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Gradualist descriptionalist set theory

David Simmons

Abstract

We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each $k \in \mathbb N$ of a sequence of ordinals $η_0 < . . . < η_k$ such that for each $i < k$, $η_i$ is $η_{i+1}$-reflecting, a notion we introduce which implies being $Π_n$-reflecting for all $n \in \mathbb N$ (and in particular being admissible and recursively Mahlo).

Gradualist descriptionalist set theory

Abstract

We introduce a formal language GDST (gradualist descriptionalist set theory) with a family of interpretations indexed by ordinals, as well as a sublanguage NMID (the language of not necessarily monotonic inductive definitions), and show that the assertion that all propositions in NMID have well-defined truth values is equivalent to the existence for each of a sequence of ordinals such that for each , is -reflecting, a notion we introduce which implies being -reflecting for all (and in particular being admissible and recursively Mahlo).

Paper Structure

This paper contains 7 sections, 20 theorems, 32 equations.

Table of Contents

  1. Introduction
  2. Definition of $\mathbf{GDST}$
  3. An axiomatization of $\mathbf{GDST}$
  4. A canonical interpretation of $\mathbf{GDST}$?
  5. Reflecting ordinals
  6. The language of not necessarily monotonic inductive definitions
  7. Proof of Theorem \ref{['theorem2']}

Key Result

Proposition 2.8

There exist set terms $\mathop{\mathrm{eval}}\nolimits_\iota$, $\mathop{\mathrm{eval}}\nolimits_j$, and $\mathop{\mathrm{eval}}\nolimits_{j'}$ such that for all limit $\eta,\tau$ with $\eta < \tau$, for all $M,N\in\mathbb N$, for all $P\in \mathcal{F}_\mathbf{GDST}(M,N)$, and for all $\mathbf x\in L and similarly for $\mathop{\mathrm{eval}}\nolimits_j$, $\mathop{\mathrm{eval}}\nolimits_{j'}$. Here

Theorems & Definitions (54)

  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Proposition 2.8
  • Corollary 2.9
  • proof
  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • ...and 44 more