On 12/14/2018 2:59 AM, Bruno Marchal wrote:
On 13 Dec 2018, at 21:24, Brent Meeker <[email protected]> wrote:
On 12/13/2018 3:25 AM, Bruno Marchal wrote:
But that is the same as saying proof=>truth.
I don’t think so. It says that []p -> p is not provable, unless p is proved.
So []([]p -> p) -> p or in other words Proof([]p -> p) => (p is true) So in
this case proof entails truth??
But “[]([]p -> p) -> p” is not a theorem of G, meaning that "[]([]p -> p) -> p”
is not true in general for any arithmetic p, with [] = Gödel’s beweisbar.
The Löb’s formula is []([]p -> p) -> []p, not []([]p -> p) -> p.
For example []f -> f (consistency) is not provable. It will belong to G* \ G.
Another example is that []<>t -> <>t is false, despite <>t being true. In fact <>t ->
~[]<>t.
Or <>t -> <>[]f. Consistency implies the consistency of inconsistency.
I'm not sure how to interpret these formulae. Are you asserting them for every
substitution of t by a true proposition (even though "true" is undefinable)?
No, only by either the constant propositional “true”, or any obvious truth you
want, like “1 = 1”.
Or are you asserting that there is at least one true proposition for which []<>t ->
<>t is false?
You can read it beweisbar (consistent(“1 = 1”)) -> (consistent (“1=1”), and
indeed that is true, but not provable by the machine too which this provability
and consistency referred to.
Nothing which is proven can be false,
Assuming consistency, which is not provable.
So consistency is hard to determine. You just assume it for arithmetic. But
finding that an axiom is false is common in argument.
Explain this to your tax inspector!
I have. Just because I spent $125,000 on my apartment building doesn't
mean it's appraised value must be $125,000 greater.
If elementary arithmetic is inconsistent, all scientific theories are false.
Not inconsistent, derived from false or inapplicable premises.
Gödel’s theorem illustrate indirectly the consistency of arithmetic, as no one
has ever been able to prove arithmetic’s consistency in arithmetic, which
confirms its consistency, given that if arithmetic is consistent, it cannot
prove its consistency.
But it can be proven in bigger systems.
Gödel’s result does not throw any doubt about arithmetic’s consistency, quite
the contrary.
Of course, if arithmetic was inconsistent, it would be able to prove (easily)
its consistency.
Only if you first found the inconsistency, i.e. proved a contradiction.
And even then there might be a question of the rules of inference.
Brent
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