|A Potential Problem for Alpha-Theory
Abstract: In a recent paper, Sylvia Wenmackers and Leon Horsten discuss how the concept of a fair infinite lottery can best be extended to denumerably infinite lotteries. Their paper uses and builds on the alpha-theory of Vieri Benci and Mauro Di Nasso. The purpose of this paper is to demonstrate a potential problem for alpha-theory.
In a recent paper, Sylvia Wenmackers and Leon Horsten (2010, p. 1) discuss ‘how the concept of a fair infinite lottery can best be extended to denumerably infinite lotteries’. Their paper uses and builds on the alpha-theory of Vieri Benci and Mauro Di Nasso (2003, p. 3) where ‘an interpretation of the ideal number α can be given as the numerosity of the set of natural numbers’. As Wenmackers and Horsten (2010, §6.1) also write ‘numerosity theory introduces α as the size of the natural numbers’. The purpose of this paper is to demonstrate a potential problem for alpha-theory. I suggest, via a discussion of two ways of performing infinitely many tasks in a finite time, that α is not the size of the natural numbers (rather, α is best thought of as the size of α).
First, imagine that Standard Sam performs a supertask in the standard manner, completing infinitely many tasks in, e.g., one hour. Let Sam perform task 1 in 1/2 hour, task 2 in 1/4 hour, task 3 in 1/8 hour, etc.
Second, imagine that Alpha Alvin, inspired by alpha-theory, reasons as follows. ‘There are α many natural numbers. And so if I perform one task every 1/α hours, then in one hour I will have performed all α tasks, just as if there were 7 tasks, and I performed one task every 1/7 hour, then I would complete all 7 tasks in one hour.’
Note that formally speaking, there is no problem. For Sam, the infinite sum 1/2 + 1/4 + 1/8… = 1. For Alvin, α * (1/ α) = 1. It appears that both finish their tasks in exactly one hour. But informally there is a problem. By assumption, Sam and Alvin are completing the same number of tasks, namely one task for each natural number. And for every task, Sam took longer to finish the task than Alvin did. For every task, Sam takes a finite, real amount of time; Alvin takes an infinitesimal time. Since Sam takes more time for every task, then Sam must take more time to complete all of the tasks.
Again it is important to note that this is an informal argument. For task 1, Sam takes 1/2 – 1/α longer than Alvin does. For task 2, Sam takes 1/4 – 1/α longer than Alvin does. And so on. To determine how much longer Sam takes overall, we might attempt to take an infinite sum of these differences, which equals the difference of the two infinite sums. The problem is that, formally speaking, an infinite sum of infinitesimals is not defined. However, I suggest that the following principle is obvious: If person B takes longer than person C to complete every one of some number of tasks (finite or infinite), then person B takes longer to complete all of the tasks. And so Sam, by taking longer on every task, must take longer overall. Put another way, it seems bizarre for Sam to argue, ‘Yes, I took longer to complete every task. Yet we took the same amount of time overall.’ At the least, if this reasoning of Sam’s is to succeed, some further explanation is needed.
In the remainder of the paper, let us attempt to diagnose the problem. α, an infinite nonstandard number, has underlying it the structure ω +(ω*+ ω)Ө + ω*. It is when completing α many tasks, of this structure, that one must complete a task every 1/α hours to complete them all in one hour. ω is a proper initial segment of ω +(ω*+ ω)Ө + ω*. It is not surprising then that Alvin seems to finish his task in less time than Sam, that is, in less than one hour. If Alvin had to perform an infinite task of structure ω +(ω*+ ω)Ө + ω*, then he would take one hour at a rate of one task every 1/α hours; but Alvin must only perform a part of this, namely, ω many tasks. And so again, via a slightly different route, informal reasoning supports the conclusion that Alvin takes less time to complete his (ω many) tasks than Sam does.
Just because there is no formal contradiction in assuming that there are α many natural numbers does not mean that this assumption makes sense. In this paper, I have suggested that the assumption is problematic. Alvin, by assuming that there are α many natural numbers, has an argument that the can complete a supertask in one hour by performing a task every 1/α hours. And yet by comparing Alvin with Sam, we see that Alvin seems to take less time than Sam does to complete his tasks, that is, less time than one hour. I suggest that the problem, simply put, is that ω is smaller than α, as ω is a proper initial segment of α. Just as it does not make sense to assume that there are finitely many natural numbers, because any finite number is a proper initial segment of ω, so too it does not make sense to assume that there are an infinite, nonstandard number of natural numbers, because ω is a proper initial segment of any infinite, nonstandard number.
Benci, V., & Di Nasso, M. (2003). Alpha-theory: An elementary axiomatic for nonstandard
analysis. Expositiones Mathematicae, 21, 355–386.
Wenmackers. S. and Horsten, L. (Published Online 2010). Fair Infinite Lotteries. Synthese.