Re: Godel cant tell us what makes a mathematical statement true

MoeBlee wrote:
On Aug 13, 11:51 pm, herbzet <herb...@xxxxxxxxx> wrote:
Nam Nguyen wrote:
herbzet wrote:
Nam Nguyen wrote:
There's no such thing as just a true formula: usually it's understood
by that we mean an interpreted-as-true formula. As such an interpreted
formula needs to have a meaning. But not the other way around. For
example, if the formula F = (0=0) /\ (0=1) has a meaning, its meaning
is *independent* from whether or not F has any truth value!
For example,
GC has some meaning but whether or not it's true or false (by whatever the
chosen underlying truth definition) is a different matter.
Well this is a technical question of some interest. I was under
the impression that interpreting a formula in a given structure
entailed a truth-value assignment also. Perhaps I am mistaken
on this point. Can you explain how a formula F can be assigned
a meaning (interpreted) without thereby arriving a truth-value
for F? I'm, admittedly, slightly fuzzy on this point.

A structure gives the truth value of SENTENCES but not of formulas in
general.

A structure PLUS an assignment for the individual variables does
provide for a formula to be satisfied or not satisfied by that
structure with that assignment for the individual variables. And a
structure alone does provide for whether a formula is satisfiable or
not satisfiable per that structure. But for truth or falsehood in a
structure onto itself, we have that only for sentences.

I don't opine as to what herbzet has in mind by 'meaning', but, aside
from intensional meaning, at least the extensional meaning of a
formula per a structure might be taken to be the set of assignments
for the individual variables under which the formula is satisfied by
that structure.

Are you responding to herbzet, or Nam?


MoeBlee



--
"To discover the proper approach to mathematical logic,
we must therefore examine the methods of the mathematician."
(Shoenfield, "Mathematical Logic")
.

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