an actual infinitude of finite numbers, each one unit greater than its immediate predecessor. For this, consider a special unary writing machine (UWM) capable of writing horizontal strings of 1s of any finite length. Now let UWM work according to the following conditions: a) On an empty tape, and at each of the successive instants ti, and only at them, of an ω-ordered sequence of instants htii in the real interval (ta,tb) whose limit is tb, UWM writes a first numeral 1, or a numeral 1 on the right side of the last numeral 1 written by UWM. At instant tb, UWM writes nothing and stops. Writing head Tape … … 1111 … UWM Figura 31.1 – The unary writing machine on the point of writing the fifth numeral, i.e. the number 5 in the unary numeral system. P574 From the supposed existence of the sequence of the natural numbers as a complete totality (hypothesis of the actual infinity subsumed into the Axiom of Infinity) and from the functioning of UWM, the following two theorems are immediately deduced: Theorema) P574a: At tb, the string S1 written by UWM is finite. Proof.-Let t be any instant in the interval (ta,tb). It holds: ∃v ∈ N : tv <t<tv+1. Therefore, at the instant t, the string S1 has a finite number v of 1s. So then, and being t any instant of (ta,tb), the string S1 is finite in the whole interval (ta,tb). Alternatively, if T is the set of all instants of (ta,tb) at which UWM has written only a finite string of 1s, the complementary set T of T in (ta,tb) is the empty set. And considering that at tb no numeral is written, S1 can only be finite. Theoremb) P574b : At tb, the string S1written by UWM is not finite. Proof n.-Let n be any natural number. If S1 were a finite string of n numerals 1, UME would not have written the corresponding numeral The unary table of the natural numbers 3 1 at each of the successive instants tn+1, tn+2, tn+2 . of htii, what is not the case. So then, at tb the string S1 is not finite. P575 Again a contradiction, and behind it the same cause: the actual infinity hypothesis. The belief that the infinite collections exist as complete totalities. The unary table of the natural numbers P576 Consider now the following ω-ordered table U of the natural numbers written in the UNS: Row r1: 1 (1) Row r2: 11 (2) Row r3: 111 (3) Row r4: 1111 (4) Row r5: 11111 (5) . (6) The nth row of U, symbolically rn, corresponds to the unary representation of the natural number n, which consists of a string of exactly n numerals “1”. According to the hypothesis of the actual infinity, the infinitely many rows of U, one for each natural number, do exist all at once, as a complete totality. P577 The number of rows of U is the same as the number of the natural numbers, i.e. ℵo, the cardinal of the set of the natural numbers. According to the infinitist orthodoxy, ℵo is the smallest infinite cardinal, the smallest number greater than all finite natural numbers (see Chapters 4 and 19 on the actual infinity and aleph null respectively). P578 The first column of U has ℵo elements, one for each row, one for each natural number. Since each element of this column belongs to a different row and no other column has more elements than this first column (it could easily be proved that each column of U has ℵo elements), we can say this first column defines the number of rows of U, in the sense that the first element of each row is a different element of this first column, and then a one to one correspondence f between the rows hrii of U and the elements hc1ii of its first column can be defined: f(ri)= c1i, ∀ri ∈ T (7) 4 Infinity One by One However, while the number of rows of U is completely defined by the number of 1s of its first column, the number of its columns is highly problematic, as we will immediately see. P579 Being each row rn composed of exactly n numerals “1”, and being each of those numerals an element of a different column, that row ensures the existence of at least n columns in U. It is in this sense that we will say that rn defines exactly n columns: r1 = 1 (r1 defines 1 column) (8) r2 = 11 (r2 defines 2 columns) (9) r3 = 111 (r3 defines 3 columns) (10) r4 = 1111 (r4 defines 4 columns) (11) . (12) n rn = 111. .111 (rn defines n columns) (13) . (14) P580 Let’s begin by proving the number of columns of the table U cannot be finite. In effect, let n be any natural number. U cannot have n columns because in that case the number n + 1 would not belong to the table: the unary representation of that number is a string of n + 1 numerals “1” and then a row of U that defines n + 1 columns. Thus, whatsoever be the finite number n, U cannot have n columns. P581 And now we will prove the number of columns of U cannot be infinite either. Since each row is the unary expression of a natural number and all natural numbers are finite, each row rn consists of a finite string of n numerals “1”. So, every row of U defines a finite number of columns. Or in other words, since no natural number is infinite, no row defines infinitely many columns. But if no row defines an infinite number of columns, U cannot have an infinite number of columns, unless the number of its columns is defined not by one row but by a certain number of rows. We will examine now this possibility. P582 Assume the infinite number of columns (C from now on) of the table U is not defined by a particular row but by a group of rows, even by the whole table. Evidently, if a group of rows (or the whole table) is needed in order to define C, then at least two rows of the group will contribute together to the definition of C. Where contribute together means that each row defines certain columns that the other does not and vice versa. The unary table of the natural numbers 5 P583 Let rk and rn be any two of those contributing rows. If rk and rn contribute together to define C, then rk will define certain columns that rn does not, and vice versa. Otherwise only one of them would be necessary in order to define C. P584 Now then, since k and n are natural numbers we will have either k<n or k>n.

Load more