Paradox of the Frog

          Paradox of the Frog

This paradox unites the paradox of the boundary with the paradox of infinite parity.

It’s a sunny day at the frog pond, but a large tree is casting a dark shadow. A frog is sitting a foot into the shade. Feeling too cold, the frog jumps out of the shade and lands 1/2 foot into the bright sunlight. Feeling too warm, the frog jumps out of the sunlight and lands 1/4 foot into the shade. Feeling too cold, the frog jumps out of the shade and lands 1/8 foot into the bright sunlight. Feeling too warm, the frog jumps out of the sunlight and lands 1/16 foot into the shade. And so on; moreover, the frog’s jumps accelerate geometrically, so they are all done in finite time.

When the frog finally settles on the shade line, it has alternated between warm and cold infinitely many times. On the shade line, is the frog warm or cold?

If, like Baby Bear’s porridge, the frog is ‘just right’, then that’s a third value in addition to ‘warm’ or ‘cold’. The third value arises from an infinite alternation of the other two; so this is like asking if infinity is odd or even.

Now let us take into account the movement of the shadow. Upon reaching the shade line, the frog enters Samadhi; but then awakes an hour later to find the shade line shifted. It then makes another infinity of jumps, reaches the shade line again, and goes back to sleep. Then the shadow-line moves, and the process repeats.

By appropriately adjusting the frog’s times of waking and sleeping, we can make the frog go through any countable infinite ordinal sequence of jumps in the course of the day. Let the number of frog jumps by the end of the day to be a large countable infinite ordinal; large enough, say, to exceed any recursive ordinal naming scheme. All though that busy day the frog was warm, cold or asleep. At the moment of sunset, is the frog awake or asleep? And if awake, warm or cold?