I was talking to a friend recently about the nature of infinity and Georg Cantor's Continuum Hypothesis and I recanted a statement from a Youtube video I watched:(adsbygoogle = window.adsbygoogle || []).push({});

"any interval of the continuum, now matter how small, has the same size as the contiuum itself"

from the 6:54 mark...

https://www.youtube.com/watch?v=<object width="480" height="385"><param name="movie" value="http://www.youtube.com/v/WihXin5Oxq8&hl=en_US&fs=1&"></param><param [Broken] name="allowFullScreen" value="true"></param><param name="allowscriptaccess" value="always"></param><embed src="http://www.youtube.com/v/WihXin5Oxq8&hl=en_US&fs=1&" type="application/x-shockwave-flash" allowscriptaccess="always" allowfullscreen="true" width="480" height="385"></embed></object>

The mapping of the interval from A to B onto the continuum (through 1-to-1 correspondence pairing of elements) was done by first "wrapping up" the interval into a semi-circle and then pairing.

Now my friend thinks this is cheating. That once you bend the interval line (change the geometry) you have effectively changed the property of the line and therefor can't relate it to the continuum.

How do I explain to him that this isn't cheating?

Thanks in advance!

For the record, I'm not a mathematician so please forgive my ignorance :(

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# How do I explain this?

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