Abstract
The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s (J Symb Log 53(2):481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s (in: van Heijnoort (ed) From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, and a theory of objects that theory calls “sets”. While Dedekind’s (in: Essays on the theory of numbers, pp. 14–58, 1888. http://www.gutenberg.org/ebooks/21016) argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored?
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Notes
“a system S is said to be infinite when it is similar to a proper part of itself (par. 32), in the contrary S is said to be a finite system”.
Looking into how intuitions get formed may help. Pantsar (2014) argues that our most basic conceptual metaphor regarding infinity is that of an unending process. You start counting, and could go on counting into eternity. A process which could go on indefinitely is reified into a set which is actually infinite. One moves from a possible process to an actual set. (Bear in mind that this transition is supposed to underlie the everyman's conception of infinity, acquired and entrenched ever deeper as our cognition develops.) As Pantsar explains, the transition is metaphorical. And, I add, metaphors are unfit for justificatory purposes. This quibble aside, counting indefinitely is patently not counting an infinity. If the slippage from indefiniteness to infinity underlies the everyman's thinking about infinity, elucidating it makes for a nice debunking of any epistemological role untutored intuitions of infinity were thought to play. No wonder untutored intuitions need tutelage.
To the charge that “objects cannot be conjured into existence by stipulation” (made in a different connection by Potter and Smiley (2002), p. 337), a straightforward reply is that consistent sets of set-theoretical axioms have non-empty models; objects aren't made up.
As Lavers (2016) reconstrues it, central to Carnap's approach to infinity (from as early as 1934 to as late as 1950) is the use of coordinate languages. Within syntactic frameworks which mention real numbers as coordinates, the existence of infinitely many coordinates is supposedly ensured as a matter of logic. It is unclear what all this amounts to. Saying there are infinitely many real numbers is less than saying there is a set of real numbers, and that set is infinite. Moreover, if the theory couched in terms of a particular syntactic framework (a coordinate language) is consistent, then it has models. As Lavers (2016) recounts, Carnap insisted that the infinitely many positions which correspond to real numbers are not best thought of as objects. Why not—what should incline one to think positions aren't themselves some peculiar objects? Resnik (1981, p. 530) is clear: “I view patterns and their positions as abstract entities. “Carnap scholarship aside, it is clear we don't get an ontological free lunch of positions (and an infinite one, at that) from the mere use of coordinate languages.
It seems ontological seriousness compels us to distinguish between Platonists about abstract objects called “sets” (like Quine), and Platonists about sets (like Frege). This is no threat to semantic ascent. Tailoring an example from Quine (1948), the city of Naples is properly called “Naples” even if no meanings roam the city, but a brewery could also be called “Naples” without thereby becoming the beloved city.
Attempted extrinsic justifications abound. Maddy (1988) invokes Cantorian finitism, for which “infinite sets are like finite ones”. That is, if infinite sets exist, their analogy to finite ones should be maximized. This says nothing about whether infinite sets do exist or not. Another example is Zermelo (1908), who approaches set theory in the “anything short of contradiction” view of sets; but why should the universe of sets be maximal as opposed to minimal?
Zermelo (1908, p. 200) writes: “Set theory is that branch of mathematics whose task is to investigate mathematically the fundamental notions “number”, “order”, and “function”, taking them in their pristine, simple form, and to develop thereby the logical foundations of all of arithmetic and analysis”.
Russell notes (1919, pp. 132–133) that, in the absence of the axiom of infinity, set theory “cannot deal with infinite integers or with irrationals. Thus the theory of the transfinite and the theory of real numbers fails us”. Presumably, asking if real numbers can be represented as sets, Russell isn't asking whether they can be represented as themselves. So he must think we have reason to treat numbers and sets as distinct, unless we are convinced otherwise.
True, we might devise theories about objects called “sets”. One constraint on such theories might be that real arithmetic and functional analysis should be representable in them. But looking into the axiom of infinity is ontologically serious business. We need to talk about real sets (if any there be).
Given that we have (more than) one concept of infinite set, how could there be no infinite set? The question presupposes an instance of Frege's Basic Law V (1888, Part I): (∀F)(∀G)(∀x)((Fx ↔ Gx) ↔ (εx.Fx = εx.Gx)). Taking F as G, this implies any concept has an extension. Law V leads to Russell's paradox.
Cohen (1966, pp. 22–23) shows how a finite first-order arithmetic can be developed without the assumption, or the result, that there is a simply infinite set. Cohen (1966, p. 23) develops a system of finite set-theoretical arithmetic having axioms for extensionality, the existence of an empty set, pairing, and union. In terms of these axioms, Cohen (p. 24) takes the successor of an integer x to be x∪{x} such that any two integers are comparable (either one is a member of another, or they are identical), and integers are transitive (members of members of an integer x are members of x). The point of finite arithmetic (p. 22) is that “all of traditional elementary number theory can be formulated in it [even though] in this system the elements are to be thought of as finite sets”.
Indeed, Dedekind uses a variety of terms in roughly equivalent ways: system, collection, class. Some others might be added: plurality, set, aggregate, etc. Belonging to the pre-Russell era of set theory, Dedekind could not suspect these terms would subsequently be used differently. But in the post-Russell usage, it is not clear why my possible objects of thought form a set.
For instance, Skolem (1922, p. 300) objects “sets are not generated univocally by applications of” “the principle of choice”. Many choice sets exist as long as not all sets chosen from are singletons. No concept or property is needed to uniquely define the choice set, which runs against the “demand that every set be definable”. Along with Skolem, Fregeans should be worried about this. Another worry, more widely shared, is that the Banach–Tarski paradox follows from the axiom of choice (e.g., Maddy 2011, p. 33).
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Acknowledgements
The paper is based on a talk given at the CLMPS in Helsinki (August 2015). I greatly benefited from conversations with Jim Cargile (UVa), Nora Grigore (UT Austin), Markus Pantsar (Helsinki), Gabriel Săndoiu (Sfântul Sava College), and Susan Vineberg (Wayne State).
Funding
This work was supported by the Jefferson Scholars Foundation through a John S. Lillard fellowship.
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Mărăşoiu, A. Why Believe Infinite Sets Exist?. Axiomathes 28, 447–460 (2018). https://doi.org/10.1007/s10516-018-9375-5
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DOI: https://doi.org/10.1007/s10516-018-9375-5