Finite number divided by infinity

[Dale]

I think you are confusing “computable” with “physical”. Physical locations are isomorphic to the reals (or at least we have no evidence to the contrary). So picking a physical position is isomorphic to picking a real number, even if you cannot write it down or enter it into a computer exactly.

 

[PeroK]

I think you are confusing “computable” with “physical”. Physical locations are isomorphic to the reals (or at least we have no evidence to the contrary). So picking a physical position is isomorphic to picking a real number, even if you cannot write it down or enter it into a computer exactly.

How do you choose a point in the continuum? QM explicitly avoids localising a point particle precisely (even if a particle can be a point). You cannot wave your arms and say: that particle is precisely at location $(x, y, z)$.

The mathematical model involves a continuum, but you cannot choose a point in the continuum, except mathematically.

 

[A.T.]

You can't "choose" one in the sense that the zero probability of doing so has any real meaning.

Going back to the original topic, one can choose a planet, and there might be infinitely many planets.

 

[kuruman]

Y'all missed the point of "Hitchhiker's Guide." Clearly, $$\text{lim}_{x\rightarrow\infty} \frac{1}{x}=42.$$

 

[PeroK]

Going back to the original topic, one can choose a planet, and there might be infinitely many planets.

Yes, there is a distribution on the natural numbers where, for example: $p(1) = 1$ and $p(n) = 0, \ n \ne 1$.

Or, where: $\forall n: \ p(n) = \dfrac{1}{2^n}$.

 

[PeroK]

I think you are confusing “computable” with “physical”. Physical locations are isomorphic to the reals (or at least we have no evidence to the contrary). So picking a physical position is isomorphic to picking a real number, even if you cannot write it down or enter it into a computer exactly.

PS if you can do this, then any theory that is limited to a finite number of spacetime events, say, has been disproven. If you can genuinely map spacetime isomorphically to a continuum, then discrete spacetime theories are, as fundamental theories, impossible.

 

[DaveC426913]

Mathematically, we can have a uniform distribution on the interval $[0, 1]$, where the probability density is uniform. In reality, however, it's not possible to select from more than a finite number of outcomes with equal probability.

The paradox "something with zero probability can happen" occurs by misapplying mathematics. Or, at least, imagining that there is a scenario where you can apply that mathematics. It's impossible to select real numbers uniformly. More explicitly, the positive integers cannot be selected uniformly.

Another example is Hilbert's Hotel. That's valid mathematics and a good example of the behaviour of infinite sets; but, we cannot build a hotel with an infinite number of rooms.

The Internet is awash with the fallacy that these things could be physically realised, rather than being purely mathematical constructions.

Sure but who said anything about physically realizing them as a requirement ?

Can any scenario that contains infinities be physically realized?