On 24 Jan 2015, at 19:22, Russell Standish wrote:
On Sat, Jan 24, 2015 at 09:55:19AM -0800, meekerdb wrote:
On 1/24/2015 12:20 AM, Bruno Marchal wrote:
Do you see the relationship between Gödel's second incompleteness
theorem and the modal formula
<>t -> ~[] <>t ?
I don't see it, because I don't understand what <>t means. t is a
tautology, the negation of a contradiction. Yet you seem to use t
to mean "has a model"? And I'm not clear on how one is supposed to
know the true propositions of a model.
Brent
<>t can be read as "consistent"
The theorem therefore states "consistent theories cannot prove their
own consistency".
Yes.
And, by the completeness theorem: <>t is equivalent with "I have a
model", or "there is a structure satisfying my beliefs (or consistent
with my beliefs)", or "there is a reality".
So incompleteness + completeness entails, with computationalism that
IF there is a reality, we cannot prove it exists.
It is bad luck that we use "completeness" with a so big difference of
meaning in "incompleteness".
A theory is complete means that: provable is implied by "true in all
models" (or equivalently: consistent implies at least one model). PA,
ZF, all first-order theory are complete in that sense.
Incomplete means simply that there are non provable true (in some
model, but usually we think about the standard model) sentence.
Usually, it is simple to prove that "provable implies true in all
model (soundness)". What is hard is to prove the converse
(completeness). It is already not trivial even for the simple
propositional classical calculus. People should read a proof in books
like Mendelson, or other good textbooks.
Bruno
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