REVERSE CONCEPT CALCULUS. Harvey Friedman had presented several topics to FOM listing concerned with concept calculus.[1,2] My understanding of Concept Calculus is that it begins from some natural concepts, combine them in a plausible manner and then seek to know the position of that combination in the interpret-ability hierarchy with special consideration to its place from ZFC.[3] I personally have been always seeking casting abstract mathematical theories in natural similes, which are natural interpretations that are truth persevering under various expositions of set/class theories. Of course my direction was from the abstract to the natural which is somehow opposite that of Concept calculus, however I see that this can meet with the later, the more one can bring about natural similes and even improve on them giving more natural versions then one would end up with a natural context that is really too far from the abstract theory it began with. Since natural contexts 'stronger' than ZFC are of the most interest in Concept calculus, what I'm saying here is to begin with a mathematical theory that is stronger than ZFC, then try cast it in natural dress and further refining that would also lead to a natural context that is interesting, somehow working in the reverse direction but it can give unexpected results and virtually culminate in fulfilling the same goal behind Concept Calculus. For example: we begin with the following abstract theory in first order logic with equality and membership that does prove Con(ZFC) actually with Global choice! And then see below it the rather unexpected refined natural simile of it. Define: set(x) iff Exist y. x in y Axioms: Identity axioms + Extensionality: (for all z. z in x iff z in y) -> x=y Impredicative class comprehension scheme: if phi is a formula in which x is not free, then (Exist x. for all y. y in x iff set(y) & phi) is an axiom. Binary limitation: (for all y. y in x -> y=a or y=b) -> set(x) Universal limitation: x < W -> set(x) Hereditary limitation: set(x) iff Exist y. set(y) & for all z in x (z << y) where < is strict subnumerousity, and << is hereditary subnumerousity (no necessarily strict), both defined in the conventional manner. W is the class of all sets. Extensionality can be dropped. This theory clearly proves Con(ZFC + Global choice). Now a natural simile would be the following: We consider 'collectors' and define Natural collectors as collectors that are themselves collectible, while Supernaturals would be collectors that are not collectible by any collector. A natural dichotomy. [1] We stipulate the very natural principle of having a collector for Every collection of natural collectors. This is basically saying that we can have any collection of what is collectible, which is a very natural seek for maximality. [2] We add the natural principle that any collector that collects up to maximally two collectors Is a natural collector. This is obvious since nobody would think of such weak collectors as supernaturals. [3] We add also the principle that any collector collecting a collection that is strictly smaller than the collection of all natural collectors Is a natural collector. This can be understood under the natural assumption that Deities do not allow inferior powered beings to be among them, which is somewhat a natural assumption about the supernatural. [4] We add the principle of considering any collector x for which there exist a natural collector M such that every collector that lies at the bottom of a collector chain up to x would be collecting a collection not larger in size than that of M, then x is a natural collector. This principle albeit somewhat complex yet it has some natural expectation, since nobody expects the collective power of a Deity to be so harshly dictated by that of a natural collector. [5] The reverse direction of [4]. This is the least natural of those principles. What it is saying is that the total collective action of a natural collector and all down chain collectors from it Is within the collective reach of some natural collector. The motivation rests upon the concept of non reach-ability of Deities from below. [6] Extensionality is paraphrased by stating that collectors collecting the same collection cannot be distinguished from each other. This is justified by context, since the above is discourse about collectors we know nothing about a part from their collective action. So with such focused context it is natural to distinguish them after what they collect. / Now clearly this natural simile was only reached after multiple refinements of trials to give natural contexts for the above abstract theory, but the result does seem to be far from it, and of course by far much much more natural than it. Actually when I defined the above theory I didn't have any idea of that natural simile in my mind although I did have the idea of 'collectors' being a natural simile of sets and classes, however I didn't have any shred of thought of seeing the abstract exposition of this theory interpretable so naively in such rather very natural context of thought about the supernatural. To say that ZFC + Global choice is PROVABLE from such almost naive principles of thought about natural and supernatural collectors is a point in favor of its consistency relative to such contexts. Zuhair Al-Johar Feb. 9. 2013 Referrences: [1] Friedman H.M. A divine Consistency Proof for Mathematics. http://www.cs.nyu.edu/pipermail/fom/2012-December/016866.html [2] Friedman H.M. Eight Supernatural Consistency Proofs for Mathematics. http://www.cs.nyu.edu/pipermail/fom/2013-January/016898.html [3] Friedman H.M. Concept Calculus. http://www.math.osu.edu/~friedman.8/pdf/BanffLect022407.pdf