1.10. The “no-self-defeating object” argument, revisited 23 The existence of the set B is guaranteed by the axiom schema of specifi- cation. If B were an element of itself, then by construction we would have B B, a contradiction. Thus we must have B B. From construction, this forces B A. In the usual axiomatic formulation of set theory, the axiom of foundation implies, among other things, that no set is an element of itself. With that axiom, the set B given by Proposition 1.10.9 is nothing other than A itself, which by the axiom of foundation is not an element of A. But since the axiom of foundation was not used in the proof of Proposition 1.10.9, one can also explore (counterfactually!) what happens in set theories in which one does not assume the axiom of foundation. Suppose, for instance, that one managed somehow to produce a set A that contained itself10 as its only element: A = {A}. Then the only element that A has, namely A, is an element of itself, so the set B produced by Proposition 1.10.9 is the empty set B := ∅, which is indeed not in A. One can try outrunning Proposition 1.10.9 again to see what happens. For instance, let’s add the empty set to the set A produced earlier, to give the new set A := {A, ∅}. The construction used to prove Proposition 1.10.9 then gives the set B = {∅}, which is indeed not in A . If we then try to add that set in to get a new set A := {A, ∅, {∅}}, then one gets the set B = {∅, {∅}}, which is again not in A . Iterating this, one in fact begins constructing the11 von Neumann ordinals. 1.10.2. Logic. One can also convert the no-self-defeating arguments given in the logic section of the previous section into “every object can be defeated” forms, though these were more difficult for me to locate. We first turn to the result (essentially coming from the liar paradox) that the notion of truth cannot be captured by a predicate. We begin with the easier “self-referential case”: Theorem 1.10.10 (Impredicativity of truth, self-referential case. (Informal statement)). Let L be a formal language that contains the concepts of pred- icates and allows self-reference, and let M be an interpretation of that lan- guage (i.e., a way of consistently assigning values to every constant, ranges to every variable, and truth values to every sentence in that language, obey- ing all the axioms of that language). Then there does not exist a “truth predicate” T(x) in L that takes a sentence x as input, with the property that for every sentence x in L, that T(x) is true (in M) if and only if x is true (in M). 10 Informally, one could think of A as an infinite nested chain, A := {{{. . .}}}. 11Actually, the original set A plays essentially no role in this construction one could have started with the empty set and it would have generated the same sequence of ordinals.
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