Another Infinity 'Discovered'

It's always awesome when the math ideas we read about pop up weeks, months or even years later in interesting new ways.  Take infinity, for example:

First there was her hand-made dolly Amelia writing an essay about 'what infinity means to me.'

Then there was her recent theory of multiple universes which was buoyed by set theory and the illustration of an 'infinite number of infinities of different sizes.'

And, this morning, there was this:

Her: Mama! I discovered another infinity!  Half of two is one.  Half of one is one-half.  Half of one-half is one-half of one-half.  Half of one-half of one-half is half of...

Me: Hey, I totally get it, but you wanna hear an easier way to say it?

Her: Yeah.

Me: What's half of two?

Her: One!

Me: What's half of one?

Her: One half.

Me: What's half of one half?

Her: I don't know.

Me: Ummm...you know how when we bake muffins?

Her: Yeah.

Me: Well, when we use the measuring spoons...[I talk a little more but it becomes obvious she's not getting it and I'm not equipped at 6:30 in the morning to think clearly. I abort the mission]...okay, here's the easier way to say it: Half of one is one-half.  Half of one-half is one-fourth.  Half of one-fourth is one-eighth, half of one-eight is one-sixteenth...

Her: So it gets smaller as it gets bigger!

What's remarkable to me about this particular conceptualization is that one, it mirrors thinking of done thousands of years ago by philosophers -- but she has never heard those paradoxes.  The second thing is that, obviously, she's not done much with fractions and yet, the idea of half-ness and size is fully there.

And all this before 7:00am.  Any time is math time, right?