Set Containing Itself: X and {X}

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Discussion Overview

The discussion revolves around the properties of sets in set theory, particularly focusing on whether a set can contain itself or contain a set that contains itself. The scope includes theoretical considerations in set theory, specifically within the context of Zermelo-Fraenkel (ZF) set theory and variations thereof.

Discussion Character

  • Debate/contested
  • Technical explanation

Main Points Raised

  • Some participants assert that a set cannot contain itself, stating X \notin X, but question if \{X\} \in X is permissible.
  • Others clarify that neither scenario is allowed in ZF set theory, although there are alternative theories, such as ZF without the Axiom of Foundation, where these might be permitted.
  • One participant mentions that the set {{X},X} lacks a disjoint element, implying that such constructions are not valid.
  • Another participant questions the possibility of constructing such a set in ZF without the Axiom of Foundation.
  • It is suggested that constructing such a set would lead to a theorem of anti-foundation, although specific axioms of anti-foundation may allow for certain constructions.
  • A participant raises a point of confusion regarding the designation "X," suggesting it represents both an element and a set, which complicates the interpretation of \{X\} in X and relates to the axiom of regularity.

Areas of Agreement / Disagreement

Participants generally agree that traditional ZF set theory does not allow for sets containing themselves or sets that contain themselves. However, there is disagreement regarding the implications and possibilities in alternative set theories, particularly concerning the Axiom of Foundation and anti-foundation.

Contextual Notes

The discussion highlights limitations in understanding the implications of set designators and the axioms involved, particularly the Axiom of Foundation and the Axiom of Regularity, without resolving the underlying complexities.

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A set is not allowed to has itself as a member: [tex]X \notin X[/tex]. But I wonder if this is allowed: [tex]\{X\} \in X[/tex].
 
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Outlined said:
A set is not allowed to has itself as a member: [tex]X \notin X[/tex]. But I wonder if this is allowed: [tex]\{X\} \in X[/tex].

Neither are allowed in ZF. But there are theories in which both are allowed -- for example, ZF without the Axiom of Foundation.
 
It's not allowed since the set {{X},X} have no disjoint element of itself.
 
CRGreathouse said:
Neither are allowed in ZF. But there are theories in which both are allowed -- for example, ZF without the Axiom of Foundation.

Is it possible to construct such a set in ZF without Axiom of Foundation?
 
No -- such a construction would amount to a theorem of anti-foundation.

Specific axioms of anti-foundation may provide constructions, however.
 
The confusion here, I think, is in the designator "X", where this letters is trying to stand for both an element as well as a set. {X} in X means the element X as a set is found within the set, which violates the axiom of regularity, otherwise called "axiom of foundation", designed to prevent such expressions.
 

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