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Statements like

A) A is false.

or

B1) B2 is true.
B2) B1 is false.

cannot be assigned a truth-value due to their paradoxical use of self-reference. Are all statements lacking a truth-value self-referential, or are there non-self-referential statements that also cannot be assigned a truth-value?

Phrased another way: Can every non-self-referential statement be assigned a truth-value?

Edit: I think what I mean by self-referential is a set of statements where at least one statement in the set refers to a statement in the set. But perhaps there is a better definition.

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What is the sense in which B1 and B2 are self-referential? I know what you mean, but I think it might help to clarify your question. –  Chris Taylor Feb 8 '12 at 1:04
    
@ChrisTaylor: See my edit. –  Matthew Feb 8 '12 at 1:33
3  
It's ceertainly possible to make a paradoxical statement without explicit self-reference. A Standard example is the statement '"yields a false statement when preceded by its quotation" yields a false statement when preceded by its quotation'. –  Chris Eagle Feb 8 '12 at 1:40
    
@ChrisEagle: That's really interesting. So the essential difference is that in indirect self-reference, the statement refers to part of the statement, rather than the statement itself. What if we don't allow any self-reference at all (direct or indirect)? Can we achieve a paradox then? –  Matthew Feb 8 '12 at 1:51
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One might want to note that the "cannot be assigned a truth-value" here, I think, refers to an inability to assign a classical truth value of either "true" or "false". –  Doug Spoonwood Feb 8 '12 at 4:51
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2 Answers 2

up vote 15 down vote accepted

One of the main discoveries of set-theoretic research over the past fifty years is the widespread independence phenomenon, the phenomenon by which numerous fundamental statements of set theory are independent of the principal axioms of set theory. Many instances of this ubiquitous phenomenon are described in this mathoverflow question. Not only is the continuum hypothesis independent of ZFC, but an enormous number of other natural questions arising in set theory, infinite combinatorics and many related fields are independent of ZFC, to the point that set-theorists now begin with the expectation of any given nontrivial set-theoretic statement, that it is reasonably likely to be independent of our axioms.

None of these naturally arising independent statements instantiating the independence phenomenon is self-referential, and since they are independent, in most cases set-theorists are at a loss to explain what is their correct truth value. Thus, these statements can be seen as instances of the kind you seek: non-self-referential statements, which we seem unable to assign a definite truth value.

The question of mathematical truth for such assertions runs into deeply philosophical issues on the nature of mathematical truth and existence. For a taste of this, I can recommend some of the literature we read for my recent course at NYU on the philosophy of set theory.

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This is fascinating, great answer. –  Matthew Feb 8 '12 at 4:30
    
I wonder what the downvote stands for... –  Asaf Karagila Feb 8 '12 at 17:32
    
Its not clear that these statements arent indirectly self-referential. –  user1708 Feb 8 '12 at 18:32
    
In my answer, I meant to refer only to the clearly non-self-referential statements, such as statements about CH, or Luzin's hypothesis or Souslin's Hypothesis or $\Diamond$ or MA or any of the diverse other statements which are independent of the axioms of set theory, but to which it would at the very least require substantial philosophical argument to claim to assign a definitive truth value. (Of course it is easy to produce self-referential instances of this, using the standard methods of the incompleteness theorem.) –  JDH Feb 8 '12 at 19:09
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Thanks for the link to your course. It looks very interesting. –  Kaveh Feb 9 '12 at 2:24
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JDH has given a deep, interesting answer -- but it's deep and interesting in part because it relates to ZFC, which is a deep and interesting theory. Formal mathematical theories don't have to be deep and interesting, and it is possible to answer this question using very simple and straightforward examples.

Here is an example of a mathematical theory:

There are three well-formed formulas in this theory: x, y, and z. We have an axiom that says that x is true, and another axiom that says y is false. That's it. This theory has no grammatical rules for producing more complex formulas from simpler ones, no other machinery. In this theory, z cannot be assigned a truth value.

A less trivial example is the following. Take Tarski's axioms and delete the axiom of Euclid. This is a formal system that represents the same ideas as Euclid's original formulation of plane geometry, but without the parallel postulate. In this system, we have various statements that cannot be assigned truth values. One such statement is the axiom of Euclid (i.e., basically the parallel postulate). Another would be the Pythagorean theorem.

The self-referential thing is an interpretation of a particular strategy used by Godel for constructing undecidable statements in theories that can describe a certain amount of arithmetic. Note the three parts: (1) an interpretation, (2) a particular strategy, and (3) only for theories that can describe a certain amount of arithmetic.

The examples I've given above don't require Godel's strategy. Furthermore, Godel's strategy doesn't even work for Tarski's system, because Tarski's system can't describe the necessary amount of arithmetic.

Even in examples that do use Godel's strategy, the self-referentialism is only an interpretation. The undecidable statements don't literally refer to themselves -- they can just be interpreted that way.

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