Are There More Real Numbers Than Rational Numbers Without Complex Set Theory?

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The discussion centers on proving that there are more real numbers than rational numbers without relying heavily on set theory. Participants note that while the rationals are countable, the reals are uncountable, and seek simpler proofs than Cantor's diagonal argument. There is a desire for a more intuitive or straightforward approach to demonstrate this difference in cardinality. The conversation highlights the challenge of finding such a proof while acknowledging the established mathematical consensus. Ultimately, the quest for an alternative proof remains a topic of interest.
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I am trying to figure out a way to prove this without using much set theory (i know that the rationals are countable and the reals are not). is there a way to show that there are more reals than rationals in a more straightforward proof?
 
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