Taking the Pirahã seriously

Highlights

• We provide first order theories axiomatizing a bounded universe of finite or infinite natural numbers.

• The theories are undecidable, and consistent if Nelson’s predicative arithmetic is.

• In these theories some of the basic features of the theory of grossone can be proved.

• We can prove that each set has cardinality bigger than every proper subset.

• Every series converges, and is invariant under any rearrangement of its terms.

Abstract

We propose first order formal theories ${\mathrm{\Gamma }}_n$, with $n⩾1$, and $\mathrm{\Gamma }={\bigcup }_{n⩾1}{\mathrm{\Gamma }}_n$, which can be roughly described as follows: each one of these theories axiomatizes a bounded universe, with a greatest element, modeled on Sergeyev’s so-called grossone; each such theory is consistent if predicative arithmetic $I{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_1$ is; inside each such theory one can represent (in a weak, precisely specified, sense) the partial computable functions, and thus develop computability theory; each such theory is undecidable; the consistency of ${\mathrm{\Gamma }}_n$ implies the consistency of ${\mathrm{\Gamma }}_n\cup \{{\text{Con}}_{{\mathrm{\Gamma }}_n}\}$, where ${\text{Con}}_{{\mathrm{\Gamma }}_n}$ “asserts” the consistency of ${\mathrm{\Gamma }}_n$ (this however does not conflict with Gödel’s Second Incompleteness Theorem); if $n>1$, then there is a precise way in which we can say that ${\mathrm{\Gamma }}_n$ proves that each set has cardinality bigger than every proper subset, although two sets have the same cardinality if and only if they are bijective; if $n>2$, inside ${\mathrm{\Gamma }}_n$ there is a precise sense in which we can talk about integers, rational numbers, and real numbers; in particular, we can develop some measure theory; we can show that every series converges, and is invariant under any rearrangement of its terms (at least, those series and those rearrangements we are allowed to talk about); we also give a basic example, showing that even transcendental functions can be approximated up to infinitesimals in our theories: this example seems to provide a general method to replace a significant part of the mathematics of the continuum by discrete mathematics.

Introduction

In a series of papers (just to mention some of the papers that are more detailed about foundational aspects, see for instance [1], [2], [3], [4], [5]; se also the book [6]), Sergeyev motivates the introduction of an infinite number, called the grossone, and discusses its main properties, as well as some applications, which range from numerical analysis, dynamical systems, even to logic and the theory of computation (see e.g., [7], [8]). The advantages for the use of grossone are clear: cardinalities behave better (for instance, one avoids the well-known Hilbert Grand Hotel paradox), the sum of a series always exists and is invariant under rearrangements of its terms, infinitesimals and infinite elements exist as in nonstandard analysis, etc.

Sergeyev also introduces some basic principles for the theory of the grossone. In listing below these principles, we accompany them by some comments which we hope may serve as a guide to the leading ideas worked out in our paper.

Postulate (1). There exist infinite and infinitesimal objects, but human beings and machines can only execute a finite number of operations.

Comment: we accept without comments the first half of this postulate, which is one of the basic principles of nonstandard analysis; we also agree on the second half of the postulate, to the point that in order to deal with infinite and infinitesimal objects, we will propose a very weak and resource bounded formal system.

Postulate (2). We cannot tell what the objects we deal with are, we just shall construct more powerful tools that will allow us to improve our capacities to observe and to describe properties of mathematical objects.

Comment: this is a non-technical postulate. One possible interpretation is that the general theory is not a formal axiomatic system, and hence, new tools and new objects may be introduced during the development of the theory. Although we respect this point of view, we do not adopt it in the sequel.

Postulate (3). The principle the part is less than the whole is applied to all numbers (finite, infinite and infinitesimal) and to all sets and processes (finite and infinite).

Comment: this is a very interesting point, and we fully support it.

There is yet another, more technical, principle, which is a strengthening of (1):

Postulate (4). There exists an infinite natural number, $①$, which is assumed to satisfy all properties which are shared by the usual natural numbers. (This postulate is called by Sergeyev the Infinite Unit Axiom.)

Comment: although we accept the existence of $①$ (called grossone), the principle stating that the grossone enjoys all properties which are shared by all natural numbers is, in our opinion, a little bit problematic, as will be explained later.

Although not formulated with reference to a fixed formal language, Postulate (4) is reminiscent of the so called Transfer Principle of nonstandard analysis: any first order property which holds of the usual reals, holds of nonstandard reals as well. However, the theory of grossone has at least one important difference with respect to nonstandard analysis. While in nonstandard analysis the natural numbers constitute an unbounded set, in Sergeyev’s theory of the grossone, $①$ is the greatest natural number and numbers like $①+1\text{,}①+①$ etc. are no longer natural numbers: the enlarged set of numbers forms the set of the so called extended natural numbers.

In our opinion, this principle, according to which natural numbers constitute a bounded set, is an important difference between Sergeyev’s theory of the grossone, and nonstandard analysis. Moreover, in [1], the author introduces the interesting example of the Pirahã people, whose members identify all numbers greater than 2, and call them many. One way of stretching the Pirahã point of view, would be to suggest that all numbers greater than $①$ are to be identified. As remarked above, this is not however Sergeyev’s point of view: indeed, in the light of Postulate (2), in order to do computations involving $①$, Sergeyev introduces the extended natural numbers, which may be much larger than $①$. In our opinion, this is quite natural, but in this way the difference between extended natural numbers and nonstandard natural numbers (i.e., the nonstandard natural numbers introduced by nonstandard analysis), becomes very tenuous, for, instead of working in the theory of grossone, we might as well work in the nonstandard natural numbers and stipulate that the grossone is a distinguished infinite natural number.

In this paper, we will follow a different approach. That is, we will follow more closely Sergeyev’s initial idea that the grossone is the greatest natural number, and taking the Pirahã point of view seriously, we will identify all numbers greater than the grossone. However, we will require some properties ensuring that the grossone is a really large number.

Our intuitive idea is that the grossone is a number so big that we will never need to use it. The use of the grossone would lead to paradoxes like $①+1=①$, which conflicts with the cancelation law (indeed, by cancelation we would get $1=0$). But, firstly, counterintuitive results do not mean contradictions, and, secondly, as far as our computations involve numbers smaller than the grossone we do not get counterintuitive results. Finally, as we said before, our axioms will guarantee that the grossone is so big that we have many infinite smaller “substitutes” of it, which may replace it without danger of undesired consequences.

Let us try to be more precise about this point. Our universe will be the numbers smaller than $①$, but we require the existence of many infinite natural numbers less than $①$, i.e. $\cdots {\infty }_{-n}\ll {\infty }_{-2}\ll {\infty }_{-1}\ll ①$, where $x\ll y$ means that y is exponentially larger than x. So, we can start from the subuniverse consisting of the numbers less than ${\infty }_{-n}$, with a sufficiently large $n>0$ , and then expand the universe if necessary, by using the numbers less than ${\infty }_{-n+k}$ for some $k<n$. We will see that inside the resulting sets, we can perform part of mathematical analysis, we can represent computable functions, we can code sets of natural numbers and introduce cardinalities so that there is no bijection between a set and a proper subset of it, thus satisfying Postulate (3).

As said before, while our approach fully satisfies Postulates (1) and (3), we do not follow Postulates (2) and (4). As to Postulate (2), we try to do as much mathematics as possible using only what in [1] are considered to be the natural numbers. So, our universe constitutes a bounded set with maximum $①$.

Moreover, a possible interpretation of Postulate (4) (although this postulate is not formulated with reference to any formal language) is that Sergeyev’s extended natural numbers should be viewed to be elementarily equivalent to the finite natural numbers. Indeed, let us formalize Postulate (4) as follows:

If $\mathrm{\Phi }(x)$ is a formula such that for every finite natural number n, $\mathrm{\Phi }(n)$ holds in the standard model $\mathbb{N}$ of the natural numbers, then $\mathrm{\Phi }(①)$ holds in the extended natural numbers. .

Then given a sentence $\phi$ of arithmetic, the formula $\mathrm{\Phi }(x)≕\phi \wedge x=x$ is satisfied by all natural numbers in the standard model $\mathbb{N}$ if and only if $\phi$ is true in $\mathbb{N}$, if and only if it is satisfied by $①$ in the extended natural numbers (by Postulate (4)), if and only if $\phi$ is true in the extended natural numbers. Hence, as far as we can see, the formal theory of the extended natural numbers is not even arithmetical, and its consistency seems to require infinitary methods.

To the contrary, the theory presented in this paper is much weaker: its set of theorems is computably enumerable and its consistency is implied by the consistency of Nelson’s Predicative Arithmetic $I{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_1$ (that is, Robinson’s Q, plus induction for bounded formulas plus the totality of the function $x^{|y|}$, see [9] or [10]).

Summing up, the basic ideas in this paper are the following:

• (a) We try to do mathematics using only the natural numbers in the sense of [1], that is, the numbers smaller than the grossone. We do not use other numbers. The territory beyond the grossone is completely out of reach: “hic sunt leones”, as, they say, ancient cartographers used to label in their maps unexplored territories.

• (b) We require the presence of infinite natural numbers ${\infty }_{-1}\text{,}\dots \text{,}{\infty }_{-n}$ smaller than the grossone. The numbers smaller than ${\infty }_{-n}$ play the same role as the numbers smaller than the grossone in [1], and the numbers smaller than the grossone play the role of the extended natural numbers in [1]: see Fig. 1.

• (c) Unlike Sergeyev, we give a precise axiomatization of the numbers we are going to use. Indeed, although we believe that formalism does not replace mathematical intuition, we feel safer when we can prove our results in a formal system. Moreover we take care of problems like consistency, and more generally, we try to use the finitistic means of Nelson’s Predicative Arithmetic.

• (d) Unlike Sergeyev, we do not use things like (finite or infinite) rational numbers or real numbers, we only need natural numbers. The natural numbers smaller than ${\infty }_{-n}$ for n sufficiently large are our starting point. These numbers are iterated logarithms of other numbers and hence their iterated exponentials exist. Integers are coded by natural numbers less than $2·{\infty }_{-n}^2$, and hence, less than ${\infty }_{-n+1}$. We can approximate real numbers by rational numbers with denominator ${\infty }_{-n}$. Since their denominator is fixed, we can represent these numbers using their numerator only, which will be an integer in $[-{\infty }_{-n}^2\text{,}{\infty }_{-n}^2]$, and hence may be coded by a natural number less than ${\infty }_{-n+1}$. In particular, the numbers we need are all less than the grossone. But if we want to code subsets of $[0\text{,}{\infty }_{-n+1}]$ or functions on $[0\text{,}{\infty }_{-n+1}]$ we need larger natural numbers, that is, natural numbers in $[0\text{,}{\infty }_{-n+2}]$. In any case, we can use the natural numbers less than ${\infty }_{-n}$ with n sufficiently large, as ground natural numbers and then use larger natural numbers to represent infinite or infinitesimal reals, or sets, or functions on them.

• (e) The weakness of our theory allows us to isolate those operations which are feasible, that is, polynomial time. In developing our setting our first reference has been the so-called bounded arithmetic, a formal system which was introduced by Nelson [9] for foundational purposes, aiming at a predicative arithmetic; interest in bounded arithmetic subsequently grew up, due to the work of Buss [11], who gave a proof theoretic characterization of the polynomial time computable functions. (Wilkie and Paris [10] have also shown that Gödel’s incompleteness results hold in bounded arithmetic.) Nevertheless, contrary to what happens in bounded arithmetic, where some functions like the exponential function are partial, in our case, exponential and factorial are total, but they may take value $①$. The exponential and factorial functions are fundamental in our setting, in that they allow us to jump to the next infinite number, which is not reachable by polynomial time operations only.

The paper is organized as follows. In Section 2 we exhibit the axioms of our theory. We will present many different theories ${\mathrm{\Gamma }}_n$, all interpretable in $I{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_1$, as well as their union $\mathrm{\Gamma }$. In Section 3 we prove that the computable functions are definable, in a weak sense, in ${\mathrm{\Gamma }}_n$. In Section 4 we prove that ${\mathrm{\Gamma }}_n$ is interpretable in $I{\mathrm{\Delta }}_0+{\mathrm{\Omega }}_1$, and in Section 5 we prove that some elementary mathematics can be performed inside ${\mathrm{\Gamma }}_n$. In particular, inside ${\mathrm{\Gamma }}_n$ we can construct structures which are similar to the integers and to the rationals, we can code bounded sets, and we can do some elementary calculus. We also give a basic example, showing that even transcendental functions can be approximated up to infinitesimals in our system.

As far as we know, the setting we propose, as well as the basic technical results, are new, and have not appeared earlier in the literature.