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.
Brent
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