[FOM] sound, unsound, and undecidable theory foundations

[Paul Hollander]

[Note:  final submission; typos corrected.]

Schematically, using only the symbols 'T', 'L', and 'S', one may 
distinguish between sound, unsound, and undecidable foundations for an 
object theory and its metatheory.

Let 'T' express a formal truth predicate, and let 'L' and 'S' be 
variables ranging over the sentences of a theory of first-order logic 
with identity ('L') and either naive set theory or the lambda calculus 
('S').  Then,

     |- T(L) => S,

expresses a demonstrably sound claim -- think of Loeb's theorem -- while,

     |- S => T(L),

expresses a demonstrably unsound claim -- think of Russell's paradox and 
Curry's/Loeb's paradox -- and

     |- T(S) <=> L,

expresses a demonstrably undecidable claim -- think of Goedel's 
incompleteness argument and Tarski's argument on truth.

To me, this is pedagogically useful because it explains four distinct 
meta-theoretical results, Loeb's, Russell's, Curry's/Loeb's, and 
Goedel's/Tarski's, using just three symbol schemas 'T', 'L', and 'S'.

I'd appreciate any feedback from FOM.

Cheers,

Paul Hollander