Omega’s offers are unbounded; 10^^n exceeds any finite bound with a finite n. If the Hubble distance (edge of the observable universe) recedes, then even with a fixed quantity of mass-energy the quantity of storable data increases. You have more potential configurations.
Yes, in the hypothetical situation given; I can’t consistently assert anything else. In any “real” analogue there are many issues with the premises I’d take, and would likely merely take omega up a few times with the intend of gaining Omega-style ability.
I believe you are confused about what ‘bounded’ means. Possibly you are thinking of the Busy Beaver function, which is not bounded by any computable function; this does not mean it is not bounded, merely that we cannot compute the bound on a Turing machine.
Further, ‘unbounded’ does not mean ‘infinite’; it means ‘can be made arbitrarily large’. Omega, however, has not taken this procedure to infinity; he has made a finite number of offers, hence the final lifespan is finite. Don’t take the limit at infinity where it is not required!
Finally, you are mistaken about the effects of increasing the available space: Even in a globally flat spacetime, it requires energy to move particles apart; consequently there is a maximum volume available for information storage which depends on the total energy, not on the ‘size’ of the spacetime. Consider the case of two gravitationally-attracted particles with fixed energy. There is only one piece of information in this universe: You may express it as the distance between the particles, the kinetic energy of one particle; or the potential energy of the system; but the size of the universe does not matter.
No, I mean quite simply that there is no finite bound that holds for all n; if the universe were to collapse/rip in a finite time t, then Omega could only offer you the deal some fixed number of times. We seem to disagree about the how many times Omega would offer this deal—I read the OP as Omega being willing to offer it as many times as desired.
AFAIK (I’m only a mathematician), your example only holds if the total energy of the system is negative. In a more complicated universe, having a subset of the universe with positive total energy is not unreasonable, at which point it could be distributed arbitrarily over any flat spacetime. Consider a photon moving away from a black hole; if the universe gets larger the set of possible distances increases.
I think we are both confused on what “increasing the size of the Universe” means. Consider first a flat spacetime; there is no spatial limit—space coordinates may take any value. If you know the distance of the photon from the black hole (and the other masses influencing it), you know its energy, and vice-versa. Consequently the distance is not an independent variable. Knowing the initial energy of the system tells you how many states are available; all you can do is redistribute the energy between kinetic and potential. In this universe “increasing the size” is meaningless; you can already travel to infinity.
Now consider a closed spacetime (and your “only a mathematician” seems un-necessarily modest to me; this is an area of physics where I wish to tread carefully and consult with a mathematician whenever possible). Here the distance between photon and black hole is limited, because the universe “wraps around”; travel far enough and you come back to your starting point. It follows that some of the high-distance, low-energy states available in the flat case are not available here, and you can indeed increase the information by decreasing the curvature.
Now, a closed spacetime will collapse, the time to collapse depending on the curvature, so every time Omega makes you an offer, he’s giving you information about the shape of the Universe: It becomes flatter. This increases the number of states available at a given energy. But it cannot increase above the bound imposed by a completely flat spacetime! (I’m not sure what happens in an open Universe, but since it’ll rip apart in finite time I do not think we need to care.) So, yes, whenever Omega gives you a new offer he increases your estimate of the total information in the Universe (at fixed energy), but he cannot increase it without bound—your estimate should go asymptotically towards the flat-Universe limit.
With that said, I suppose Omega could offer, instead or additionally, to increase the information available by pumping in energy from outside the Universe, on some similarly increasing scale—in effect this tells me that the energy in the Universe, which I needed fixed to bound my utility function, is not in fact fixed. In that case I don’t know what to do. :-) But on the plus side, at least now Omega is breaking conservation of energy rather than merely giving me new information within known physics, so perhaps I’m entitled to consider the offers a bit less plausible?
I think we’re talking on slightly different terms. I was thinking of the Hubble radius, which in the limit equates to Open/Flat/Closed iff there is no cosmological constant (Dark energy). This does not seem to be the case. With a cosmological constant, the Hubble radius is relevant because of results on black hole entropy, which would limit the entropy content of a patch of the universe which had a finitely bounded Hubble radius. I was referring to the regression of the boundary as the “expansion of the universe”. The two work roughly similarly in cases where there is a cosmological constant.
I have no formal training in cosmology. In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation. In a flat universe, your causal boundary recedes at a constant c, and thus peak entropy in the patch containing your past light cone goes as t^2. It is not clear that there is a finite bound on the whole of a flat spacetime. I agree entirely on your closed/open comments.
Omega could alternatively assert that the majority of the universe is open with a negative cosmological constant, which would be both stable and have the energy in your cosmological horizon unbounded by any constant.
In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation.
No; the energy is quantized and finite, which disallows some distance-basis states.
But in any case, it does seem that the physical constraint on maximum fun does not apply to Omega, so I must concede that this doesn’t repair the paradox.
Omega’s offers are unbounded; 10^^n exceeds any finite bound with a finite n. If the Hubble distance (edge of the observable universe) recedes, then even with a fixed quantity of mass-energy the quantity of storable data increases. You have more potential configurations.
Yes, in the hypothetical situation given; I can’t consistently assert anything else. In any “real” analogue there are many issues with the premises I’d take, and would likely merely take omega up a few times with the intend of gaining Omega-style ability.
I believe you are confused about what ‘bounded’ means. Possibly you are thinking of the Busy Beaver function, which is not bounded by any computable function; this does not mean it is not bounded, merely that we cannot compute the bound on a Turing machine.
Further, ‘unbounded’ does not mean ‘infinite’; it means ‘can be made arbitrarily large’. Omega, however, has not taken this procedure to infinity; he has made a finite number of offers, hence the final lifespan is finite. Don’t take the limit at infinity where it is not required!
Finally, you are mistaken about the effects of increasing the available space: Even in a globally flat spacetime, it requires energy to move particles apart; consequently there is a maximum volume available for information storage which depends on the total energy, not on the ‘size’ of the spacetime. Consider the case of two gravitationally-attracted particles with fixed energy. There is only one piece of information in this universe: You may express it as the distance between the particles, the kinetic energy of one particle; or the potential energy of the system; but the size of the universe does not matter.
No, I mean quite simply that there is no finite bound that holds for all n; if the universe were to collapse/rip in a finite time t, then Omega could only offer you the deal some fixed number of times. We seem to disagree about the how many times Omega would offer this deal—I read the OP as Omega being willing to offer it as many times as desired.
AFAIK (I’m only a mathematician), your example only holds if the total energy of the system is negative. In a more complicated universe, having a subset of the universe with positive total energy is not unreasonable, at which point it could be distributed arbitrarily over any flat spacetime. Consider a photon moving away from a black hole; if the universe gets larger the set of possible distances increases.
I think we are both confused on what “increasing the size of the Universe” means. Consider first a flat spacetime; there is no spatial limit—space coordinates may take any value. If you know the distance of the photon from the black hole (and the other masses influencing it), you know its energy, and vice-versa. Consequently the distance is not an independent variable. Knowing the initial energy of the system tells you how many states are available; all you can do is redistribute the energy between kinetic and potential. In this universe “increasing the size” is meaningless; you can already travel to infinity.
Now consider a closed spacetime (and your “only a mathematician” seems un-necessarily modest to me; this is an area of physics where I wish to tread carefully and consult with a mathematician whenever possible). Here the distance between photon and black hole is limited, because the universe “wraps around”; travel far enough and you come back to your starting point. It follows that some of the high-distance, low-energy states available in the flat case are not available here, and you can indeed increase the information by decreasing the curvature.
Now, a closed spacetime will collapse, the time to collapse depending on the curvature, so every time Omega makes you an offer, he’s giving you information about the shape of the Universe: It becomes flatter. This increases the number of states available at a given energy. But it cannot increase above the bound imposed by a completely flat spacetime! (I’m not sure what happens in an open Universe, but since it’ll rip apart in finite time I do not think we need to care.) So, yes, whenever Omega gives you a new offer he increases your estimate of the total information in the Universe (at fixed energy), but he cannot increase it without bound—your estimate should go asymptotically towards the flat-Universe limit.
With that said, I suppose Omega could offer, instead or additionally, to increase the information available by pumping in energy from outside the Universe, on some similarly increasing scale—in effect this tells me that the energy in the Universe, which I needed fixed to bound my utility function, is not in fact fixed. In that case I don’t know what to do. :-) But on the plus side, at least now Omega is breaking conservation of energy rather than merely giving me new information within known physics, so perhaps I’m entitled to consider the offers a bit less plausible?
I think we’re talking on slightly different terms. I was thinking of the Hubble radius, which in the limit equates to Open/Flat/Closed iff there is no cosmological constant (Dark energy). This does not seem to be the case. With a cosmological constant, the Hubble radius is relevant because of results on black hole entropy, which would limit the entropy content of a patch of the universe which had a finitely bounded Hubble radius. I was referring to the regression of the boundary as the “expansion of the universe”. The two work roughly similarly in cases where there is a cosmological constant.
I have no formal training in cosmology. In a flat spacetime as you suggest, the number of potential states seems infinite; you have an infinite maximum distance and can have any multiple of the plank distance as a separation. In a flat universe, your causal boundary recedes at a constant c, and thus peak entropy in the patch containing your past light cone goes as t^2. It is not clear that there is a finite bound on the whole of a flat spacetime. I agree entirely on your closed/open comments.
Omega could alternatively assert that the majority of the universe is open with a negative cosmological constant, which would be both stable and have the energy in your cosmological horizon unbounded by any constant.
As to attacking the premises; I entirely agree.
No; the energy is quantized and finite, which disallows some distance-basis states.
But in any case, it does seem that the physical constraint on maximum fun does not apply to Omega, so I must concede that this doesn’t repair the paradox.