Is it ever really Pi Time?

Walking with my son at 3:14pm the other day, I mentioned to him, “Hey, it’s Pi Time”. My son knows 35 digits of $\pi$ (don’t ask), and knows that it’s transcendental. He replied, “is it exactly $\pi$ time?”

This led to a discussion about whether there is ever a time each afternoon that is exactly $\pi$, meaning 3:14:15.926535…

This feels like some kind of Zeno’s Paradox. I told him that (assuming time is continuous) it had to be $\pi$ time at some point between 3:14:00 and 3:15:00, but the length of that moment was 0. However, this discussion left him confused.

Can anyone suggest a good way to explain this to a child?

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Pi time is not like a concrete slab on the sidewalk that you can stand on; it’s not even like the crack between the slabs, which have a width. It is like the line precisely down the middle of that crack. When you’re walking on the sidewalk, you cross right over it without stopping on it.

And so is every other precise time: like exactly noon or midnight.

Of course, the analogy fails a little bit because when you stop on the sidewalk, you cover a whole range of positions. As far as we can tell in everyday life, that’s not true of time… not that we can stop in time, anyway…

That is how I would explain it to a non-mathematician.