Godel’s Second Incompleteness Theorem
July 29, 2010 2 Comments
A formal system is characterized by its axiom set. In this post, I talk about systems with only finitely many axioms. A system is said to be consistent if its axioms do not contradict any of its axioms. In a system, a statement is said to be independent of the system if it is not derivable from its axioms logically.
Godel’s Second Incompleteness Theorem states: If a system is consistent, then the consistency of the system is not provable within the system.
More formally one may state the theorem as: If a system 
It follows that, if there is a proof of consistency of a system within the system then it is necessarily inconsistent ( contra positive of the theorem statement ). The surprise element related to this theorem is, say some postulates such as Euclid’s postulates to do 2-d geometry are consistent, then this very fact forbids us from giving a proof of postulates being consistent. One may think of proving consistency of a system 
A related puzzle problem,
Consider a system 
Consistency proof:-
Since nothing is derivable with just one axiom, hence the only axiom
Isn’t this a counter example to Second Incompleteness Theorem? Why not ?
Godel’s Theorems talk about logical systems that try to prove their own consistency within the System S. But, this means, statements regarding provability and consistency are required within the System. Hence, if we are asking about consistency of S, A cannot be the only rule within S.
‘ ” For any formal effectively generated theory T including basic arithmetical truths and also certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent. ” A technical subtlety in the second incompleteness theorem is how to express the consistency of T as a formula in the language of T. There are many ways to do this, and not all of them lead to the same result. ‘
– from wikipedia:- http://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems
One example is, the Rule: R(s) = the statement ‘s’ is not provable in S.
Right.
In the consistency proof I wrote, I went out of the system by axiomatizing the fact that nothing is derivable with just one axiom.
No way one can prove consistency of a consistent system within the system.
@quoted theorem statement from Wikipedia: Many web sources don’t say the theorem is ‘if and only if’, probably because the fact that: If a system is inconsistent then it’s inconsistency proof can be given within the system, is trivial. I’m still not sure about that. Neither could I construct a counter example to this. Looks like theorem is ‘if and only if’.
I’m struck constructing a proof to the theorem. Any leads, let me know.