Quick question about cardinals

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SUMMARY

The discussion centers on defining uncountable sets of cardinal numbers, specifically questioning the assertion that all cardinals are countable. It establishes that an uncountable ordinal, denoted as w, exists and is not in bijection with any countable initial segment. This implies that w defines an uncountable cardinal, challenging the notion that all cardinals are countable. The conversation emphasizes the importance of understanding the relationship between ordinals and cardinality in set theory.

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  • Understanding of cardinal and ordinal numbers in set theory
  • Familiarity with bijection concepts
  • Knowledge of uncountable sets and their properties
  • Basic principles of mathematical proofs
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  • Study the properties of uncountable ordinals in set theory
  • Learn about the concept of bijections and their implications for cardinality
  • Explore proofs related to the countability of cardinals
  • Investigate the axioms of set theory that govern cardinal and ordinal relationships
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Mathematicians, students of set theory, and anyone interested in the foundations of mathematics and the properties of cardinality.

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Assuming we define the cardinal number for a set A as the least ordinal number b such that A and b are equipollent, how would you define an uncountable set of cardinal numbers?
 
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It's just an uncountable set. That it happens to be a set of cardinals is immaterial.
 
What I mean is that under this model every set of cardinals "up to x" seems to be countable. So how to define an uncountable one?
 
How does that imply that all cardinals are countable? Why not post a proof of that statement if it 'seems' to be so. Hint, let w be an uncountable ordinal. Such exist. It is not in bijection with any countable initial segment, so it must define an uncountable cardinal too.
 
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