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Hi Tim,
In each of his books, my father wrote at length about what happens
in mathematics when all semantics are removed and replaced with purely
syntactical rules. His belief was that mathematics is a complex system in its
own right. That means mathematics as a system is relational and has
its own organization which we cannot understand when we fractionate it or reduce
it. That means that the inherent semantics generated by such organization is
indispensable. When Hillbert, et al, tried to "improve mathematics" by divorcing
the system from all semantics, it deformed mathematics into a system
characterized by a weird collection of paradoxes which make it useless
as a modeling tool for much of anything.
Judith
----- Original Message -----
From: Tim Gwinn To: *** Sent: Sunday, December 12, 2004 7:27 PM Subject: Re: [ROSEN] Is it sure that any mechanism has a largest model? (LI,8C, p.205) Judith, I don't understand what you mean by "semantics being dispensed with" in reference to the theory of transfinite numbers. Regards, Tim -----Original Message----- From: ROSEN Forum [mailto:*** Behalf Of Judith Rosen Sent: Sunday, December 12, 2004 11:29 AM To: *** Subject: Re: Is it sure that any mechanism has a largest model? (LI,8C, p.205) Hi Tim, What you refer to with transfinite numbers and different sizes of infinities is exactly what happens when the semantics are dispensed with. Regarding "THE intersection of an infinite set of models"... think about it: There is no singular "largest model" in an infinite set. There is no one "intersection". However, I can answer one aspect of that question: Any intersection of two (or more) models is also a model. In an infinite set, there will be infinite intersections. Judith |