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Inverse View
It is not the case that Zermelo-Fraenkel set theory blocks paradox via the axiom schema of separation without presupposing any notion of self-presupposing collections.
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Reasons For
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Reason for
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1.
Separation presupposes given sets to separate within, leaving unexplained how the initial universe of sets avoids self-presupposing circularity.
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2.
The cumulative hierarchy itself embodies a notion of stages that arguably presupposes implicit understanding of hierarchical self-limitation.
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3.
ZF avoids explicit self-reference but may merely displace rather than eliminate the conceptual problem underlying paradox formation.
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Reasons Against
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Reason against
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1.
Separation restricts set formation to subsets of existing sets, avoiding the unrestricted comprehension that generates Russell's paradox.
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2.
ZF requires no antecedent commitment to self-membering collections; it builds sets bottom-up from empty set using only restricted operations.
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3.
The axiom schema's restriction to definable properties prevents self-referential collection without invoking prior philosophical notions of totality.
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