Abstract

This paper examines the role of observer in the context of provability within logical systems and the inherent limitations imposed by G¨odel’s Incompleteness Theorems. With the law of the Excluded Middle (LEM) and the concept of undecidable propositions being separate entities, a sequence of logical systems {sn} is constructed to represent the idea that every proposition may eventually be provable within some sufficiently strong system {sn+} and assessing whether such model can overcome the fundamental limitations identified by G¨odel. An hierarchy of observers {On}, each capable of perceiving statements that previous observers cannot, is introduced, and its relation to the provability of propositions is analyzed. Ultimately, undecidability remains an intrinsic feature of these observed systems.