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Paradox and potential infinity. (English) Zbl 1277.03059
The article describes some models for systems of predicative arithmetic inside intuitionistic set theory to show the crucial behaviours of potentially infinite sets in the foundational literature back to Aristotle. A discussion of the strictly related Church’s thesis concludes the paper.
In detail, starting from Zeno’s paradox about the impossibility of movement and the way Aristotle confuted Zeno’s argument, the idea of potential infinity is introduced. Formalising Zeno’s argument, the core in the reasoning is reduced to prove the existence of a set \(D\) which is, at the same time, not finite and not infinite – an obvious falsity in classical mathematics, but possible in the intuitionistic framework.
Using a variant of realizability, the formal machinery to synthesize \(D\)-like sets within the domain of intuitionistic set theory is constructed. Then, by providing a sensibly positive definition of finite and infinite sets, the existence of a third class of potentially infinite sets becomes clear, as the classification is not exhaustive. The article explores in some depth the consequences of this fact with respect to many philosophical approaches to mathematical infinity, culminating with an analysis of Zeno’s paradox.
Then, the author analyses how the very same pattern (not finite implies infinite) appears in Bolzano and Chomsky, leading to fallacies in their way of reasoning, at least from the point of view of constructive mathematics.
Since, under the realizability interpretation, each potentially infinite set provides a natural model of predicative arithmetic, the retraceable sets playing a distinguished role, it is natural to ask whether such models are special. In the paper, the author proves that, apart from \(\omega\), the set of ordinary natural numbers, Zenonian numbers provide the only fully inductive number structures that exist in the set-theoretic universe under Kleene’s number realizability.
Finally, the article proves that the only functions represented in potentially infinite, retraceable realizability structures for predicative arithmetic are recursive functions with weakly bounded ranges, directly relating to the weak Church’s thesis.
MSC:
03F55 Intuitionistic mathematics
03A05 Philosophical and critical aspects of logic and foundations
03F30 First-order arithmetic and fragments
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