Abstract
Transfinite recursion is an essential component of set theory. In this paper, we seek intrinsically justified reasons for believing in recursion and the notions of higher computation that surround it. In doing this, we consider several kinds of recursion principles and prove results concerning their relation to one another. We then consider philosophical motivations for these formal principles coming from the idea that computational notions lie at the core of our conception of set. This is significant because, while the iterative conception of set has been widely recognized as insufficient to establish replacement and recursion, its supplementation by considerations pertaining to algorithms suggests a new and philosophically well-motivated reason to believe in such principles.
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Notes
For a modern treatment, see Jech (2003, p. 40). This result is sometimes called the perfect set theorem.
A real number within a set \(X\) is isolated in X if none of its neighbors are in \(X\)—that is, if there exists an open interval of reals containing it such that no other numbers in the interval are in \(X\).
See, e.g., Jech (2003, p. 23).
I.e., Def\((Y)\!=\!\Bigl \{ \{ y \mid y \in Y \text { and } (Y,\in ) \models \Phi (y,z_1,\ldots ,z_n) \} ~ \Big | ~ \Phi \text { is a formula and } z_{1},\ldots ,z_{n} \in Y. \Bigr \}\)
See, e.g., Jech (2003, p. 175).
There are other equivalent formulations, and in first-order set theory the axiom is standardly represented as an infinite schema.
See Kanamori (2012).
Returning to the motivating examples earlier in this section, one may notice that there is actually a certain asymmetry between the Cantor-Bendixson example and the others. In the Cantor-Bendixson case, we are going “down” and not “up”, in the sense that the sets being defined at each stage are smaller than those at previous stages. In contrast, the definitions of \(V_\alpha \) and \(L_\alpha \), as well as the operations of transfinite arithmetic, result in objects that are unboundedly larger or higher ranked than their predecessors in the recursive construction. One may want to distinguish between these two types of cases, which arguably warrant different treatment. In 1922, the same year that replacement was introduced, Kuratowski (1922) actually gave a general procedure for how to do proofs with “downward” examples like the Cantor-Bendixson function without using recursion (see Hallett (1984, p. 253ff.)). This method certainly does not work for cases like \(V_\alpha \).
In Sect. 3, we prove some results showing that replacement is actually provable by route of certain types of recursion and other principles. This, of course, is useful. But the point is that even if we did not have results like these, what we are ultimately interested in are reasons for believing in such principles. These we talk about in Sects. 4 and 5.
See also Potter (2004, p. 231).
An important distinction to keep in mind here is between limitation of size and the related but alternative principle Hallett has called Cantorian finitism. Roughly, Cantorian finitism is the idea that infinite sets should be treated as much as possible in the same way that we treat finite sets. One of the tenets is that any set, even an infinite one, should be conceived as a unit. Another one is extensionalism: sets (even infinite ones) consist precisely of their members. (For more on this, see Hallett (1984, pp. 7–9 and 32–40).) Withrespect to the present discussion, Cantorian finitism can justify the axiom of power set on the grounds that since finite sets always have power sets, so should infinite sets. This sort of argument is independent of arguments based on limitation of size, and so should not be viewed as providing reason for them.
We actually will not be appealing to the axiom of choice as such, but rather the well-ordering theorem. For convenience, we will abbreviate this by C.
While first-order treatments of set theory are more common, our aim is to formalize principles yielding recursion, which is then used to justify part of replacement. Recursion is most naturally formulated as a second-order axiom. In future work, I hope to investigate whether these results can be replicated in first-order logic or related systems like the schematic logic of Lavine (1999). This would be important because there are at least two well-known concerns with the invocation of second-order logic in the context of set theory. First, some suggest that the standard semantics for second-order logic is already set-theoretically “entangled”. (The metaphor of mathematical entanglement is originally from Charles Parsons. For further use of it, see Koellner (2010)). Second, there is the concern that an appeal to second-order logic is inconsistent with the iterative conception (see Burgess (1985)).
See, e.g., Fine (1998).
Though we do not do so in this paper, one might also be interested in defining a variation of \(\Omega \)-Rec that works with general well-orderings instead of just ordinals, in such a way that the axiom \(\Omega \)-Abs is not needed.
For reference, the Von Neumann ordinal \(\alpha +1 =_{\text {df}} \alpha \cup \{ \alpha \}\), whereas the Zermelo ordinal \(\alpha +1 =_{\text {df}} \{\alpha \}\). So, e.g., the set of finite Zermelo ordinals is \(\{\{\}, \{\{\}\}, \{\{\{\}\}\},\ \ldots \) }.
Turing’s original definition of a ‘computer’ in Turing (1936) depicted a human being, not a machine, following the rote instructions of a program in “a desultory manner”. For this reason, Sieg (2005) writes ‘computor’ to indicate a human agent calculating in this way and ‘computer’ to indicate a machine in the word’s contemporary sense. In these terms, Kreisel’s challenge is to provide a general concept of computation and an associated Church’s Thesis, while also offering an account of the phenomenology of such a computor.
An OTM works similarly to a classical Turing machine at successor ordinal-numbered steps of computation, but at limit ordinal steps the cells’ contents must be calculated by taking some sort of limit of previous values, implemented here as the lim inf. This means the machine must be able to answer for each cell the \(\Sigma _2\) question of whether there exists a step before the limit such that, for all steps after, the cell value was stabilized at \(1\). If so, the cell’s limit value becomes \(1\), and otherwise (i.e., if it diverged or it stabilized at \(0\)) it is set to \(0\).
Since not every computation halts, it is more precise really to say that some algorithms determine functions.
See, e.g., Soare (1987, Chaps. 2 and 4).
The phrase chosen later in Boolos (1989) is “And so it goes,” but the meaning is much the same.
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Acknowledgments
I would like to acknowledge Kyle Banick, Philip Ehrlich, Ethan Galebach, Peter Koepke, Christopher Menzel, Charles Parsons, Erich Reck, Sam Roberts, Sam Sanders, Claudio Ternullo, Sean Walsh, Kai Wehmeier, and John Wigglesworth, for their numerous helpful suggestions. I would also like to thank the referees for their many written comments. Finally, I wish to thank Benedikt Löwe, Dirk Schlimm, and the other organizers of FotFS VIII. The research for this paper was partly funded by a grant from the Analysis Trust.
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Rin, B. Transfinite recursion and computation in the iterative conception of set. Synthese 192, 2437–2462 (2015). https://doi.org/10.1007/s11229-014-0560-9
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DOI: https://doi.org/10.1007/s11229-014-0560-9