| The "{}" symbol above can represent the beautiful image of two praying hands pressed together with nothing in between. (Or is it two ears?) It's a firm grip on nothing. It’s the sign of the empty set of which "there can be only one"- -(Highlander dude talking about the empty set). The empty set is the set of everything nothing. All nonexistent is unreal and nothing unreal exists. Perhaps it can be said to be the religious symbol of atheists as the empty set contains no existing elements. Perhaps it can be said to be the ultimate religious symbol 'to' atheists because the empty set "contains" everything nonexistent, i.e. all that is imaginary and unreal, but has no real and existing elements, thus "god" (or rather "all gods") is the king of the empty set. He wields supreme imaginary power over all unreal, such as all seven-legged zebras and all spherical cubes. "He" can be omniscient and omnipotent about all that is imaginary. He can create an unreal rock so heavy that he can't lift it and then lift it anyway because the rules of reality only apply to the real, not the unreal. Topologically, the empty set is both closed and open, thus "clopen".*
-------------- "Considered as a subset of the real number line (or more generally any topological space), the empty set is both closed and open. All its boundary points (of which there are none) are in the empty set, and the set is therefore closed; while for every one of its points (of which there are again none), there is an open neighborhood in the empty set, and the set is therefore open." "By the definition of subset, the empty set is a subset of any set A, as every element x of {} belongs to A. If it is not true that every element of {} is in A, there must be at least one element of {} that is not present in A. Since there are no elements of {} at all, there is no element of {} that is not in A, leading us to conclude that every element of {} is in A and that {} is a subset of A. Any statement that begins "for every element of {}" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set." "Any axiom that states the existence of any set will imply the axiom of empty set, using the axiom schema of separation. For example, if A is a set then the axiom schema of separation allows the construction of the set B = {x in A | x 'doesn't equal' x}, which can be defined to be the empty set." "The empty set"; wikipedia. http://en.wikipedia.org/wiki/Open_set * http://en.wikipedia.org/wiki/Clopen_set ----------- Where "All S are P" and "All S are not P" is stated and where 'S' is the empty set, (like "All seven legged zebras eat concrete" and "all seven legged zebras don't eat concrete") then both statements are vacuously true because universal statements don't imply existence, and because there are no seven legged zebras to falsify either statement, and because the empty set is a subset of "P" being that the empty set is a subset of ALL sets, and likewise, it is also a subset of the set called "not P", which of course is the set of everything that isn't "P". The Atheist prayer: "Un-father, who art lord of the empty set, "Hollow", be thy name. Thy magic kingdom is in the kidney of the enchanted forest and zero therefore will be done, for it is the magic number of thine existing elements and is the only number that is both real and imaginary. Give us this day our unreal threads, unless they are from the '70's. And grant us unlimited wishes and magical dreams as is the 'likely' story about leprechauns, fairy goblins, dragons and sea serpents. Forgive us for thinking that torturing and crucifying your unreal son is a groovy idea, (not to mention drinking his blood), but you handed us this unreal idea when you dictated your imaginary words to the nonexistent Moses Inc. Tell us your vacuous truth, oh king of nothing, for truly everything is logically true of your all your existing elements, of which there are none. You, who has the omnipotent power to do anything imaginary and otherwise absurd within the magic kingdom of the empty set, can even make "All gods are A" and "All gods are not A" both vacuously true statements and not a logical contradiction, for only the existent have properties (contradictory or otherwise) and being that nothing unreal exists, you, oh master of nothing, art the stuff dreams are made of." "Amen-Ra" (The other un-god). {} |
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| (some of the following is tongue in cheek and some of the following is merely an unorthodox, yet consistent and 'legal' consideration of modern symbolic logic) |