Assuming a >0 value density throughout the universe, then an infinite space will contain infinite value as no matter how small you make the value density, the universe is infinitely large, so no matter how much value you want, you can just look further out and consider a larger space that will contain as much value as needed, ad infinitum.
The >0 value density assumption is the important part, and basically means that “for each X-large expansion of the space under consideration, there exists at least one means of generating even infinitesimally small value from using this space as opposed to not using it, and hence you can arbitrarily expand how much of the universe you want to take into account in order to obtain infinite value (assuming you actually do use these spaces)”.
However, to fully answer the question: It is not necessarily so, but if it does have both infinite value and infinite space, then as I mention in the grandparent it is fully plausible to increase local value without decreasing value anywhere at all and without changing the total value (by property of infinity).
Side-note:
I guess the grandparent was poorly phrased in that regards. I can indeed conceive of various kinds of infinite-space universes without infinite value (for various definitions of “value” or “infinite value”).
For instance, if you add the somewhat-contrived possible condition that it is impossible to configure a mind that will not evaluate with diminishing returns for astronomically high amounts of value until the returns reach zero at some factor correlating with the amount of nodes/matter/whatever of which the mind is made of, then it becomes obvious that you would need a mind infinitely big in order for it to gain infinite value from an infinite universe—something that will never, ever be built within finite time without breaking a bunch of other implicit rules.
Of course, there could-in-principle already exist such a mind, if the mind only occupies space in one direction towards infinity, but humans are unlikely to ever come into contact with such a mind, let alone integrate themselves into it, within finite time, so humans would not benefit from this infinite value being.
These things are fun to think about, but seem to have little value beyond that.
Assuming a >0 value density throughout the universe, then an infinite space will contain infinite value as no matter how small you make the value density, the universe is infinitely large, so no matter how much value you want, you can just look further out and consider a larger space that will contain as much value as needed, ad infinitum.
You are simply wrong about the math there. I can construct an infinite sequence of terms >0 which sum to a finite number.
sum_{nrightarrowinfty}{n=1}1/2n=1
If you meant that there is some sigma, such that every X-large portion of the universe had value of at least sigma, you would be technically correct. You are already setting a lower bound on value, which precludes the possibility of there being an x-large area of net negative value.
Your math looks wrong. The sum from 1 to infinity of 1/n does not converge, and as a simple visualization 1 + 1⁄2 + 1⁄3 + 1⁄4 is already greater than 2.
Assuming a >0 value density throughout the universe, then an infinite space will contain infinite value as no matter how small you make the value density, the universe is infinitely large, so no matter how much value you want, you can just look further out and consider a larger space that will contain as much value as needed, ad infinitum.
The >0 value density assumption is the important part, and basically means that “for each X-large expansion of the space under consideration, there exists at least one means of generating even infinitesimally small value from using this space as opposed to not using it, and hence you can arbitrarily expand how much of the universe you want to take into account in order to obtain infinite value (assuming you actually do use these spaces)”.
However, to fully answer the question: It is not necessarily so, but if it does have both infinite value and infinite space, then as I mention in the grandparent it is fully plausible to increase local value without decreasing value anywhere at all and without changing the total value (by property of infinity).
Side-note:
I guess the grandparent was poorly phrased in that regards. I can indeed conceive of various kinds of infinite-space universes without infinite value (for various definitions of “value” or “infinite value”).
For instance, if you add the somewhat-contrived possible condition that it is impossible to configure a mind that will not evaluate with diminishing returns for astronomically high amounts of value until the returns reach zero at some factor correlating with the amount of nodes/matter/whatever of which the mind is made of, then it becomes obvious that you would need a mind infinitely big in order for it to gain infinite value from an infinite universe—something that will never, ever be built within finite time without breaking a bunch of other implicit rules.
Of course, there could-in-principle already exist such a mind, if the mind only occupies space in one direction towards infinity, but humans are unlikely to ever come into contact with such a mind, let alone integrate themselves into it, within finite time, so humans would not benefit from this infinite value being.
These things are fun to think about, but seem to have little value beyond that.
You are simply wrong about the math there. I can construct an infinite sequence of terms >0 which sum to a finite number.
sum_{nrightarrowinfty}{n=1}1/2n=1
If you meant that there is some sigma, such that every X-large portion of the universe had value of at least sigma, you would be technically correct. You are already setting a lower bound on value, which precludes the possibility of there being an x-large area of net negative value.
EDIT: Corrected from 1/n to 1/2^n
Your math looks wrong. The sum from 1 to infinity of 1/n does not converge, and as a simple visualization 1 + 1⁄2 + 1⁄3 + 1⁄4 is already greater than 2.
Brainfart: My math was wrong. Corrected to 1/n^2
That’s 1/2+1/4+1/8… or Zeno’s sum.