Re: Definitions, Proofs, Derivations

2003-01-09 Thread Jim Choate 

On Wed, 8 Jan 2003, Ken Hirsch wrote:

> In general you have to consider the whole system, including derivation
> rules, not just the axioms, although you can certain start with a set of
> axioms like:
>
> { x=1,  x=2}
> or, come to think of it,
> { 1=2 }

You'd first have to define what '=' means, that would be your axiom. 'x',
'1', and '2' would simply be symbols derived from the definition of '='.


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Re: CDR: Re: Definitions, Proofs, Derivations

2003-01-08 Thread Jim Choate 

On Wed, 8 Jan 2003, Sarad AV wrote:

> there will be no inconsistency in a formal axiomatic
> systems

Can't prove a negative, even in a formal system.


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  We are all interested in the future for that is where you and I
  are going to spend the rest of our lives.

  Criswell, "Plan 9 from Outer Space"

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Re: Definitions, Proofs, Derivations

2003-01-08 Thread Ken Hirsch 
"Sarad AV" writes:
> there will be no inconsistency in a formal axiomatic
> systems

Huh?

>-but can any one point me to a contradicting
> set of axioms in an axiomatic system?

In general you have to consider the whole system, including derivation
rules, not just the axioms, although you can certain start with a set of
axioms like:

{ x=1,  x=2}
or, come to think of it,
{ 1=2 }

Most famously, Frege's system was shown to be inconsistent by Russel.  More
recently, the first edition of Quine's Mathematical Logic (1940) was shown
to be inconsistent by Rosser.

For Frege, see "From Frege to Gvdel: A Source Book in Mathematical Logic,
1879-1931" by Jean van Heijenoort

Re: Definitions, Proofs, Derivations

2003-01-08 Thread Sarad AV 
hi,

> > Then, if any two or more axioms of an
> > alleged mathematical
> > theory are found to be inconsistent with each
> other,
> > the whole theory
> > collapses."
> 

there will be no inconsistency in a formal axiomatic
systems-but can any one point me to a contradicting
set of axioms in an axiomatic system?


Regards Sarath.

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Re: Definitions, Proofs, Derivations

2003-01-05 Thread Sarad AV 
hi,

Thats a beautiful one.
--- Jim Choate <[EMAIL PROTECTED]> wrote:
To assert that a theorem is
> false means to deny
> one or more of the axioms. However, to assert that a
> theorem is true does
> not necessarily mean to assert the truth of all
> axioms.

yes-it only means its time to update  our mathametical
model for certain observations which does not agree
with the model we made for it.

> Some theorems
> remain true even if some of the axioms of a
> mathematical theory are
> rejected.

yes-our  original model still works for the domain it
still agrees with our observations.

> 
> To accept the truth of the axioms is simply to agree
> to assume them to be
> true.
yes-we will have to,other wise we won't have models
which work.

> However, the consistency of the axioms with
> each other is sometimes
> in question.

Then its time to modify  our axioms.

> Then, if any two or more axioms of an
> alleged mathematical
> theory are found to be inconsistent with each other,
> the whole theory
> collapses."

It will be a proposition then-won't be an axiom.If we
find that the two axioms are inconsistent with each
other-its time we update our axioms and they will
become propositions and no more axioms.The theory as
such doesn't collapse.


Regards Sarath.
 


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