Fragments of Frege's Grundgesetze and G\"odel's Constructible Universe
Sean Walsh

TL;DR
This paper explores how much set theory can be developed within consistent fragments of Frege's Grundgesetze, using G"odel's constructible universe, and provides conditions for the consistency of abstraction principles.
Contribution
It demonstrates a model of a fragment of Grundgesetze that satisfies most of ZF set theory, and offers a sufficient condition for the consistency of abstraction principles with limited comprehension.
Findings
A model of a Grundgesetze fragment satisfies all ZF axioms except power set.
Uses G"odel's constructible universe and projectum concepts in the proof.
Provides a condition ensuring the consistency of abstraction principles.
Abstract
Frege's Grundgesetze was one of the 19th century forerunners to contemporary set theory which was plagued by the Russell paradox. In recent years, it has been shown that subsystems of the Grundgesetze formed by restricting the comprehension schema are consistent. One aim of this paper is to ascertain how much set theory can be developed within these consistent fragments of the Grundgesetze, and our main theorem shows that there is a model of a fragment of the Grundgesetze which defines a model of all the axioms of Zermelo-Fraenkel set theory with the exception of the power set axiom. The proof of this result appeals to G\"odel's constructible universe of sets, which G\"odel famously used to show the relative consistency of the continuum hypothesis. More specifically, our proofs appeal to Kripke and Platek's idea of the projectum within the constructible universe as well as to a weak…
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Taxonomy
TopicsComputability, Logic, AI Algorithms · Advanced Topology and Set Theory · Logic, Reasoning, and Knowledge
