Abstract
In this article we investigate similarities between the role that ineffability of Absolute Infinity plays in class theory and in theology.
How can I talk to you, I have no words
Virgin Prunes, I am God
Versions of this article have been presented at various conferences and workshops, including at the First Conference of the Italian Network for the Philosophy of Mathematics (Milan 2014). I am grateful to the audiences at these events for invaluable questions, comments and suggestions. Among these, I am especially indebted to Anthony Anderson, Sam Roberts, Øystein Linnebo, Mark van Atten, Christian Tapp. Thanks also to two anonymous referees, for giving careful suggestions for improvement. But above all I am grateful to my colleague Philip Welch: I could never have written this article without the discussions that I have had with him about proper classes and reflection.
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Notes
- 1.
For one expression of this view, see Jané (1995).
- 2.
Zermelo here means a series of initial segments \(V_{\alpha }\) of the set theoretic universe V, for \(\alpha \) ranging over the strongly inaccessible ordinals, and the membership relation restricted \(V_{\alpha }\).
- 3.
In this quotation, Cantor speaks of the necessity of ‘knowing’ the domain of variation through a ‘definition’. Surely Cantor is merely sloppy here, and we should discount the epistemological overtones. Another slip can be detected in Cantor’s use of the word ‘set’ in this quotation: Cantor means the argument to be applicable not just to sets but also to absolute infinities. For a discussion of Cantor’s sometimes sloppy uses of the term ‘set’, see Jané (2010), footnote 60.
- 4.
The connection between Cantor’s conception of the mathematical absolutely infinite and his conception of God is explored in van der Veen and Horsten (2014).
- 5.
- 6.
This notion goes back to Anaximander, and is variously translated as ‘limitless’,’boundless’, ‘formless’, ‘the void’....
- 7.
Cantor’s 1899 argument that the ordinals form an inconsistent totality is critically discussed in Jané (1995), pp. 395–396.
- 8.
Admittedly this passage is sufficiently vague as to be open to multiple interpretations. The view that this passage should be seen as an application of reflection is defended in Hallett (1984), pp. 117–118.
- 9.
The Montague-Levy reflection principle is discussed in Drake (1991), Chap. 3, Sect. 6.
- 10.
Ranks \(V_{\alpha }\), for \(\alpha \) strongly inaccessible, are models of \(ZFC^2\).
- 11.
NBG differs from full \(ZFC^2\) in that the second-order comprehension scheme is restricted to formulae that do not contain bound occurrences of second-order quantifiers.
- 12.
See von Neumann (1967).
- 13.
- 14.
Parameter free sentences of higher orders are unproblematic.
- 15.
We will see later (Sect. 7.6.3) that the expression “any assertions” in this statement may need to be qualified.
- 16.
A philosophical defence of GRP is given in Horsten and Welch (forthcoming).
- 17.
The large cardinal strength of versions of GRP is discussed in Horsten and Welch (forthcoming) and in Welch (2012).
- 18.
“Quomodo potest finitum attingere ad infinitum? Propter hoc dixerunt alii quod deus contemperatum se exhibebit nobis, et quod ostendet se nobis non in sua essentia, sed in creatura”.
- 19.
As quoted in Segal (1977), p. 163.
- 20.
I am indebted to an anonymous referee for these points.
- 21.
See also Horsten and Welch (forthcoming).
- 22.
Even the Augustinian idea that sets are ideas in God’s mind is compatible with this view. Within such a framework, the mereological conception of classes would result in conceiving of classes (proper and improper) as parts of God’s mind.
- 23.
It seems to me that Maddy’s own view of classes does not completely satisfy the first desideratum. The reason is that she takes the class membership relation to be governed by partial logic. According to her theory, there is in many cases no fact of the matter whether a given class is an element of another given class.
- 24.
See the quotation of Dionysius in Sect. 7.5.
- 25.
This argument does not go through if instead j is only \(\Sigma ^0_{1}\) elementary: there is then not enough elementarity to preserve the impredicative second order comprehension scheme upwards. Nonetheless, since MK holds at \((V_{\kappa }, V_{\kappa +1}, \in \)), accepting \(GRP_{\Sigma ^0_{1}}\) still commits one to believing that impredicative second-order logic is at least coherent.
- 26.
See Gödel (1984).
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Horsten, L. (2016). Absolute Infinity in Class Theory and in Theology. In: Boccuni, F., Sereni, A. (eds) Objectivity, Realism, and Proof . Boston Studies in the Philosophy and History of Science, vol 318. Springer, Cham. https://doi.org/10.1007/978-3-319-31644-4_7
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