Re: A question about FOL theories and models

In article <uhnGg.445553$IK3.279192@pd7tw1no>,
Nam Nguyen <namducnguyen@xxxxxxx> wrote:
<Let me explain my motivations here. What if abstractly we could *not*
<come up with any model for T = G + ~(5th postulate)? In such case,
<how would we know for certain whether T is or is not inconsistent.
<Conversely, if we syntactically formulate a theory T and have proven
<a "handful" numbers of theorems, and have a structure M where these
<theorems are true, is *always guaranteed* that if we *assume* M is
<a model of T, M would genuinely be model of T?
[...]
<The issue here, imho, is the 2 "dualistic"
<components of FOL reasoning: the concreteness/objectivity of anything
<finite, such as formulae, proofs, inference rules; and the
<abstraction/subjectivity such as semantic, interpretation, truth,
<and models. To the extend that in general we could not infer one
<component from the other, our ability to face with some of the
<"severe" problems in FOL would be insufficient (or "incomplete"?).

I don't fully understand what you're driving at; however, it sounds to
me that your concerns are addressed by Goedel's completeness theorem,
which establishes the equivalence of "consistent" and "has a model."
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
.

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