Path: utzoo!utgpu!news-server.csri.toronto.edu!rpi!zaphod.mps.ohio-state.edu!mips!sgi!rpw3@rigden.wpd.sgi.com From: rpw3@rigden.wpd.sgi.com (Rob Warnock) Newsgroups: comp.compression Subject: Re: Integer not expressable in less than 13 words Message-ID: <101603@sgi.sgi.com> Date: 5 May 91 04:24:07 GMT References: <15901@smoke.brl.mil> <1991Apr25.234539.24276@unislc.uucp> <15989@smoke.brl.mil> Sender: guest@sgi.sgi.com Reply-To: rpw3@sgi.com (Rob Warnock) Organization: Silicon Graphics, Inc., Mountain View, CA Lines: 39 In article <15989@smoke.brl.mil> gwyn@smoke.brl.mil (Doug Gwyn) writes: +--------------- | ttobler@unislc.uucp (Trent Tobler) writes: | >No, no paradox. The number simply does not exist. | | Oh, good -- then we can conclude that all positive integers are | expressable in fewer than 13 English words. That has ramifications | for data compression.. | | In case you didn't detect the sarcasm, I'm well aware of Godel's | work etc. but I don't think it helps answer the question "What is | wrong with this argument?". Self-reference is not automatically | invalid; consider "This is a sentence." There's nothing wrong | with that. Even "This is not a sentence." is meaningful, albeit | false. "The set of all sets that do not include themselves" is | not meaningful, but you can't lay the entire blame on self-reference. +--------------- Doug, the problem *is* self-reference, but the sticking point is simply that this discussion thread hasn't quoted the "right" authority yet... ;-} We need not refer to Godel (yet), but to Russell. Using Russell's Theory of Types, if there is a thing pointed to by the phrase "the smallest positive integer not expressable in 13 English words", whatever that thing is, it's not itself a "positive integer" -- it's of a higher type, since it talks *about* "integers expressable by words". Thus there is no immediate paradox (though there may be one later in the investigation). If I remember it correctly, in the Theory of Types, one would oddly enough say that "This is not a sentence" is *true*, since it is *not* a "sentence" but a "sentence-about-sentences" (a higher-order type). -Rob ----- Rob Warnock, MS-1L/515 rpw3@sgi.com rpw3@pei.com Silicon Graphics, Inc. (415)335-1673 Protocol Engines, Inc. 2011 N. Shoreline Blvd. Mountain View, CA 94039-7311