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Uncountable Infinitude, Illogically Concluded

Date: 11/21/2010 at 17:15:31
From: Boaz
Subject: Rational and Irrational Numbers

Regarding the question that I have seen here: Which set is bigger, the set
of rational or irrational numbers?

I understand both proofs: the countability of rationals and the
uncountability of the irrationals. However, between each two irrational
number we can always find a rational number. So -- to follow the same kind
of reasoning that says there are the same infinite number of odd and even
integers because between each two odd numbers there is an even number -- I
would expect that there will be the same number (infinite of course) of
rational and irrational.

What is the solution to this contradiction? Where is my mistake?

Thank you very much.

Boaz Castro

Date: 11/21/2010 at 22:42:45
From: Doctor Peterson
Subject: Re: Rational and Irrational Numbers

Hi, Boaz.

You seem to be assuming that because there is AT LEAST ONE rational
between any two irrationals, and vice versa, that there are THE SAME
NUMBER of each between the other.

There are in fact countably (infinitely) many rationals between any two
irrationals, but UNcountably many irrationals between any two rationals!

- Doctor Peterson, The Math Forum

Date: 11/22/2010 at 01:36:11
From: Boaz
Subject: Rational and Irrational Numbers

Dear Doctor Peterson,

Many thanks for your reply.

Between any two of your UNcountable irrationals, I can find a rational
number. Doesn't that mean that the rationals are also UNcountable?

I have seen some proofs regarding the countability of rationals, but they
were not convincing. Maybe I need a more accurate proof.... (actually, I
think I have two proofs, but they are probably not valid.)

I hope I managed to explain my difficulty correct.

Again, thanks for the prompt response!!!


Date: 11/22/2010 at 10:35:49
From: Doctor Peterson
Subject: Re: Rational and Irrational Numbers

Hi, Boaz.

You're assuming that if every member of a set A lies between two members
of an uncountable set B, then set A is also uncountable. Do you have a
proof of that?

An uncountable set is one that CAN'T be counted. All you've done is
essentially to show ONE way you could TRY to count the rationals and fail
-- namely, by putting them between irrationals and realizing that you
can't count those. That doesn't mean that another way does not exist. And
in fact, proofs do provide a way to count the rationals.

Infinities can't be trusted not to surprise you. You have to watch out for
hidden assumptions such as the kind you've made.

- Doctor Peterson, The Math Forum

Date: 11/25/2010 at 07:50:30
From: Boaz
Subject: Thank you (Rational and Irrational Numbers)

Many thanks.

I think I can sleep now :-) 

Associated Topics:
High School Logic
High School Number Theory

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