Abstract
To the Pythagoreans, to apeiron, that which has no bounds, the infinite, “was something abhorrent” (Moore, The infinite, Routledge, p 19, 2001). Nevertheless the infinite thrust itself upon them once they became aware of the fact that there was no common length for which the sum of finitely many instances of it would be equal to both the length of the side and the length of the diagonal of a square. Again and again mathematicians have engaged warily in infinitary reasoning, and most often the methods were found to be useful and robust even when rigorous justification was only to be obtained at a later time. In this essay I will survey some of these encounters with the infinite, beginning with Torricelli’s horn that was so shocking in the seventeenth century, and ending with the very enormous infinite sets that contemporary set theorists study without any hope of a proof that their existence is consistent with the axioms of ordinary mathematics.
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Notes
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The method of exhaustion worked by showing that the quantity concerned could be neither smaller nor larger than the suggested result. Therefore it was necessary to already have that result at hand, whereas with indivisibles the answer could be computed directly.
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Gerhardt (1978, I, p. 338). The translation from Latin is by Alexis Manaster Ramer.
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Although Euler obtained this infinite product by using a doubtful analogy, Karl Weierstrass (1815–1897) eventually obtained it quite rigorously as a special case of a general theorem about complex analytic functions.
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Details can be found in a number of excellent text books, e.g., see Enderton (1977).
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For full details of this work see Jech (1991).
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A directed graph or digraph is a set of objects called vertices together with a set of ordered pairs of these vertices called edges. When such a digraph is used to model the axioms of set theory, the vertices represent sets and the existence of an edge \(\left\langle {a,b} \right\rangle\) is taken to represent the membership relation \(a \in b\).
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A primer of the terminology of descriptive set theory: Writing \({\mathcal{R}}\) for the set of real numbers and \({\mathcal{R}}^{n}\) for \(n\)-dimensional Euclidean space, the Borel sets of \({\mathcal{R}}^{n}\) are all the subsets of \({\mathcal{R}}^{n}\) that can be obtained from the open sets by applying a finite numbers of times the operations of countable union, countable intersection, and complement with respect to \({\mathcal{R}}^{n}\). For a set \(C \subseteq {\mathcal{R}}^{n + 1}\), Proj(C) is defined as the set \(A \subseteq {\mathcal{R}}^{n}\) defined by
$$ \left\langle {x_{1} , \ldots ,x_{n}} \right\rangle \in A \Leftrightarrow \exists y \in {\mathcal{R}}\left[ { \left\langle {x_{1} , \ldots ,x_{n} ,y} \right\rangle \in C} \right]$$The hierarchy of projective sets is defined simultaneously in all \({\mathcal{R}}^{n}\) as follows:
- \(\sum\nolimits_{0}^{1}\):
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The set of Borel sets
- \(\prod\nolimits_{m}^{1} \):
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\(\left\{ {A \subseteq {\mathcal{R}}^{n} |{\mathcal{R}}^{n} - A \in \sum_{m}^{1} } \right\}\)
- \(\sum\nolimits_{m + 1}^{1} \):
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\(\left\{ {A \subseteq {\mathcal{R}}^{n} |\exists \mathop \prod \nolimits_{m}^{1} \left[ {A = {\text{Proj}}\left( B \right)} \right]} \right\}\)
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Davis, M. (2024). Encounters with Infinity: From Torricelli to Gödel. In: Homage to Evangelista Torricelli’s Opera Geometrica 1644–2024. Logic, Epistemology, and the Unity of Science, vol 55. Springer, Cham. https://doi.org/10.1007/978-3-031-06963-5_3
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