Proving R is Bigger Than N Set Elements

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Discussion Overview

The discussion centers around the concept of cardinality, specifically addressing the claim that the set of real numbers \(\mathbb{R}\) has a greater cardinality than the set of natural numbers \(\mathbb{N}\). The scope includes theoretical aspects of set theory and the implications of Cantor's work on bijections.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant, Daniel, questions how to prove that the cardinality of \(\mathbb{R}\) is greater than that of \(\mathbb{N}\).
  • Another participant, George, states that two sets have the same cardinality if there exists a bijection between them and references Cantor's proof that no such bijection exists between \(\mathbb{R}\) and \(\mathbb{N}\).
  • Several participants express appreciation for the proof mentioned, indicating a positive reception to the explanation provided by George.

Areas of Agreement / Disagreement

While there is acknowledgment of Cantor's result regarding the lack of a bijection, the discussion does not reach a consensus on the implications or the understanding of the proof itself, as some participants express unfamiliarity with it.

Contextual Notes

The discussion does not delve into specific details of the proof or the assumptions underlying the concepts of cardinality and bijections, leaving some aspects unresolved.

Who May Find This Useful

This discussion may be of interest to those studying set theory, particularly concepts related to cardinality and the foundational work of Cantor.

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