On 12/16/2018 9:30 PM, Jason Resch wrote:
On Sun, Dec 16, 2018 at 9:39 PM Bruce Kellett <[email protected]
<mailto:[email protected]>> wrote:
On Mon, Dec 17, 2018 at 1:50 PM Jason Resch <[email protected]
<mailto:[email protected]>> wrote:
On Sun, Dec 16, 2018 at 7:21 PM Bruce Kellett
<[email protected] <mailto:[email protected]>> wrote:
Are you claiming that there is an objective arithmetical
realm that is independent of any set of axioms?
Yes. This is partly why Gödel's result was so shocking, and so
important.
And our axiomatisations are attempts to provide a theory
of this realm? In which case any particular set of axioms
might not be true of "real" mathematics?
It will be either incomplete or inconsistent.
Sorry, but that is silly. The realm of integers is
completely defined by a set of simple axioms -- there is
no arithmetic "reality" beyond this.
The integers can be defined, but no axiomatic system can prove
everything that happens to be true about them. This fact is
not commonly known and appreciated outside of some esoteric
branches of mathematics, but it is the case.
All that this means is that theorems do not encapsulate all "truth".
Where does truth come from, if not the formalism of the axioms? Do
you agree that arithmetical truth has an existence independent of the
axiomatic system?
No. You are assuming that arithmetic exists apart from axioms that
define it. There are true things about arithmetic that are not provable
/within arithmetic/. But that is not the same as being independent of
the axioms. Some axioms are necessary to define what is meant by
arithmetic.
Brent
There are syntactically correct statements in the system that are
not theorems, and neither are their negation theorems.
Yes.
Godel's theorem merely shows that some of these statements may be
true in a more general system.
So isn't this like scientific theories attempting to better describe
the physical world, with ever more general and more powerful theories?
That does not mean that the integers are not completely defined by
some simple axioms. It means no more than that 'truth' and
'theorem' are not synonyms.
I agree with this.
Jason
--
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 [email protected]
<mailto:[email protected]>.
To post to this group, send email to [email protected]
<mailto:[email protected]>.
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
--
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 [email protected].
To post to this group, send email to [email protected].
Visit this group at https://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
