Path: utzoo!utgpu!news-server.csri.toronto.edu!bonnie.concordia.ca!uunet!samsung!zaphod.mps.ohio-state.edu!rpi!uupsi!sunic!sics.se!sics.se!torkel From: torkel@sics.se (Torkel Franzen) Newsgroups: comp.ai.philosophy Subject: Re: Minds, machines, and Godel Message-ID: <1991Jan21.130344.19720@sics.se> Date: 21 Jan 91 13:03:44 GMT References: <703.9101211032@s4.sys.uea.ac.uk> Sender: news@sics.se Organization: Swedish Institute of Computer Science, Kista Lines: 98 In-Reply-To: jrk@information-systems.east-anglia.ac.uk's message of 21 Jan 91 10:32:27 GMT In article <703.9101211032@s4.sys.uea.ac.uk> jrk@information-systems. east-anglia.ac.uk (Richard Kennaway CMP RA) writes: [quoting me:] >I am not claiming any such thing. I thought you did, in saying that "This is strong enough evidence that I can regard its consistency as 'uncontroversial'." Anyway, the "statistical" argument for the consistency of elementary arithmetic or ZFC is a very poor one. Take some classical open problem A in mathematics, one on which lots of mathematicians have been working for maybe two hundred years. The "statistical" argument would lead us to the conclusion that A is undecidable in current mathematics. Still every now and then such a classical open problem is settled using perfectly ordinary mathematics. In the case of the question of the consistency of strong set theories, the statistical argument is further weakened by the fact that very few people try to exploit and explore the full strength of these theories. On the other hand, there is nothing mysterious about the claim that e.g. first order arithmetic is obviously consistent. There is no need to speak poetically about intuition: the theory is obviously consistent because its axioms are obviously true. The mere occurrence of the word "true" in an observation is often, it seems, taken to imply that the observation must be metaphysical or otherwise deplorable. To say that the axioms are true, however, is to say no more than what is said in asserting the recursion equations for addition and multiplication, and the principle of mathematical induction. Now if somebody does in fact find these assertions obscure or doubtful or false, he will have no reason to believe that first order arithmetic is consistent. It is mere ritual however, to speak of the consistency of arithmetic as dubious without attempting to establish just what is obscure, false, or doubtful about the axioms. If one does in fact have serious doubts concerning the consistency of arithmetic, one must have serious doubts concerning the validity of practically every theorem of mathematics. It is perhaps worth pointing out that the trivial consistency proof for arithmetic - the axioms are true, so the theory is consistent - is easily formalizable in ordinary mathematics (second order arithmetic). The proof is mathematically unsatisfactory not because it is invalid, but because it has no interesting mathematical content whatever. If one regards the conclusion of this particular proof as so extremely dubious, why should one have confidence in any theorem of elementary analysis? Again the supposed mystique of consistency appears to be mostly a matter of ritual. Nor is it strange that many people who regard the consistency of elementary arithmetic as unproblematic find the consistency of ZFC highly problematic. The axioms of ZFC, when put forward as true assertions, are simply very much more abstract and problematic and open to objections of various kinds. You raised another point: >>consider the Godel sentence of ZFC+ 'there is an uncountable >>measurable cardinal'. Is this sentence (in its ordinary >>interpretation) true or not? Nobody knows, and quite possibly nobody >>ever will. >What would count as evidence? What is this "ordinary interpretation" of >ZFC+'tiaumc'? What would it mean to find out that the sentence was true or >false? I'm not speaking of any interpretation of ZFC+UM, but of the ordinary interpretation of a particular set-theoretical formula. This is an arithmetical formula in the set-theoretical sense, i.e. all its quantifiers are restricted to the hereditarily finite sets. So we need to agree on the interpretation of such formulas. But of course we also need to agree on the interpretation of the formula defining the hereditarily finite sets. The most direct way of finding out that the Godel statement for ZFC+UM is false is to establish that ZFC+UM is inconsistent. Evidence for the truth of the Godel sentence is a rather more difficult matter, and it may well be that we will never arrive at anything that anybody regards as at all convincing evidence for the consistency of ZFC + UM (unlike the situation with less problematic axioms of infinity). My adding "in its ordinary interpretation" had a point: suppose somebody presents us with a theory, one that he associates with a particular interpretation. We don't need to know what his interpretation is, or introduce any interpretation of the theory, in order to establish that Godel's theorem applies to it. What we do is to interpret arithmetic in his theory (this a purely proof-theoretic matter), and establish that the translation G* in his theory of a particular arithmetical sentence G is not provable in his theory, provided it is consistent. We know nothing about the truth-value of G* under his interpretation, and indeed in the case of the Godel sentence, G* may well be refutable in his theory, without this implying that anything is wrong with the theory (in particular, it need not be inconsistent). In the case of the Rosser sentence R, we know that R* is undecidable in the theory if it is consistent. However, it may still well be the case that R* on his own, perfectly good interpretation is a false statement.