> The following sentence is true. > The previous sentence is false. > > Oh and by the way this sentence is also false.
The Liar's Paradox would not be a good example of useful mathematical systems being mutually inconsistent, or of formal language being imprecise or expressing non-absolute ideas. A string of characters in a language is not necessarily evaluable. So "=+4()F" is not a well-formed statement, and neither are those sentences (Nor the simpler version, "This sentence is false."). When you think of how you would formalize something like that (i.e. how you would construct a system where a sentence could discuss its own truth value), you come to realize that there is no way to do so. Which makes sense, since sentences like that don't contain any information anyway. On the other hand, well-formed statements can talk about some of their properties in certain systems. If worse comes to worse, you can simply use a different system to evaluate the statement. This really does make sense and there is information conveyed--a parallel would be Raymond Smullyan's example of a sign that reads, "This sign was made my Cellini." That sign is actually telling you something. The famous sentence, "This sentence cannot be proved in system S," can be a well-formed statement in some systems. If it can be expressed in system S then system S is either incomplete (there is a true non-theorem) or inconsistent (there is a false theorem). This is Godel's result. The non-obvious and surprising part about Godel's Incompleteness Theorem is that it turns out that any mathematical system powerful enough to provide for basic arithmetic is also in effect powerful enough to express such a statement. Hence there is no single mathematical system that can prove all true statements and disprove all false ones. (Having to do with constraints on statementhood, intuitionist logicians might disagree with that claim--I'm not sure. But then, I don't think intuitionists accept Godel's proof anyway, because it is a reductio.) This seriously calls the notion of absolute mathematical truth into question. And yet, that no mathematical system of useful complexity is both complete and consistent does not diminish the precise nature of mathemtical formalism, nor does it ensure that there be multiple inconsistent systems that are simultaneously useful. Those are for other reasons. Mathematical precision is limited because ultimately any definition is understood on the basis of ideas that are not themselves defined, much as a written tradition cannot exist without an oral tradition that consisting at least of literacy skills. Inconsistent systems are simultaneously useful because it is valuable to take different assumptions seriously and explore their results, and also because when one is describing only part of reality--which is all that anybody has ever been able to do with formal mathematical systems anyway--it is useful to use assumptions different from those most useful describe another part of reality. Similarly, in the world of informal (or less formal) communication, I think it is inherently valuable when people disagree about things, have different perspectives, embrace different worldviews, subscribe to different religions, to have different cultural backgrounds, and so forth. The whole *point* is that they are inconsistent, and not merely that they are different. -Eliah
