Results 1 
2 of
2
Reflections on Skolem's Paradox
"... In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the Lö ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the LöwenheimSkolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...
The Philosophical Significance of Tennenbaum’s Theorem
"... Notice. This paper is due to appear in Philosophia Mathematica. This paper may be subject to minor changes. The authoritative version should be obtained from Philosophia Mathematica. Tennenbaum’s Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have rec ..."
Abstract
 Add to MetaCart
Notice. This paper is due to appear in Philosophia Mathematica. This paper may be subject to minor changes. The authoritative version should be obtained from Philosophia Mathematica. Tennenbaum’s Theorem yields an elegant characterisation of the standard model of arithmetic. Several authors have recently claimed that this result has important philosophical consequences: in particular, that it offers us a way of responding to modeltheoretic worries about how we manage to grasp the standard model. We disagree. If there ever was such a problem about how we come to grasp the standard model, then Tennenbaum’s Theorem doesn’t help. 1 Appealing to Tennenbaum’s Theorem Tennenbaum’s Theorem tells us that the only model of PA (firstorder Peano Arithmetic) whose interpretation of addition and/or multiplication is recursive is the standard model. To explain a bit further. It is (relatively) standard practice to distinguish models of a theory only up to isomorphism. Adopting this practice will beg no question in the debate which interests us here. So for the purposes of this paper, ‘the ’ standard model for PA is, indifferently, ⟨N, 0, 1, +, ×, < ⟩ – whatever exactly you think that is! – or any isomorphic model.