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Summary: Errata for ``The Power of Temporal Proofs''
September 8, 1989
The proof of Theorem 6.2 in ``The Power of Temporal Proofs'' is seriously
flawed. The claim was that all valid temporal formulas are arithmetical in
the proof system T 2 . The claim does not appear to be true. Indeed, Andr'eka,
N'emeti, and Sain, who uncovered the problem, recently sent me a paper with
a counterexample, ``On the Strength of Temporal Proofs.''
The main role of the claim was as a stepping stone towards Theorem 7.2.
This claim, in turn, says that T 1
is complete for arithmetical formulas while
T 2 is complete for all formulas. With the failure of Theorem 6.2, the result
proved in 7.2 is weaker. It holds simply that both T 1
and T 2
are complete
for arithmetical formulas. More precisely, Theorem 7.2 should read:
For every formula u, `O P (u) ) ` T 1
u if u is arithmetical in ` T 1
.
For every formula u, ` P P (u) ) ` T 2
u if u is arithmetical in ` T 2
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