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- In 1908 Ernst Zermelo published an alternative system designed to avoid the known paradoxical statements of the time, even though absolute consistency remained unproven. In Zermelo's set theory, a mathematician first starts with an existing set, and then applies the Axiom of Separation using definite properties to partition out subsets <RT·endnote>Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I," <em>Mathematische Annalen</em> 65 (1908): 261–281.</RT·endnote>. To see how this works, consider the expression <RT·math>\{x \mid P(x)\}</RT·math>. Under unrestricted comprehension, a logician is permitted to define the predicate <RT·math>P(x)</RT·math> as <RT·math>x ∉ x</RT·math>. This produces Russell's Paradox, so the set fails to be defined. In contrast, consider the same predicate, though restricted by Zermelo's Axiom of Separation over a predefined set <RT·math>S</RT·math>, written as <RT·math>\dot{R} = \{x \mid x ∈ S ∧ x ∉ x\}</RT·math>. The only thing a person needs to know about <RT·math>S</RT·math> here is that it has already been successfully defined. So let us ask, is <RT·math>\dot{R}</RT·math> in <RT·math>\dot{R}</RT·math>? If we assume <RT·math>\dot{R}</RT·math> is a member of <RT·math>S</RT·math>, evaluating the second term forces the familiar fatal loop: if <RT·math>\dot{R}</RT·math> is in <RT·math>\dot{R}</RT·math>, it shouldn't be, and if it isn't, it should be. Thus if we assume that <RT·math>\dot{R}</RT·math> is in <RT·math>S</RT·math>, then <RT·math>\dot{R}</RT·math> can not be defined, but by definition, <RT·math>S</RT·math> is defined, and thus its members are defined. As we arrived at a contradiction, the original assumption must be false, i.e. it is wrong to assume that <RT·math>\dot{R}</RT·math> is in <RT·math>S</RT·math>. As <RT·math>\dot{R}</RT·math> is definitively not a member of <RT·math>S</RT·math>, the first term of the set comprehension rule, <RT·math>x ∈ S</RT·math>, is false, and the paradox vanishes.
+ In 1908 Ernst Zermelo published an alternative system designed to avoid the known paradoxical statements of the time, even though absolute consistency remained unproven. In Zermelo's set theory, a mathematician first starts with an existing set, and then applies the Axiom of Separation using definite properties to partition out subsets<RT·endnote>Ernst Zermelo, "Untersuchungen über die Grundlagen der Mengenlehre I," <em>Mathematische Annalen</em> 65 (1908): 261–281.</RT·endnote>.
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- A person might suggest defining <RT·math>S</RT·math> as the set of all definable mathematical objects, forming a universal set. However, if such a universal set <RT·math>S</RT·math> existed, the Axiom of Separation could be applied as per the proof in the prior paragraph to show <RT·math>\dot{R}</RT·math> is not in <RT·math>S</RT·math>. However, as <RT·math>\dot{R}</RT·math> is a valid, definable set, it must reside within <RT·math>S</RT·math> by the very definition of a universal set. This contradicts the premise that <RT·math>S</RT·math> contains everything. Therefore, within any system governed by the Axiom of Separation, a universal set cannot exist.
+ To see how this works, begin with the unrestricted case. A logician is permitted to define the predicate <RT·math>P(x)</RT·math> as <RT·math>x ∉ x</RT·math>. Then <RT·math>\{x \mid P(x)\}</RT·math> produces Russell's paradox, so the set fails to be defined. This is inconsistent with the founding assumption that any predicate would work, so it is a problem. In contrast, consider the same condition, though restricted by Zermelo's Axiom of Separation over a predefined set <RT·math>S</RT·math>, written as <RT·math>\dot{R} = \{x \mid x ∈ S ∧ x ∉ x\}</RT·math>. Now the predicate has two terms. The only thing a person needs to know about <RT·math>S</RT·math> here is that it has already been successfully defined; we don't need to know what that definition is. Now assume <RT·math>\dot{R}</RT·math> is in <RT·math>S</RT·math>. That gives the second term of the condition authority, which enables the familiar fatal loop: if <RT·math>\dot{R}</RT·math> is in <RT·math>\dot{R}</RT·math>, it shouldn't be, and if it isn't, it should be. And thus it is clear that the initial assumption, that <RT·math>\dot{R}</RT·math> is in <RT·math>S</RT·math>, must be wrong, and <RT·math>\dot{R}</RT·math> is not in <RT·math>S</RT·math>. Authority returns to the first term, <RT·math>x ∈ S</RT·math>, which is false, so the condition is false. No contradiction follows, and the paradox vanishes.
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- The authority to remove Russell's Paradox set formulation comes from the set <RT·math>S</RT·math>. If we know its definition, then the authority comes through that definition. However, if we merely stipulate that <RT·math>S</RT·math> must be defined, then we are expressing our authority through <RT·math>S</RT·math> by declaring, "Undefined sets are not allowed." In the explanation above, it is only after discovering a set is undefined that we conclude it is not a member of <RT·math>S</RT·math>. I sometimes wonder how mathematics might have evolved had Frege simply taken that approach. We take this question up again in chapter <RT·Counter·read snapshot="Section·computational-naturalism"></RT·Counter·read>, Computational Naturalism, and discover there is a deeper issue.
+ A person might suggest defining <RT·math>S</RT·math> as the set of all definable mathematical objects, forming a universal set that can be used in any set formulation by this method. However, if such a universal set <RT·math>S</RT·math> existed, the Axiom of Separation could be applied as per the proof in the prior paragraph to show <RT·math>\dot{R}</RT·math> is not in <RT·math>S</RT·math>. That contradicts the premise that <RT·math>S</RT·math> contains everything. Therefore, within any system governed by the Axiom of Separation, a universal set cannot exist<RT·endnote>ibid</RT·endnote>.
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+ The authority to remove Russell's Paradox set formulation comes from the set <RT·math>S</RT·math>. If we know its definition, then the authority comes through that definition. However, if we merely stipulate that <RT·math>S</RT·math> must be defined, then we are expressing our authority through <RT·math>S</RT·math> by declaring, "Undefined sets are not allowed." In the explanation above, it is only after discovering a set is undefined that we conclude it is not a member of <RT·math>S</RT·math><RT·endnote>ibid</RT·endnote>. I sometimes wonder how mathematics might have evolved had Frege simply taken that approach. We take this question up again in chapter <RT·Counter·read snapshot="Section·computational-naturalism"></RT·Counter·read>, Computational Naturalism, and discover there is a deeper issue.
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