Re: Continuum hypothesis

On Sep 12, 7:04 am, "R. Srinivasan" <sradh...@xxxxxxxxxx> wrote:
The simplest case to consider is the theory corresponding to the null
set of axioms, T_0.

This case IS NOT simple.
This case is MAXIMALLY HARD
in the CLASSICAL paradigm!
The absence of axioms entails absence of a signature, which
entails absence of any inputs to the first-order-language construction
algorithm! UNTIL AFTER you have a language, you simply have NOTHING
to say! This creates a HARD PHILOSOPHICAL problem, something very
much akin to questions, in ZFC, like "is the empty relation
reflexive"?
The vacuousness question is VERY thorny.
You will get DIFFERENT REACTIONS FROM EXPERTS regarding whether,
for example, P v ~P follows from NO axioms, the point being that, IF
THERE
IS NO P IN THE SIGNATURE, how can you get P into the language?
Other experts will simply refuse to worry about any particular
language and
insist that OF COURSE P v ~P is ALWAYS tautologous; THERE ARE EXPERTS
out there who will INSIST that having NO axioms means you are leaving
the
language COMPLETELY UNrestricted and that e v ~e, f v ~f, and tons of
other
tautologies ARE STILL ALWAYS ALREADY IN the language!


.

Relevant Pages

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