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Re: [ontolog-forum] Current Semantic Web Layer Cake

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, Peter F Brown <peter@xxxxxxxxxx>
Cc: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Kathryn Blackmond Laskey <klaskey@xxxxxxx>
Date: Tue, 31 Jul 2007 18:09:37 -0400
Message-id: <p06110438c2d562672e5d@[]>
>theory is ..., usually in the form of
>Zermelo-Fraenkel (Z-F) set theory    (01)

or the equally powerful, somewhat less intuitive, but finitely 
axiomatizable von Neuman / Godel / Bernays (NGB) set theory    (02)

>  ... still the widely
>accepted mathematical foundational language that
>is as near to consistent as anything can be.    (03)

It either is or is not consistent.  You can't be "near to consistent" 
any more than you can be "almost pregnant."    (04)

If it is not consistent, then a sufficiently persistent attempt to 
establish whether it is consistent will eventually uncover the 
inconsistency.  If it is consistent, then no inconsistency will ever 
be discovered by a sound theorem prover -- but at no point in time 
can it be proven to be consistent.    (05)

Human mathematics has not yet discovered an inconsistency, and most 
mathematicians believe it is consistent.    (06)

>... the
>question of how to ground such theories in the
>actual world (either of physical entities or of
>experiences) is not even remotely relevant to, or
>influenced by, such matters as axiomatic set
>theory or Goedel's second theorem.    (07)

Well said.    (08)

>By the way, Im not sure what you are referring to
>by "vicious circularity", but what links
>Russell's paradox and Goedel's undecideability
>result (and Turing's uncomputability theorem) is
>that they all derive fairly directly from the
>classical Liar paradox "This sentence is false",
>which itself is made possible only by
>self-reference. If this is what you mean, it also
>has nothing much to do with ontologies and their
>grounding.    (09)

Well said.    (010)

Kathy    (011)

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