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Replies: 26   Last Post: May 11, 2011 6:43 PM

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 Jack Markan Posts: 6,964 Registered: 2/26/05
Posted: May 8, 2011 5:26 PM

On May 7, 9:05 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote:

> 1. There is no surjection from N to R.
> 2. There is no injection from R to N.
>
> Cantor's theorem proves 1. Using classical logic,
> we can show that 1 implies 2. But is 2 provable
> constructively?

Toward a contradiction, let f be an injection from R into N.

Define g, a function from N onto R, as follows:

If k in the range of g, then g(k) = inverse_f(k).
If k not in the range of g, then let g(k) = the real number 0.

Oops, yeah, we used excluded middle to conclude that the domain of g
is N.

I see your point, I think.

MoeBlee

Date Subject Author
5/7/11 Daryl McCullough
5/7/11 W. Dale Hall
5/7/11 Jack Markan
5/7/11 Daryl McCullough
5/7/11 W. Dale Hall
5/8/11 tommyrjensen@gmail.com
5/8/11 Jack Markan
5/8/11 W. Dale Hall
5/9/11 Jack Markan
5/8/11 Daryl McCullough
5/9/11 Daryl McCullough
5/9/11 Jack Markan
5/7/11 David Yen
5/7/11 David Yen
5/8/11 Jack Markan
5/9/11 Bill Taylor
5/9/11 Daryl McCullough
5/10/11 Daryl McCullough
5/9/11 Michael Stemper
5/9/11 Daryl McCullough
5/11/11 Michael Stemper
5/11/11 Daryl McCullough
5/11/11 hagman
5/11/11 Daryl McCullough
5/11/11 Daryl McCullough
5/9/11 David Yen
5/9/11 David Yen