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S-denying a theory
We can introduce a threshold Thres = (dthres, ithres, uthres) , where dthres, ithres, uthres are crisp numbers in [0, 1], for decidability, indeterminate-decidability, and undecidability respectively, whence d ≥ dthres, i ≤ ithres, u ≤ uthres respectively, when d, i, u are crisp numbers in [0, 1]; but if d, i, u are subsets of [0, 1], we may consider either max(d), max(i), max(u), or min(d), min(i), min(u), or mid(d), mid(i), mid(u) with mid(.) being the midpoint of the set, or other function: f: P([0, 1]) → [0, 1], where P([0, 1]) is the power set of the interval [0, 1], depending on the application and experts, as f(d), f(i), f(u) respectively. Open Question. Is it possible from a subgroup G of an axiomatic system, using the deducibility methods, to get a set of deducible propositions D1, and from this to get D2, and so on, until one obtains (after a finite or infinite) number of steps Dn+1 = Dn? We mean fixed points, or the sequence of resulted theories D1, D2, … Dn converging to a maximum theory?
S-denying a theory
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It means to take an axiom of a theory and S-deny it, i.e. validate and invalidate it in the same space, or only invalidate it but in many ways.
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