Simmons' Paradox

Keith Simmons told this story about ten years ago:

Suppose I’ve just passed by a colleague’s office, and I see denoting phrases on the board there. That puts me in the mood to write denoting phrases of my own, and so I enter an adjacent room, and write on the board the following expressions:

......pi
......six
......the sum of the numbers denoted by expressions on the board in room 213.

Now I am in fact in room 213, though I believe that room 213 is my colleague's office. I set you the task of providing the denotations of these expressions.

The question is, what is X when X = pi + 6 + X, and the obvious answer is “infinity”. More precise possibilities include omega, aleph-null, beth-null, aleph-one et cetera; and because there are lots of possible answers, the “the” at the start of the third expression is a little deceptive. It is a bit like asking for the name of the king of France: the obvious reply is “which king of France?”
......What if “numbers” (in that expression) was replaced with “finite numbers” though? If the third expression with “finite” added does not denote anything (much as the expression “the present king of France” denotes nobody) then the sum of the numbers denoted by expressions on the board would be pi + 6, so the third expression should then denote pi + 6, whence it should denote pi + 6 + pi + 6, and so forth. If the third expression denotes anything, then it denotes something else, and if it denotes nothing then it denotes something. We seem to be stuck with a logical impossibility if the third expression is a denoting expression. Should we conclude that it is not a denoting expression?
......The problem is that the third expression is obviously a denoting expression. The light on the horizon is that reference is in general a matter of degree. To why, it may help to imagine a man staggering through a desert and seeing a mirage, which he takes to be a pool. Coincidentally there is a pool just where he takes one to be, but it is obscured from his view by the mirage. As he staggers on he is constantly thinking “that pool looks cool” rather obsessively. And as he nears the pool its image gradually replaces the illusory one without him noticing. So, the referent of “that pool” gradually changes to the pool, and so he will at some point have referred to the pool only as much as not.
......What did “that pool” denote when the man was referring to the pool only as much as not? I would say that it denoted the pool, but only vaguely. And similarly, the third expression with “finite” refers us to pi + 6 insofar as it does not, so it refers us to pi + 6 as much as not. It denotes pi + 6, but only vaguely.