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Re: distinguishability - in context, according to definitions
Posted:
Feb 19, 2013 7:50 AM
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In <soOdnWuz7siUXbzMnZ2dnUVZ_rqdnZ2d@giganews.com>, on 02/17/2013
at 11:45 PM, fom <fomJUNK@nyms.net> said:
>In that book, the reflexive axiom is implemented by simply writing
>down the identity. As an axiom of identity, it will not require
>discharge. Thus, when one wants to say,
>"Let a and b be such that not a=b."
>One has in the derivation,
>a=a
>b=b
>|-(a=b)
That's not a derivation.
>0.999...=0.999...
>1.000...=1.000...
>|1.000...=0.999...
>since the assertion is that the two
>symbols are equal.
Then that assertion has to be the first step in the derivation and you
get the trivial
1.000...=0.999...
|- 1.000...=0.999...
>If one is interested in proofs that begin with quantified
>statements and that end with quantified statements, then the
>presumption is that every constant used in the proof
>is definable relative to a description.
It is quite common for a theory to include constants. They are
different from notational conventions that one defines in terms of
other symbols; their nature is constrained only by the axioms.
>It is just that the other axioms for a set theory need
>to take the universal class into account.
There are set theories in which there is no such class.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
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