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

--
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.

Reply via email to