Welcome to MathOverflow

Question

22

10

This is a question about self-reference: Has anyone established an abstract framework, maybe a certain kind of formal language with some extra structure, which makes it possible to define what is a self-referential statement?

flag	asked Sep 22 2010 at 15:30 Peter Arndt 6,888●18●42

9

If there is, it would be interesting to use it to decide if your question is self referential. :-) – Dick Palais Sep 22 2010 at 15:40

1

If it turns out that not, I would have to rephrase my question :-) – Peter Arndt Sep 22 2010 at 16:06

1

Are you familiar with Vicious Circles by Barwise and Moss? – Noam Zeilberger Sep 22 2010 at 16:24

2

The "Recursion Theorem" of classical Recursion Theory, covered e.g. in Hartley Roger's "The Theory of Recusive Functions and Effective Computability" gives the beginnings of such an abstract framework. Also check out Smullyan's book "Diagonalization and Self-Reference", which tries to abstract as much as possible from Godel numbering. – SJR Sep 22 2010 at 16:46

2

Now Smullyan together with Lawvere's diagonalisation argument, as exposed at arxiv.org/abs/1006.0992, leaves me quite satisfied... – Peter Arndt Sep 23 2010 at 14:41

show 2 more comments

G. Rodrigues · Accepted Answer · 2010-09-23 17:24:37Z UTC

7

I am not quite sure if it fits the bill but you can also check out:

N. Yanofsky - A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points

answered Sep 23 2010 at 17:24

G. Rodrigues
995●4●8

Ah, another exposition of Lawvere's argument - it certainly fits the bill, see my last comment to the question. Thanks! – Peter Arndt Sep 23 2010 at 23:40

Jérôme JEAN-CHARLES · Answer 2 · 2010-09-25 16:28:23Z UTC

8

For pleasure only I can at least give you the shortest definition of self reference.

You need only to look in a good dictionary ( from Borges world of course) it says:

Self-reference : see self-reference.

answered Sep 25 2010 at 16:28

Jérôme JEAN-CHARLES
608●7●15

1

+1 :) – Peter Arndt Sep 25 2010 at 20:08

dfranke · Answer 3 · 2010-09-22 15:50:32Z UTC

4

Perhaps the right question to ask is if the statement is expressible in any system whose proof-theoretic ordinal is smaller than the Feferman–Schütte ordinal.

answered Sep 22 2010 at 15:50

dfranke
73●7

Hm, could you clarify that? So, to make sense of proof-theoretic ordinals I need a language in which I can talk about some fragment of arithmetic. So maybe your proposal is to say a statement in a formal language is selfreferential if under some/any translation (to be defined, maybe via a Gödelization?) into the language of arithmetic it becomes self-referential (this depends on a Gödelization of the language of Arithmetic)? – Peter Arndt Sep 22 2010 at 16:11

I don't quite like the dependence on a Gödelization, but maybe there is no other way. Maybe there is always a choice of Gödelization that renders a given statement self-referential... – Peter Arndt Sep 22 2010 at 16:11

1

I think that before I can make a more rigorous proposal about what a self-referential statement is, we first need to set boundaries on what qualifies as a statement. If we need to take into account such sentences as "colorless green ideas sleep furiously", then I wouldn't know where to begin. – dfranke Sep 22 2010 at 16:18

Kaveh · Answer 4 · 2010-09-23 14:03:40Z UTC

3

Raymond Smullyan, "Diagonalization and Self-Reference", 1994

answered Sep 23 2010 at 14:03

Kaveh
2,187●5●24

	Thanks, I already got it, following the advice of SJR in the comments. It is indeed excellent! – Peter Arndt Sep 23 2010 at 14:19
	@Peter: Sorry, I didn't see his comment. – Kaveh Sep 23 2010 at 15:10
	No "sorry": It deserves an answer of its own:) – Peter Arndt Sep 25 2010 at 20:06

Carl Mummert · Answer 5 · 2012-10-10 15:40:38Z UTC

You might also be interested in Graham Priest's article "The Structure of the Paradoxes of Self-Reference", Mind 103 (1994) pp. 25-34. (Journal page ; JStor) and similar work by Priest. He has a general framework that he argues captures the various self-referential paradoxes. I believe he also discusses this in some of his other work and monographs.

Peter Arndt · Answer 6 · 2012-10-07 20:02:51Z UTC

2

I found another one:

John Bell, “Incompleteness in a General Setting”. Bulletin of Symbolic Logic 13, 2007. It's paper number 66 here

answered Oct 7 at 20:02

Peter Arndt
6,888●18●42

o a · Answer 7 · 2011-08-11 20:55:32Z UTC

1

I have heard that Kapranov once said that he really wants to understand what is self-reference (i.e. is/was working on the question).

answered Aug 11 2011 at 20:55

o a
53●4

Mahmud · Answer 8 · 2012-10-08 05:41:31Z UTC

1

Have you checked Craig Smorynski's work such as "Modal Logic and Self-Reference" (Google Books)?

answered Oct 8 at 5:41

Mahmud
280●1●3●13

none · Answer 9 · 2012-10-11 01:42:09Z UTC

There's a bunch of writing on this topic by the philosopher of mathematics Charles Chihara, includng a book called "Ontology and the Vicious Circle Principle". I haven't read that one but he also discusses the topic in his later book "Constructability and Mathematical Existence". You can probably find reviews of these books online that would help you decide if they are relevant to your interests.

Is there a general setting for self-reference?

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

9 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.