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

Thread starter dmatador
Start date Aug 24, 2010

AI Thread Summary

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.

Aug 24, 2010

dmatador

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?

Mathematics news on Phys.org

Aug 24, 2010

LCKurtz

Science Advisor

Homework Helper

You don't like the Cantor diagonal proof?

http://en.wikipedia.org/wiki/Cantor's_diagonal_argument

Thread 'Non-orthogonal bases'

Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...

View full post »

Thread 'Fixing Things Which Can Go Wrong With Complex Numbers'

Greg tells me the feature to generate a new insight announcement is broken, so I am doing this: https://www.physicsforums.com/insights/fixing-things-which-can-go-wrong-with-complex-numbers/

View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

View full post »