Add 2 Dense Sets for Non-Dense Set Result

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Discussion Overview

The discussion revolves around the properties of dense sets in the context of set theory and topology, particularly focusing on whether the union of two dense sets can result in a non-dense set. Participants explore definitions of dense sets, uncountability, and the nature of infinity, while also addressing misconceptions and clarifying concepts.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that the union of two dense sets cannot be non-dense, while others reference a book that claims otherwise.
  • There is a discussion about the definition of a dense set, with some participants asserting it relates to topology rather than countability.
  • Uncountability is discussed, with participants noting that uncountable sets cannot be put into a one-to-one correspondence with natural numbers, but there is some confusion about the implications of this definition.
  • Participants explore the nature of infinity, with one noting that the set of natural numbers is considered the smallest infinity.
  • There is a clarification that there are different sizes of infinity, and that the cardinality of the reals is strictly greater than that of the naturals.

Areas of Agreement / Disagreement

Participants express differing views on whether the union of two dense sets can be non-dense, with some asserting it cannot while others reference conflicting information from external sources. The discussion remains unresolved regarding the implications of certain definitions and the nature of dense sets.

Contextual Notes

Some definitions and concepts discussed, such as "density" and "uncountability," depend on specific mathematical contexts, which may lead to misunderstandings. The discussion also highlights the complexity of infinity and cardinality, which are not universally agreed upon.

Who May Find This Useful

This discussion may be of interest to students and enthusiasts of mathematics, particularly those studying set theory, topology, or the foundations of mathematics.

  • #31
cragar said:
Ok I see , I am very much enjoying this conversation .

I'm glad you find this forum informative! :biggrin:
 
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  • #32
You may want to read "The pea and the sun" by Wapner. It has some very informative things on infinity and it's paradoxes...
 
  • #33
thanks for the recommendation
 

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