throopw@dg_rtp.UUCP (Wayne Throop) (10/08/86)
> dwyer@vixie.UUCP > G asserts that a certain statement has no rational proof. What > statement? The statement that G is making that a certain statement has > no rational proof. Thus, the statement that G refers to is referring to > G referring to G referring Not quite right. When cast into easily-understood English, the "statement G" is, indeed, vulnerable to something like this counter-argument. However, in it's mathematical formulation, G refers unambiguously to certain facts of number theory, which in turn can be seen to have certain implications about the statement G itself (namely, that it can never be proven). The statement "This statement is false." might possibly be guilty of infinite regress. But what about "'Forms a false statement when preceeded by its quotation' forms a false statement when preceeded by its quotation." ? No infinite regress here. The statement is talking about a well-defined typographical construction of a statement, and asserting that that statement will be false. The sentence "G" is more akin to this form of indirect self-reference, and is *not* subject to the infinite regress quibble. > Templeton states that if G is false, NOT-G must be true -- which then > leads to a contradiction. Regardless of what Templeton said, incompleteness results do *not* lead to a contradiction. They simply show that for any given formal system, either the formal system is inconsistant (and one can prove both X and NOT X for some statement X), *OR* there exist true statements that cannot be proven in the formal system. (That is, that is *one* of the incompleteness results.) Finding that formal reasoning must either be contradictory or incomplete, most folks tend to conclude that it must be incomplete. > In short, an alleged statement that has no meaningful referent -- that > is neither true nor false -- that does not serve as a cognitive > instrument -- cannot be used to prove anything, least of all that reason > is inefficacious. As outlined above, "G" *does* have a meaningful referent. Further, "G" doesn't show reason to be "inefficacious". It simply shows that reason has a certain, nonintuitive (but well defined) limit. > (Besides, the whole idea of using reason to prove reason inefficacious > is ab- surd. [...] After all, the whole concept of reason rests on the > law of non-contradiction. If contradictions are possible, then no > rational argument can prove anything.) Again, incompleteness yields *NO* contradiction, and does *NOT* result in the conclusion that reason is "inefficacious". It merely states that "truth" cannot be totally captured by formal reasoning within any given formal system. -- It is better to know some of the questions than all of the answers. --- James Thurber -- Wayne Throop <the-known-world>!mcnc!rti-sel!dg_rtp!throopw