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## Re: [ontolog-forum] Ontological Means for Systems Engineering

 To: "[ontolog-forum]" "John F. Sowa" Tue, 27 Jan 2009 09:57:16 -0500 <497F20CC.1070703@xxxxxxxxxxx>
 ```Pat,    (01) I agree with the technical points you've made, but the following claim is still controversial:    (02) PH> But set theory is not particularly discrete. Sets are the > basis for all mathematics, including continuous mathematics.    (03) For example, what happens to the midpoint when you break a line segment in two equal parts?    (04) 1. In classical Euclidean geometry, the two halves are isomorphic.    (05) 2. But if the line is considered isomorphic to a set of points that are in a one-to-one correspondence with the real numbers, the former midpoint must belong to one side or the other.    (06) Assumption #2 implies that the two "halves" cannot be isomorphic because the one that contains the former midpoint will be closed at that end. But the one from which the former midpoint was removed will be open at that end.    (07) This puzzle has troubled many brilliant mathematicians, including Kurt Gödel.    (08) Tarski's solution to that puzzle (and related issues) was to use a version of mereology combined with an ontology whose basic constructs were not points, but 3-dimensional spheres of arbitrary finite size.    (09) Tarski then showed that it was possible to define a Euclidean point as the limit of an infinite sequence of decreasing concentric spheres. This solution implies that points are not truly "parts" of a line (or any other geometric structure). Instead, they are limits and each half of the line segment can have its own limit point. (For that matter, even lines and planes are limits of constructions made of spheres.)    (010) For the fourth volume on geometry of the _Principia Mathematica_. Whitehead independently developed a version of mereology in which the basic construct was a 4-D blob and points were decreasing sequences of nested blobs. He never completed that volume, but he published some of the material in other writings.    (011) It's significant that Gödel couldn't find a solution that used only set theory and that Tarski and Whitehead adopted mereology rather than set theory to address such issues.    (012) And by the way, the Whitehead-Tarski approach can be adapted to relate a time ontology based on intervals to one that is based on instants. In effect, an instant is the limit of a decreasing nest of intervals. With this mapping, it is possible to map any observation statement made in terms of one ontology to an equivalent statement in terms of the other.    (013) John    (014) _________________________________________________________________ Message Archives: http://ontolog.cim3.net/forum/ontolog-forum/ Config Subscr: http://ontolog.cim3.net/mailman/listinfo/ontolog-forum/ Unsubscribe: mailto:ontolog-forum-leave@xxxxxxxxxxxxxxxx Shared Files: http://ontolog.cim3.net/file/ Community Wiki: http://ontolog.cim3.net/wiki/ To join: http://ontolog.cim3.net/cgi-bin/wiki.pl?WikiHomePage#nid1J To Post: mailto:ontolog-forum@xxxxxxxxxxxxxxxx    (015) ```
 Current Thread Re: [ontolog-forum] Ontological Means for Systems Engineering, (continued) Re: [ontolog-forum] Ontological Means for Systems Engineering, Schiffel, Jeffrey A Re: [ontolog-forum] Ontological Means for Systems Engineering, Pat Hayes Re: [ontolog-forum] Ontological Means for Systems Engineering, Schiffel, Jeffrey A Re: [ontolog-forum] Ontological Means for Systems Engineering, paola . dimaio Re: [ontolog-forum] Ontological Means for Systems Engineering, Jack Ring Re: [ontolog-forum] Ontological Means for Systems Engineering, Howard Mason Re: [ontolog-forum] Ontological Means for Systems Engineering, Jack Ring Re: [ontolog-forum] Ontological Means for Systems Engineering, Tolk, Andreas Re: [ontolog-forum] Ontological Means for Systems Engineering, Ed Barkmeyer Re: [ontolog-forum] Ontological Means for Systems Engineering, Azamat Re: [ontolog-forum] Ontological Means for Systems Engineering, John F. Sowa <= Re: [ontolog-forum] Ontological Means for Systems Engineering, Pat Hayes Re: [ontolog-forum] Ontological Means for Systems Engineering, John F. Sowa Re: [ontolog-forum] Ontological Means for Systems Engineering, Pat Hayes Re: [ontolog-forum] Ontological Means for Systems Engineering, John F. Sowa Re: [ontolog-forum] Ontological Means for Systems Engineering, Ron Wheeler [ontolog-forum] Ontological Means for Systems Engineering, Tolk, Andreas Re: [ontolog-forum] Ontological Means for Systems Engineering, paola . dimaio Re: [ontolog-forum] Ontological Means for Systems Engineering, Anders W.Tell