Banach-Tarski is a consequence not just of infinities but also uses non-measurable sets, which depend on the axiom of choice. You can just limit or reject the AoC to prevent it. There are weaker versions of the AoC that don’t result in Banach-Tarski.
I actually think Banach-Tarski paradox could actually be done if Conservation of Energy is allowed to be removed. The real issue is that conservation of energy prevents you from making new energy, which is likely key to why Banach-Tarski’s works.
Thanks for clarifying. I find the axiom of infinity much more unintuitive than the axiom of choice. So, for me, rejecting Banach-Tarski implies rejecting the axiom of infinity.
Also note that Banach-Tarski uses a different kind of (and larger) infinity, specifically the continuum (real numbers) to model objects. If the universe is discrete but infinite in extent, then you can still avoid Banach-Tarski.
Thanks for asking. Not sure I have much more to say, but:
I find the concept of infinity quite unintuitive because in everyday life the size of the whole is not equal to the size of each of its parts.
As any non-null number multiplied by infinity equals infinity, 10^-10^10^10^10^10^10^10^10^10 of infinity is exactly equal to infinity.
This applying to something in the real world seems non-sensical to me.
According to this page of Wikipedia, the axiom of choice is equivalent to the following. “Given any family of nonempty sets, their Cartesian product is a nonempty set”.
In the same way that we cannot arrive to null energy from positive energy (since that would break conservation), we cannot arrive to empty sets multiplying non-empty sets.
That being said, my intuitions with respect to the axiom of choice are more or less agnostic.
Banach-Tarski is a consequence not just of infinities but also uses non-measurable sets, which depend on the axiom of choice. You can just limit or reject the AoC to prevent it. There are weaker versions of the AoC that don’t result in Banach-Tarski.
I actually think Banach-Tarski paradox could actually be done if Conservation of Energy is allowed to be removed. The real issue is that conservation of energy prevents you from making new energy, which is likely key to why Banach-Tarski’s works.
Interesting. I also find lack of conservation of energy quite unintuitive, so it looks my intuitions are internally consistent if that is the case.
Thanks for clarifying. I find the axiom of infinity much more unintuitive than the axiom of choice. So, for me, rejecting Banach-Tarski implies rejecting the axiom of infinity.
The axiom of choice is already true in ZF for finite families of sets without the full axiom of choice: https://mathoverflow.net/questions/32538/finite-axiom-of-choice-how-do-you-prove-it-from-just-zf
Confusingly (to me, at least; I haven’t spent the time to understand this), AoC and Banach-Tarski are also true generally in the constructible universe, which is nice as a model of ZF, but reasonably defined sets are still measurable, so you don’t get the Banach-Tarski paradox if you only use them: https://math.stackexchange.com/questions/142499/are-sets-constructed-using-only-zf-measurable-using-zfc
Also note that Banach-Tarski uses a different kind of (and larger) infinity, specifically the continuum (real numbers) to model objects. If the universe is discrete but infinite in extent, then you can still avoid Banach-Tarski.
Thanks! I have replaced “They allow for the Banach–Tarski paradox” by “Some types of infinity allow for the Banach–Tarski paradox”.
Huh, this is interesting to me—can you go into a bit more detail here?
Hi Bruce,
Thanks for asking. Not sure I have much more to say, but:
I find the concept of infinity quite unintuitive because in everyday life the size of the whole is not equal to the size of each of its parts.
As any non-null number multiplied by infinity equals infinity, 10^-10^10^10^10^10^10^10^10^10 of infinity is exactly equal to infinity.
This applying to something in the real world seems non-sensical to me.
According to this page of Wikipedia, the axiom of choice is equivalent to the following. “Given any family of nonempty sets, their Cartesian product is a nonempty set”.
I find this decently intuitive due to similarities with conservation of energy.
In the same way that we cannot arrive to null energy from positive energy (since that would break conservation), we cannot arrive to empty sets multiplying non-empty sets.
That being said, my intuitions with respect to the axiom of choice are more or less agnostic.