On 12/16/2018 11:02 PM, Jason Resch wrote:
On Mon, Dec 17, 2018 at 12:05 AM Brent Meeker <meeke...@verizon.net
<mailto:meeke...@verizon.net>> wrote:
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.
How about this:
Arithmetic soundness[edit
<https://en.wikipedia.org/w/index.php?title=Soundness&action=edit§ion=5>]
If/T/is a theory whose objects of discourse can be interpreted
asnatural numbers <https://en.wikipedia.org/wiki/Natural_numbers>,
we say/T/is/arithmetically sound/if all theorems of/T/are actually
true about the standard mathematical integers. For further
information, seeω-consistent theory
<https://en.wikipedia.org/wiki/%CE%A9-consistent_theory>.
OK. But note that it assumes the natural numbers exist independently of
T and in order to be applied it assumes one can know that every theorem
of T is true of the natural numbers. If the natural numbers exist
independently of T then a theorem of T may or may not apply to them.
But in fact the natural numbers are only defined as that which satisfies
some theory T'.
Brent
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.