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by dcminter 1753 days ago
If I remember rightly there'a a Feynman anecdote where he points out that as the real universe is quantised this is a purely mathematical notion.

I used to riff with a friend that we were "the two members of the Banach-Tarski quartet." :)
2 comments
tsimionescu 1753 days ago
Current physical models don't have the universe itself (space-time) quantized - only matter is quantized. Even the planck time and planck length only represent minimal measurable distances/durations - the maths still assume that two things can be separated by fractional multiples of these.

That's not to say that physics requires infinities, but current models also don't disallow infinity.

Of course, actual infinity is outside the purview of science - there is no way to differentiate between infinity and something too big/small to measure, even in principle. Apparent paradoxes related to infinity, such as Banach-Tarski, don't change this, as they also require infinite precision to realize, making them impossible to test as well - even if a sphere is indeed made up of an infinity of space-time points, and even if we could manipulate those, we wouldn't be able, in finite time, to extract the necessary infinite subsets of points to create the two spheres from one.

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rob_c 1753 days ago
Physics doesn't require infinities, but it's scary how well QED/QFT approximates the g-factor (https://en.m.wikipedia.org/wiki/G-factor_(physics) ) for electrons given the amount of renormalization (cancelling of infinities) needed to estimate the true value in nature.
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dcminter 1753 days ago
I didn't remember quite correctly - here it is, from the section "A Different Box of Tools" in "Surely you're joking Mr Feynman". He doesn't state it explicitly, but I think it's clear they must have been talking about Banach-Tarski:

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...It often went like this: They would explain to me, "You've got an orange, OK? Now you cut the orange into a finite number of pieces, put it back together, and it's as big as the sun. True or false?"

"No holes?"

"No holes."

"Impossible! There ain't no such thing."

"Ha! We got him! Everybody gather around! It's So-and-so's theorem of immeasurable measure!"

Just when they think they've got me, I remind them, "But you said an orange! You can't cut the orange peel any thinner than the atoms."

"But we have the condition of continuity: We can keep on cutting!"

"No, you said an orange, so I assumed that you meant a real orange."

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