Proving R is Bigger Than N Set Elements

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SUMMARY

The discussion centers on the mathematical proof that the cardinality of the set of real numbers, denoted as \mathbb{R}, is greater than that of the natural numbers, \mathbb{N}. Daniel references Cantor's theorem, which establishes that no bijection exists between these two sets, thereby confirming their differing cardinalities. George expresses appreciation for the clarity of the proof, indicating its simplicity once understood. This highlights the foundational concept of cardinality in set theory.

PREREQUISITES
  • Understanding of set theory concepts, particularly cardinality.
  • Familiarity with bijections and their role in comparing set sizes.
  • Knowledge of Cantor's theorem and its implications.
  • Basic mathematical notation, including symbols like \mathbb{R} and \mathbb{N}.
NEXT STEPS
  • Study Cantor's diagonal argument to grasp the proof of \mathbb{R} being uncountable.
  • Explore the concept of cardinality in more depth, focusing on different sizes of infinity.
  • Investigate other proofs of the uncountability of \mathbb{R}.
  • Learn about the implications of Cantor's theorem in modern mathematics.
USEFUL FOR

Mathematicians, students of mathematics, and anyone interested in set theory and the foundations of mathematics will benefit from this discussion.

dextercioby
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that the number of elements of \mathbb{R} (seen as a set, obviously) is bigger than the number of elements of \mathbb{N} ...? :bugeye:

Daniel.
 
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Two sets have the same cardinality iff there exists a bijection between the sets. Cantor showed that there is no bijection between \mathbb{R} and \mathbb{N}. A "[URL proof[/URL] of this involves a very simple idea - simple once one has seen it, but not until then.

Regards,
George
 
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Very nice. I've never seen that proof before.
 
Wow...that's good!
 

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