|From:||Alex Shkotin <alex.shkotin@xxxxxxxxx>|
|Date:||Tue, 21 Sep 2010 11:00:16 +0400|
you are right:
- I should be more careful using FOL term abbreviation. It should be "In First order language to get definition..." or even "In First order language with definition feature to get definition...".
- and we do need theorem of existence and uniqueness.
This is why I have mentioned transitive closure: If we keep in mind that systems we describe are algebraic systems, then in set theory (as you mentioned) we may prove that for that kind of models we do have definition.
- and in FOL we do can only introduce an axiom
R_(x,y) <-> R(x,y) or there_exists z such_that R_(x,z) and R(z,y)
as representative of our definition.
and anyway may I summarize that definition is more then just axiom;-)
Even context of "definition" may be beyond just language (and logic) - may be we need to keep in mind referent itself or at least class of models (math speaking).
For ex- if an infinite in one direction, properly directed chain (node -> node ->...) is an existing thing for us then "+", "*" arithmetic operations has recursive definition on it.
2010/9/20 Christopher Menzel <cmenzel@xxxxxxxx>
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