Is a Von Neumann Universe possible in ZC without the Axiom of Choice?

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SUMMARY

The discussion centers on the impossibility of a Von Neumann Universe existing within ZC (Zermelo-Fraenkel set theory without the Axiom of Choice), specifically up to the ordinal \(\omega_2\). Participants reference a related question on Math Stack Exchange that addresses this topic. The consensus is that the absence of the Axiom of Choice in ZC fundamentally restricts the formation of such universes.

PREREQUISITES
  • Understanding of Zermelo-Fraenkel set theory (ZFC)
  • Familiarity with the Axiom of Choice
  • Knowledge of ordinal numbers, particularly \(\omega_2\)
  • Basic comprehension of set theory terminology and notation
NEXT STEPS
  • Research the implications of the Axiom of Choice in set theory
  • Study the structure and properties of ordinals, focusing on \(\omega_2\)
  • Explore the differences between ZC and ZFC set theories
  • Investigate related mathematical discussions on Math Stack Exchange regarding set theory
USEFUL FOR

Mathematicians, logicians, and students of set theory seeking to deepen their understanding of the limitations imposed by the Axiom of Choice and the structure of Von Neumann Universes.

Garrulo
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Why in ZC (ZFC-reemplacement+separation) can´t exist Von Neumann Universet not even till \omega2. Sorry, I am new in the forum and I dont´know use Latex
 
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Thanks. Yeah, it was my question
 

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