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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?

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    $\begingroup$ If there is, it would be interesting to use it to decide if your question is self referential. :-) $\endgroup$ – Dick Palais Sep 22 '10 at 15:40
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    $\begingroup$ If it turns out that not, I would have to rephrase my question :-) $\endgroup$ – Peter Arndt Sep 22 '10 at 16:06
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    $\begingroup$ Are you familiar with Vicious Circles by Barwise and Moss? $\endgroup$ – Noam Zeilberger Sep 22 '10 at 16:24
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    $\begingroup$ 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. $\endgroup$ – Sidney Raffer Sep 22 '10 at 16:46
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    $\begingroup$ Now Smullyan together with Lawvere's diagonalisation argument, as exposed at arxiv.org/abs/1006.0992, leaves me quite satisfied... $\endgroup$ – Peter Arndt Sep 23 '10 at 14:41

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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

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  • $\begingroup$ Ah, another exposition of Lawvere's argument - it certainly fits the bill, see my last comment to the question. Thanks! $\endgroup$ – Peter Arndt Sep 23 '10 at 23:40
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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.

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    $\begingroup$ +1 :) $\endgroup$ – Peter Arndt Sep 25 '10 at 20:08
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Raymond Smullyan, "Diagonalization and Self-Reference", 1994

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

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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.

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  • $\begingroup$ 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)? $\endgroup$ – Peter Arndt Sep 22 '10 at 16:11
  • $\begingroup$ 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... $\endgroup$ – Peter Arndt Sep 22 '10 at 16:11
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    $\begingroup$ 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. $\endgroup$ – dfranke Sep 22 '10 at 16:18
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I found another one:

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

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

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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).

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How relevant this is to "an abstract framework, maybe a certain kind of formal language with some extra structure" in the OP, I am not quite sure, and I am only beginning to read Halbach's works, yet I think they are general enough to properly belong to this thread, and they weren't yet mentioned. Very very roughly, what I gather from Halbach's work on this topic so far is that (my interpretation)

the 'say' in the old idea of formulae which 'say' about themselves they are not provable is not an idea which is synthetic a priori (like e.g. the idea of necessity), rather is an idea consisting of many sub-ideas. (For example, by filtering according to where the relevant formulae are placed in the arithmetical hierarchy.)

Philosophically, the approach to self-reference I am pointing to here could be called an 'analytic' approach to self-reference, as opposed to the 'synthetic' approach of Lawvere. (Both 'analytic' and 'synthetic' to be taken in their neutral technical sense, i.e. breaking a concept into parts in the former, combining the concept with other concepts in the latter.)

Some references on Halbach's work:

  • so new that it hasn't yet appeared: Volker Halbach has announced a book (with Graham Leigh) whose draft has the title Syntax and Circularity: A Study in Self-Reference and Paradox.

  • so new that it lies after the answers given so far:

Volker Halbach, Albert Visser: Self-reference in arithmetic I. Vol. 7(4), 2014 , pp. 671-691

Volker Halbach, Albert Visser: Self-reference in arithmetic II. Vol. 7(4), 2014 , pp. 692-712

  • a relevant lecture of Halbach's with a very general title, containing explanations on the work of Halbach and Visser:

V. Halbach: Self-reference. Talk at the Workshop in Mathematical Philosophy. Ludwig-Maximilians-Universität München. September 13, 2011

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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.

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1) Let $C$,$P$, $T$, $R$ be sets. (as a mnemonic, you can think $C$ as the set of "claims", $P$ the set of "properties", $R$ the set of "binary relations" and $T$ the set of "terms").

2) We assume there is also

2.1) a map $\alpha_1: P\times T \to C $; (obviously builds a claim from a property and a term: this is purely syntactic)

2.2) a map $\alpha_2: R \times T \times T \to C$; (builds a claim form a relation and two terms; syntactic too)

2.3) a map $\#_0: C \to T$ (a Gödel numbering).

2.4) a map $\#_1: P \to T$ (idem)

2.5) an equivalence relationship $\equiv$ in $C$ (it will be often equality itself, or for instance when $a,b$ are logical formulas, $a \equiv b$ if and only if $\vdash a \leftrightarrow b$ ).

3) We also assume the following : for every $f \in P$, there exists $\overline f\in R$ such that for every $(u,x)\in P \times T$, $\alpha_1 \big (f , \#_0 \alpha_1 (u,x) \big)\equiv \alpha_2 \left (\overline f, \#_1 u,x \right)$

4) Finally we assume that for every $g\in R$, there exists $g^* \in P$ such that for all $v\in P$, $\alpha_2(g,\#_1 v,\#_1 v)\equiv \alpha_1(g^*,\#_1 v)$.


We have the following fixed point (recursion...) theorem: under all the assumptions 1) to 4) above, For every $h\in P$, there exists $e\in C$ such that $\alpha_1(h,\#_0 e)\equiv e$.

Proof: from 3) and 4), it follows that for every $w\in P$, we have:

$$\begin{align} \alpha_1 \big(h, \#_0 \alpha_1 (w,\#_1 w)\big) & \equiv \alpha_2 (\overline h ,\#_1 w, \#_1 w) \\ & \equiv \alpha_1 \big( (\overline h)^*, \#_1 w \big)\tag{I} \end{align}$$

now define $e:= \alpha_1 \big((\overline h)^*, \#_1 (\overline h)^*\big)$. We get immediately $\alpha_1(h,\#_0 e)\equiv e$ by replacing $w$ by $(\overline h)^*$ in $\text{(I)}$. Qed.

$e$ says the same as "my Godel number has property $h$".

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