From owner-qed Thu Oct 27 04:46:04 1994 Received: from localhost (listserv@localhost) by antares.mcs.anl.gov (8.6.4/8.6.4) id EAA20956 for qed-out; Thu, 27 Oct 1994 04:45:53 -0500 Received: from SAIL.Stanford.EDU (SAIL.Stanford.EDU [36.28.0.130]) by antares.mcs.anl.gov (8.6.4/8.6.4) with SMTP id EAA20951 for ; Thu, 27 Oct 1994 04:45:40 -0500 Received: by SAIL.Stanford.EDU (5.65/25-SAIL-eef) id AA18920; Thu, 27 Oct 1994 02:44:28 -0700 Date: Thu, 27 Oct 1994 02:44:28 -0700 From: John McCarthy Message-Id: <9410270944.AA18920@SAIL.Stanford.EDU> To: dahn@mathematik.hu-berlin.de Cc: qed@mcs.anl.gov In-Reply-To: Dahn's message of Thu, 27 Oct 94 09:47:43 +0100 <9410270847.AA01413@mathematik.hu-berlin.de> Subject: Semantics Reply-To: jmc@cs.stanford.edu Sender: owner-qed@mcs.anl.gov Precedence: bulk It is too narrow a view of mathematics to consider it as only concerning the consequences of an initially chosen set of axioms. Consider Chris Freling's paper "Evidence Against the Continuum Hypothesis", JSL 1982. As Goedel knew, Cohen (1965) had proved the independence of the continuum hypothesis from the axioms of set theory, complementing Goedel's 1940 result that the continuum hypothesis was consistent with ZF. Nevertheless, Goedel believed that the continuum hypothesis was false and believed that someone would come up with more intuitive acceptable axioms and would then be able to prove it false. Freiling did it only a few years after Goedel died. Freiling's ideas was that people have intuitions not merely about numbers but also about the real line, including the unit interval. Here is the build-up to Freiling's axiom. Suppose you have a denumerably infinite set A on the unit interval. Now you throw a dart at the unit interval getting a number x. What is the probability that the x is in A? 0, you are supposed to answer. Fine says Freiling, let's elaborate a little. Let f be a function from the unit interval to the set of deunumeraable sets in the unit interval, i.e. for each x, f(x) is a denumerable set of points in the unit interval. Now throw a dart to determine x. Now throw a second dart to to determine y.. What is the probability y is a member of f(x)? 0, you should say. What is the probability x is a member of f(y)? 0, you should say. Gotcha! say Freiling. We'll weaken what you have agreed to and express it as axiom. Axiom (Freiling): Let f be any function from the unit interval to denumerable sets in the unit interval. Then there exist x and y in the unit interval such that y is not in f(x) and x is not in f(y). >From this inuitive axiom the falsity of the continuum hypothesis follows in two lines. The coninuum hypothesis states that the cardinality of the unit interval is the same as that of the set of denumerable ordinals. Choose a 1-1 correspondence. Now for any x, let f(x) be the set of smaller points in the ordering given by the correspondence. If we now choose y, we msut have either y in f(x) or x in f(y), according to whether y or x corresponds to a smaller ordinal. I think Goedel would have been pleased with this result of Freiling's and with Freiling's other results. Determinined syntacticists, including unfortunately the editor of Goedel's collected papers, find Freiling's work of no interest. I suppose they don't like the idea of anyone having inuitions about the unit interval worthy of being made into axioms. The whole discussion given above should make a determined syntacticist nervous. He would have to ask, "Precisely what symbol manipulations were you talking aobut." Exercise: Put the above discussionion into PRA. See if another PRA fan can figure out what the hell you are talking aobut from reading it.