On 12/16/2018 9:36 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 10:22 PM Brent Meeker <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 12/16/2018 4:39 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 5:53 PM Brent Meeker
<meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:
On 12/16/2018 1:56 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 3:28 PM Brent Meeker
<meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:
On 12/15/2018 10:24 PM, Jason Resch wrote:
On Sat, Dec 15, 2018 at 11:35 PM Brent Meeker
<meeke...@verizon.net <mailto:meeke...@verizon.net>> wrote:
On 12/15/2018 6:07 PM, Jason Resch wrote:
On Sat, Dec 15, 2018 at 7:57 PM Brent Meeker
<meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
On 12/15/2018 5:42 PM, Jason Resch wrote:
hh, but diophantine equations only need
integers, addition, and multiplication,
and can define any computable function.
Therefore the question of whether or not
some diophantine equation has a solution
can be made equivalent to the question
of whether some Turing machine halts.
So you face this problem of getting at
all the truth once you can define
integers, addition and multiplication.
There's no surprise that you can't get at
all true statements about a system that
is defined to be infinite.
But you can always prove more true statements
with a better system of axioms. So clearly
the axioms are not the driving force behind
truth.
And you can prove more false statements with a
"better" system of axioms...which was my
original point. So axioms are not a "force
behind truth"; they are a force behind what is
provable.
There are objectively better systems which prove
nothing false, but allow you to prove more things
than weaker systems of axioms.
By that criterion an inconsistent system is the
objectively best of all.
The problem with an inconsistent system is that it does
prove things that are false i.e. "not true".
However we can never prove that the system doesn't
prove anything false (within the theory itself).
You're confusing mathematically consistency with
not proving something false.
They're related. A system that is inconsistent can
prove a statement as well as its converse. Therefore it
is proving things that are false.
But a system that is consistent can also prove a
statement that is false:
axiom 1: Trump is a genius.
axiom 2: Trump is stable.
theorem: Trump is a stable genius.
So how is this different from flawed physical theories?
The difference is that mathematicians can't test their theories.
Sure they can: A set of axioms predicts a Diophantine equation
has no solutions. We happen to find it does have a solution. We
can reject that set of axioms.
Then the axioms must have also included enough to include
Diophantine equations (e.g. PA) so you have added axioms making
the system inconsistent and every proposition is a theorem. The
only test of the theory was that it is inconsistent.
There is also soundness
<https://en.wikipedia.org/wiki/Soundness> which I think more
accurately reflects my example above.
"...a system is sound when all of its theorems are tautologies." Which
is to say it is true that the theorem follows from the axioms. Not that
it is true simpliciter.
Brent
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