Definitions, Proofs, Derivations

[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