arXiv subjects

M. Randall Holmes

Publications and source records attributed to M. Randall Holmes.

3 recordsLinked to original sources

arXiv2025

NF is Consistent

In this paper we will present a proof of the consistency of Quine's set theory "New Foundations" (hereinafter NF), so-called after the title of the 1937 paper in which it was introduced. This version takes the approach of building a model of tangled type theory rather than a model of the usual set theory without choice with a tangled web of cardinals; further, details of the construction are refined due to interaction with the now complete verification in Lean by the second author.

math.LO↗

arXiv2020

Acyclic Comprehension is equal to Stratified Comprehension

A new criterion of comprehension is defined, initially termed by myself as "connected" and finally as "Acyclic" by Mr. Randall Holmes. Acyclic comprehension simply asserts that for any acyclic formula phi, the set {x:phi} exists. I first presented this criterion semi-formally to Mr. Randall Holmes, who further made the first rigorous definition of it, a definition that I finally simplified to the one presented here. Later Mr. Holmes made another presentation of the definition which is also mentioned here. He pointed to me that acyclic comprehension is implied by stratification, and posed the question as to whether it is equivalent to full stratification or strictly weaker. He and initially I myself thought that it was strictly weaker; Mr. Randall Holmes actually conjectured that it is very weak. Surprisingly it turned to be equivalent to full stratification as I proved here

math.LO↗

arXiv2016

A System of Dependent Types, with an Implementation and a Philosophy

This is my working paper on a proposed logical framework for the practice of mathematics, which is paralleled by philosophical considerations and a computer implementation (a variant of Automath). Updated 10/27/2016 with a version from 10/22/2016. New versions are regularly posted on the author's web page at http://math.boisestate.edu/%7Eholmes/automath/ which is a directory containing various related files.

math.LO↗