#
I have a question about Godel_ey type stuff:
Here goes:
(where T is a theory that is sufficiently powerful to express statements about itself )
if a theory is consistent, con(T), then it cannot prove its own consistency.
¬( T_provable(con(T) ).
(1) con(T) => ¬( T_provable( con(T) )
But an INconsistent theory can prove ANYTHING, including its own consistency.
¬con(T) => T_provable( con(T) )
Therefore if a theory can NOT prove its own consistency, it definitely ISN'T an INconsistent Theory. ( assuming it is otherwise sufficiently expressive )
¬( T_provable( con(T) ) =>¬( ¬ con(T) )
(2) ¬( T_provable( con(T) ) => con(T)
since:
¬( ¬ con(T) ) => con(T)
therefore:
by (1) and (2)
con(T) < = > ¬( T_provable( con(T) ) )
This is exactly of the form:
G < = > ¬( T_provable(G) )
where G is the/a Godel statement for the theory.
My question is:
How does one show that G is DIFFERENT from merely the statement "con(T)".
and that it does not 'contain' con(T) as any sub-part of G ?
since the statement "con(T)" fulfills exactly the same criteria.
"con(T)" is just such a statement.
I probably won't understand your answer, but it might be a good excercise for your logic students!

I have a question about Godel_ey type stuff:
Here goes:
(where T is a theory that is sufficiently powerful to express statements about itself )
if a theory is consistent, con(T), then it cannot prove its own consistency.
¬( T_provable(con(T) ).
(1) con(T) => ¬( T_provable( con(T) )
But an INconsistent theory can prove ANYTHING, including its own consistency.
¬con(T) => T_provable( con(T) )
Therefore if a theory can NOT prove its own consistency, it definitely ISN'T an INconsistent Theory. ( assuming it is otherwise sufficiently expressive )
¬( T_provable( con(T) ) =>¬( ¬ con(T) )
(2) ¬( T_provable( con(T) ) => con(T)
since:
¬( ¬ con(T) ) => con(T)
therefore:
by (1) and (2)
con(T) < = > ¬( T_provable( con(T) ) )
This is exactly of the form:
G < = > ¬( T_provable(G) )
where G is the/a Godel statement for the theory.
My question is:
How does one show that G is DIFFERENT from merely the statement "con(T)".
and that it does not 'contain' con(T) as any sub-part of G ?
since the statement "con(T)" fulfills exactly the same criteria.
"con(T)" is just such a statement.
I probably won't understand your answer, but it might be a good excercise for your logic students!

Read another response about Logic