Tuesday, May 15, 2012

One for You, Two for Me

We're all familiar with the "one for you, two for me" expression.  It occurred to me that a paradox akin to the Tristram Shandy paradox is within our reach.  Imagine we share an infinite bag of cookies and I'm the one who is in charge of the distribution.  I give you one, then keep two, give you another one, then keep three, and so on out to infinity.  As time advances, my share of cookies will grow continuously larger than your own.  Now, this wouldn't be very fair of me, but it also provides the background of the paradox.

Suppose instead that we have been sharing the bag of cookies from eternity past.  For every minute that has already passed, you have received a cookie.  Given that the past is infinite in this hypothetical scenario, it follows that you have infinitely-many cookies corresponding to infinitely-many minutes.  Therefore, there is a one-to-one correspondence between your share of cookies and mine.  Yet, hasn't the number of cookies I'm keeping for myself been growing exponentially larger than the cookies I'm giving to you?

This is just another fun way of stating what we already know: that in set-theory, a part may be equal to its whole.  Infinite sets may be consistent within an abstract mathematical realm, but I see no reason to accept that they're part of nature/physical reality.

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