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The Cheese Paradox

What follows is a simple proof that the universe is entirely filled with cheese. Can you find the error(s)?

Claim: All points of space-time are simultaneously occupied by a humongous block of cheese.

Proof:

Let FST(x) denote the amount of space-time that is not occupied by an object x. This function evaluates to a positive real number or to some form of infinity, in units of meter-cubed-seconds (m3s). This function can be thought of as being equivalent to the size of the universe minus the size of the object x, though it need not be defined that way (in fact, it should not be defined that way, since it would be undefined if either the size of the universe or the object were infinite).

Let c be the largest conceivable block of cheese i.e. with FST(c) being minimal. In other words, there is no conceivable block of cheese c' such that FST(c') < FST(c). We can imagine a block of cheese that leaves all but a finite amount of space-time unoccupied. Clearly, then, FST(c) is finite, since all positive infinities are larger than all positive finite reals. So there exists at most a finite amount of space-time that is not occupied by this block of cheese, c.

Further, every point in space-time must be occupied by c, since if it were not, we could conceive of a block of cheese c' with FST(c') < FST(c) by placing cheese in the space not occupied by c and then melting it into c. But there is no such block of cheese c' for which FST(c') < FST(c), so this is a contradiction, and therefore it must be the case that every point in space-time is occupied by c. So FST(c) = 0.

So far it has been shown that there is a conceivable, or theoretical, block of cheese that occupies every point of space-time. But this object must in fact be real, since if it did not exist in the real world, our universe, there would be some points in space-time which it does not occupy, so FST(c) > 0. But FST(c) = 0, so that would be a contradiction. Therefore c really exists in our physical world.

Therefore, every point in space-time, including those in the real world, are occupied by cheese. So, in the past, present, and future, every point of space, including the space of which our universe consists, has been, is, and forever will be occupied by cheese.

Open Question: While the existence of this block of cheese has been deductively established, it remains unknown what kind of cheese it is. Is it cheddar, gouda, or havarti? Or is it a mix of different kinds of cheese?

Proposed Resolutions:

Think you know what's wrong? Send me an email or tweet me and I'll add your response here.

Here's one response from ctamblyn. I haven't had time to check it, but it sounds right:

Here’s my take on it:

In the first three paragraphs, the function *FST* was clearly meant to
denote something which maps objects *whether imagined or real* onto
space-time volumes.  The mere fact that we can compute *FST*(*x*) for some
*x* does not require that *x* exists.  Thus, the step in the second
sentence of the fourth paragraph that goes “if it did not exist in the real
world … *FST*(*c*) > 0” is invalid, and the attempted *reductio ad
absurdum*falls apart.

Alternatively, if *FST* was intended only to apply to actually existing
objects, the first paragraph is already assuming the conclusion, namely
that *c* exists, and the argument is circular.