A Theory That Proves Its Own Inconsistency · Yan Sheng's site
I recently read this mind-bending blog post by Joel David Hamkins that proves this result: there is a Turing machine program P P such that for any function f:N→N f:N→N—possibly uncomputable!—there is a model of Peano arithmetic (PA) in which P P computes f f on the standard natural numbers. If this statement doesn’t surprise you, I’m not sure what else would (except maybe those who work in logic or set theory; I’m convinced that these people routinely believe as many as six impossible things before breakfast).
A Theory That Proves Its Own Inconsistency · Yan Sheng's site Welcome to my site! © 2019. All rights reserved. Yan Sheng's site A math blog A Theory That Proves Its Own Inconsistency 06 Dec 2018 logic I recently read this mind-bending blog post by Joel David Hamkins that proves this result: there is a Turing machine program $P$ such that for any function $f:\bb N\to\bb N$—possibly uncomputable!—there is a model of Peano arithmetic (PA) in which $P$ computes $f$ on the standard natural numbers. If this statement doesn’t surprise you, I’m not sure what else would (except maybe those
saved by
related reading
- Definability of Truth in Probabilistic Logic - Christiano 2013intelligence.org
- On “compilation errors” in mathematical reading, and how to resolve them | What's newterrytao.wordpress.com
- Working through a small tiling result — LessWronglesswrong.com
- Arnold's proof of Abel-Ruffiniweb.williams.edu
- Gödel's incompleteness theorems - Wikipediaen.wikipedia.org
- [2501.02693] Any function I can actually write down is measurable, right?arxiv.org
- Eat. Sleep. Math.eatsleepmath.tumblr.com
- Shtetl-Optimized >> Blog Archive >> Ten Signs a Claimed Mathematical Breakthrough is Wrongscottaaronson.blog
- How Gödel’s Proof Works | Quanta Magazinequantamagazine.org
- Reasons to believescottaaronson.blog
- Two infinities that are surprisingly equal | Gowers's Webloggowers.wordpress.com
- canon00-goedel.pdfhirzels.com