Question

How can the Gödel sentence, expressed informally as This sentence can't be proved in system X, where X is appropriately specified, give rise to self-unprovable sentences within the system, leading to the inability of the system to completely prove itself? In the context of erotetic logic, is the statement This question cannot be answered in system Y ill-stated, and how does the concept of abstract problems, which is inherently erotetic, relate to the notion that This problem is unsolvable in system Y? Can a sentence declare itself problematic by stipulative definition, and if so, what implications does this have for understanding unsolvability within a system? If the statement This problem is unsolvable in system Y is considered well-stated, does it maintain an erotetic nature while existing in substantively factive terms, and is it possible to express the first incompleteness theorem in similar terms? How does the analogy between the Gödel sentence and incompleteness issues and the knower-paradox sentence and Fitch's paradox of knowability unfold, especially concerning the knowability of axioms and the expansive inclusivity of to solve a problem over to prove a point? Lastly, considering the potential truth of the unsolvable sentence and its self-defining nature, does it emerge as a shadow of the knower-paradox sentence, and does the difficulty in defining itself hint at its unsolvability, favoring an analogy with the unknown sentence rather than the unprovable one?

Answer 1

The Gödel sentence and concepts like the knower-paradox show limitations within logical systems, where self-reference leads to unprovable truths. These insights extend to erotetic logic, as questions or problems can similarly declare their own unsolvability, reflecting on system capabilities.

Step-by-step explanation:

The Gödel sentence represents a profound insight into the nature of mathematical systems, showing that any sufficiently powerful and complex system will contain sentences that cannot be proved within the system itself. This concept, discussed by Kurt Gödel in his first incompleteness theorem, illustrates the limitations of self-referential systems. When a sentence claims its own unprovability, it introduces a unique paradox: if the system is consistent, the sentence is true but unprovable; if the system can prove the sentence, the system must be inconsistent. In the context of erotetic logic, which focuses on questions and problems rather than statements, the idea that a question or a problem declares itself unsolvable within a system Y works analogously to the Gödel sentence. It generates a meta-level reflection, questioning the foundational capabilities of the system concerned. This exploration leads to a deeper analysis of the nature of questions and problems, cementing their roles as crucial to understanding logic and the limitations of systems.

The knower-paradox and Fitch’s paradox of knowability relate to the Gödel sentence in highlighting epistemological boundaries. The knower-paradox discusses the challenges in asserting knowability of truths within a system, analogous to Gödel's exploration of provability. The notion of solving a problem being more inclusive than proving a point reflects the wider applicability of these insights beyond formal mathematical proof to encompass broader epistemic considerations. Finally, the notion that a problem can define itself as unsolvable reveals the inherent complexity in attempting to demarcate the boundaries of what is knowable or provable within a particular logical or epistemological framework. It shows that some answers or truths may indeed constitute 'shadows'—present and real, yet inaccessible through the system they arise in.