The main idea here is that one can always derive a “greater” set (in terms of cardinality) from any given set, even if the given set is already infinite, because there are higher degrees of infinity. There is no greatest infinity, just like there is no largest number. So even if (hypothetically) a Being with infinite knowledge exists, there could be Beings with greater knowledge than that. No matter which god you choose, there could be one greater than that, meaning there are things the god you chose doesn’t know (and hence He isn’t “omniscient”, and therefore isn’t “God”, because this was a required attribute.)
I don’t know how to interpret “all existing objects”, because I don’t know what counts as an “object” in your definition. Set theory doesn’t require ur-objects (although those are known variations) and just starts with the empty set, meaning all “objects” are themselves sets. The powerset operation evaluates to the set of all subsets of a set. The powerset of a set always has greater cardinality than the set you started with. That is, for any given collection of “objects”, the number of possible groupings of those objects is always a greater number than the number of objects, even if the collection of objects you started with had an infinite number to begin with. So no, this doesn’t prove that an infinite universe cannot exist, just that there are degrees of infinities (and no “greatest” one).
Naiive set theory leads to paradoxes when defining self-referential sets. The idea of “infinite” gods seem to have similar problems. There are various ways to resolve this. The typical one used in foundations of mathematics is the notion of a collection that is too large to be a set, a “proper class”. (“Class” used to be synonymous with “set”.) But later on in the discussion it was pointed out that this isn’t the only possible resolution.
“An infinite universe can exist.” ″A greatest infinity cannot exist.” I think there is some kind of logical contradiction here. If the Universe exists and if it is infinite, then it must correspond to the concept of “the greatest infinity.” True, Bertrand Russell once expressed doubt that one can correctly reason about the “Universe as a whole.” I don’t know. It seems strange to me. As if we recognize the existence of individual things, but not of all things as a whole. It seems like some kind of arbitrary crutch, a private “ad hoc” solution, conditioned by the weakness of our brain. As for God or Gods, then, hypothetically, in the case of the coincidence of their value systems and the mental interaction between them according to a common agreed protocol, these problems should not be very important.
I’m hearing intuitions, not arguments here. Do you understand Cantor’s Diagonalization argument? This proves that the set of all integers is “smaller” (in a well-defined way) than the set of all real numbers, despite the set of all integers being already infinite in size. And it doesn’t end there. There is no largest set.
Russell’s paradox arises when a set definition refers to itself. For example, in a certain town, the barber is the one who shaves all those (and only those) who do not shave themselves. This seems to make sense its face. But who shaves the barber? Contradiction! Not all set definitions are valid, and this includes the universal one, which can be proved to not exist in many ways, at least in the usual ZFC (and similar).
There are two ways to construct a universal object. Either make it a non-set notion like a “proper class”, which can’t be an element of a set (and thus can’t contain itself or any other proper class), or restrict the axiom of comprehension in a way which results in a non-well-founded set theory. Cantor’s Theorem doesn’t hold for all sets in NF. The diagonal set argument can’t be constructed (in all cases) under its rules. NF has a universal set that contains itself, but it accomplishes this by restricting comprehension to stratified formulas. I’m not a set theorist, so I’m still not sure I understand this properly, but it looks like an infinite hierarchy of set types, each with its own universal set. Again, no end to the hierarchy, but in practice all the copies behave the same way. So instead of strictly two types of classes, the proper class and the small class, you have some kind of hyperset that can contain sets, but not other hypersets, and hyper-hypersets that can contain both, but not other hyper-hypersets, and so forth, ad infinitum.
Personally, I’m rather sympathetic to the ultrafinitists, and might be a finitist myself. I can accept the slope of a vertical line being “infinite” in the limit. That’s just an artifact of how we chose to measure something. Measure it differently, and the infinity disappears. I can also accept a potential infinity, like not having a largest integer, because the successor function can make a bigger one. We can make an abstract algorithm run on an abstract machine that can count, and it has a finite description. But taking the “completed” set of all integers as an object itself rubs me the wrong way. That had to be tacked on as a separate axiom. It’s unphysical. No operation could possibly construct a physical model of such a thing. It’s an impossible object. One could try to point to a pre-existing model, but we physically cannot verify it. It would take infinite time, space, or precision, which is again unphysical.
Similarly, there is no physical way to verify an infinite God exists, because we physically cannot distinguish it from a (sufficiently large, but) finite one. I might be willing to call such an alien a small-g “god”, but it’s not the big-G omni-everything one in valentinslepukhin’s definition. That only leaves some kind of a priori logical argument, because it can’t be an empirical one, but it has to be based on axioms I can accept, doesn’t it? I can entertain weird axioms for the sake of argument, but I’m not seeing one short of “God exists”, which is blatant question begging.
We can leave theology. It is not so important. I am more concerned with the questions of finitism and infinitism in relation to paradox of sets.
Finitism is logically consistent. However, it seems to me that it suffers from the same problem as the ontological proof of the existence of God. It is an attempt to make a global prediction about the nature of the Universe based on a small thought experiment. Predictions like “Time cannot be infinite”, “Space cannot be infinite” follow directly from finitism. It turns out that we make these predictions based on our mathematical problems with the paradox of sets. At the same time, the paradox of sets itself resembles the paradox “I’m telling a lie now”. and, it seems, should look for a solution somewhere in the same area. If we think off the cuff, it seems to me naively that the very concept of “ordinary set” is composed in such a way as to lead to paradoxes. This is the problem of the concept of “ordinary set”. This is not the problem of the existence/non-existence of physical infinity.
Oh, okay. I don’t really understand this topic. But as far as I know, not all mathematicians are finitists. So it seems that the proofs of finitism are not flawless.
On the other hand, how is the problem of the set paradox solved in cosmological infinitism? Something like “The Infinite Universe may exist, but it is forbidden to talk about it as an object”? Because any attempt to do so will bring you back to the set paradox, if you take it seriously. “Talk about any particular part of the Universe as much as you like, but don’t even think about the Universe as a whole”? This risks forming a somewhat patchwork model of the worldview. “It may exist, but you cannot think about it intelligently and rationally.” One is reminded of Zeno’s attempts to prove that one cannot think about motion without contradictions.
Finitism doesn’t reject the existence of any given natural number (although ultrafinitism might), nor the validity of the successor function (counting), nor even the notion of a “potential” infinity (like time), just the idea of a completed one being an object in its own right (which can be put into a set). The Axiom of Infinity doesn’t let you escape the notion of classes which can’t themselves be an element of a set. Set theory runs into paradoxes if we allow it. Is it such an invalid move to disallow the class of Naturals as an element of a set, when even ZFC must disallow the Surreals for similar reasons?
Before Cantor, all mathematicians were finitists. It’s not a weird position historically.
We do model physics with “real” numbers, but that doesn’t mean the underlying reality is infinite or even infinitely divisible. My finitism is motivated by my understanding of physics and cosmology, not the other way around. Nature seems to cut us off from any access to any completed infinity, and it’s not clear that even potential infinities are allowed (hence my sympathy with ultrafinitism). I have no need of that axiom.
Quantum Field Theory, though traditionally modeled using continuous mathematics, implies the Bekenstein bound: a finite region of space contains a finite amount of information. There are no “infinite bits” available to build the real numbers with. However densely you store information, eventually, at some point, your media collapses into a black hole, and packing in more must take up more space.
Physical space can’t be a continuum like the “reals”. It’s not infinitely divisible. Measuring distance with increasing precision requires higher frequency waves, and thus higher energies, which eventually has enough effective mass to gravitationally distort the very space you are measuring, eventually collapsing into a black hole.
Below a certain limit, distance isn’t physically meaningful. If you assume an electron is a point particle with “infinitesimal” size and you zoom in enough, you should be able to get arbitrarily high electric field strength. But at some point, high enough field strength results in vacuum polarization: virtual electron/positron pairs get pushed around and finally one of the positrons annihilates whatever you thought the real electron was, and then one of the virtual electrons doesn’t have anything to pair with and becomes the real one. It’s as if the electron is jumping around. You can’t nail it down. It doesn’t physically have a position down below a certain scale in time and space. There are no infinite bits. All the fundamental particle types are like this. There are no infinitesimal point particles. They’re just waves.
There’s also a cosmological horizon limiting how much of the Universe we can see. There’s also a (related) past temporal horizon at the Big Bang. We can’t see a completed past-temporal or spacial infinity, in any direction. We’re not sure of the Ultimate Fate of the Universe, but it looks like Heat Death is probably it, given our current understanding of physics. So there’s a future limit as well. The other likely candidate Fates are finite in time as well.
But even supposing finite information content in a finite region seems to be enough to make potential-infinite time not really meaningful. There’s a finite number of states possible, so eventually all reachable states are reached. If physics is deterministic (it seems to be), then we get into a cycle. So time is better modeled as a finite circle, rather than an infinite line. And if it’s not deterministic? Then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no phycial way to tell the difference.
Potential-infinite space is the same way. Any accessible region has a finite number of states, so at least some of them must repeat exactly in other regions. If there’s some determinism to the pattern, then it’s maybe better modeled as some curled-up finite space (although aperiodic tilings are also possible). If it’s random, then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no physical way to tell the difference. Once all reachable states have been saturated, why does it matter if they appear only once or a googol or infinity times?
The main idea here is that one can always derive a “greater” set (in terms of cardinality) from any given set, even if the given set is already infinite, because there are higher degrees of infinity. There is no greatest infinity, just like there is no largest number. So even if (hypothetically) a Being with infinite knowledge exists, there could be Beings with greater knowledge than that. No matter which god you choose, there could be one greater than that, meaning there are things the god you chose doesn’t know (and hence He isn’t “omniscient”, and therefore isn’t “God”, because this was a required attribute.)
I don’t know how to interpret “all existing objects”, because I don’t know what counts as an “object” in your definition. Set theory doesn’t require ur-objects (although those are known variations) and just starts with the empty set, meaning all “objects” are themselves sets. The powerset operation evaluates to the set of all subsets of a set. The powerset of a set always has greater cardinality than the set you started with. That is, for any given collection of “objects”, the number of possible groupings of those objects is always a greater number than the number of objects, even if the collection of objects you started with had an infinite number to begin with. So no, this doesn’t prove that an infinite universe cannot exist, just that there are degrees of infinities (and no “greatest” one).
Naiive set theory leads to paradoxes when defining self-referential sets. The idea of “infinite” gods seem to have similar problems. There are various ways to resolve this. The typical one used in foundations of mathematics is the notion of a collection that is too large to be a set, a “proper class”. (“Class” used to be synonymous with “set”.) But later on in the discussion it was pointed out that this isn’t the only possible resolution.
“An infinite universe can exist.”
″A greatest infinity cannot exist.”
I think there is some kind of logical contradiction here. If the Universe exists and if it is infinite, then it must correspond to the concept of “the greatest infinity.” True, Bertrand Russell once expressed doubt that one can correctly reason about the “Universe as a whole.” I don’t know. It seems strange to me. As if we recognize the existence of individual things, but not of all things as a whole. It seems like some kind of arbitrary crutch, a private “ad hoc” solution, conditioned by the weakness of our brain.
As for God or Gods, then, hypothetically, in the case of the coincidence of their value systems and the mental interaction between them according to a common agreed protocol, these problems should not be very important.
I’m hearing intuitions, not arguments here. Do you understand Cantor’s Diagonalization argument? This proves that the set of all integers is “smaller” (in a well-defined way) than the set of all real numbers, despite the set of all integers being already infinite in size. And it doesn’t end there. There is no largest set.
Russell’s paradox arises when a set definition refers to itself. For example, in a certain town, the barber is the one who shaves all those (and only those) who do not shave themselves. This seems to make sense its face. But who shaves the barber? Contradiction! Not all set definitions are valid, and this includes the universal one, which can be proved to not exist in many ways, at least in the usual ZFC (and similar).
There are two ways to construct a universal object. Either make it a non-set notion like a “proper class”, which can’t be an element of a set (and thus can’t contain itself or any other proper class), or restrict the axiom of comprehension in a way which results in a non-well-founded set theory. Cantor’s Theorem doesn’t hold for all sets in NF. The diagonal set argument can’t be constructed (in all cases) under its rules. NF has a universal set that contains itself, but it accomplishes this by restricting comprehension to stratified formulas. I’m not a set theorist, so I’m still not sure I understand this properly, but it looks like an infinite hierarchy of set types, each with its own universal set. Again, no end to the hierarchy, but in practice all the copies behave the same way. So instead of strictly two types of classes, the proper class and the small class, you have some kind of hyperset that can contain sets, but not other hypersets, and hyper-hypersets that can contain both, but not other hyper-hypersets, and so forth, ad infinitum.
Personally, I’m rather sympathetic to the ultrafinitists, and might be a finitist myself. I can accept the slope of a vertical line being “infinite” in the limit. That’s just an artifact of how we chose to measure something. Measure it differently, and the infinity disappears. I can also accept a potential infinity, like not having a largest integer, because the successor function can make a bigger one. We can make an abstract algorithm run on an abstract machine that can count, and it has a finite description. But taking the “completed” set of all integers as an object itself rubs me the wrong way. That had to be tacked on as a separate axiom. It’s unphysical. No operation could possibly construct a physical model of such a thing. It’s an impossible object. One could try to point to a pre-existing model, but we physically cannot verify it. It would take infinite time, space, or precision, which is again unphysical.
Similarly, there is no physical way to verify an infinite God exists, because we physically cannot distinguish it from a (sufficiently large, but) finite one. I might be willing to call such an alien a small-g “god”, but it’s not the big-G omni-everything one in valentinslepukhin’s definition. That only leaves some kind of a priori logical argument, because it can’t be an empirical one, but it has to be based on axioms I can accept, doesn’t it? I can entertain weird axioms for the sake of argument, but I’m not seeing one short of “God exists”, which is blatant question begging.
We can leave theology. It is not so important. I am more concerned with the questions of finitism and infinitism in relation to paradox of sets.
Finitism is logically consistent. However, it seems to me that it suffers from the same problem as the ontological proof of the existence of God. It is an attempt to make a global prediction about the nature of the Universe based on a small thought experiment. Predictions like “Time cannot be infinite”, “Space cannot be infinite” follow directly from finitism. It turns out that we make these predictions based on our mathematical problems with the paradox of sets. At the same time, the paradox of sets itself resembles the paradox “I’m telling a lie now”. and, it seems, should look for a solution somewhere in the same area. If we think off the cuff, it seems to me naively that the very concept of “ordinary set” is composed in such a way as to lead to paradoxes. This is the problem of the concept of “ordinary set”. This is not the problem of the existence/non-existence of physical infinity.
Oh, okay. I don’t really understand this topic. But as far as I know, not all mathematicians are finitists. So it seems that the proofs of finitism are not flawless.
On the other hand, how is the problem of the set paradox solved in cosmological infinitism? Something like “The Infinite Universe may exist, but it is forbidden to talk about it as an object”? Because any attempt to do so will bring you back to the set paradox, if you take it seriously. “Talk about any particular part of the Universe as much as you like, but don’t even think about the Universe as a whole”? This risks forming a somewhat patchwork model of the worldview. “It may exist, but you cannot think about it intelligently and rationally.” One is reminded of Zeno’s attempts to prove that one cannot think about motion without contradictions.
Finitism doesn’t reject the existence of any given natural number (although ultrafinitism might), nor the validity of the successor function (counting), nor even the notion of a “potential” infinity (like time), just the idea of a completed one being an object in its own right (which can be put into a set). The Axiom of Infinity doesn’t let you escape the notion of classes which can’t themselves be an element of a set. Set theory runs into paradoxes if we allow it. Is it such an invalid move to disallow the class of Naturals as an element of a set, when even ZFC must disallow the Surreals for similar reasons?
Before Cantor, all mathematicians were finitists. It’s not a weird position historically.
We do model physics with “real” numbers, but that doesn’t mean the underlying reality is infinite or even infinitely divisible. My finitism is motivated by my understanding of physics and cosmology, not the other way around. Nature seems to cut us off from any access to any completed infinity, and it’s not clear that even potential infinities are allowed (hence my sympathy with ultrafinitism). I have no need of that axiom.
Quantum Field Theory, though traditionally modeled using continuous mathematics, implies the Bekenstein bound: a finite region of space contains a finite amount of information. There are no “infinite bits” available to build the real numbers with. However densely you store information, eventually, at some point, your media collapses into a black hole, and packing in more must take up more space.
Physical space can’t be a continuum like the “reals”. It’s not infinitely divisible. Measuring distance with increasing precision requires higher frequency waves, and thus higher energies, which eventually has enough effective mass to gravitationally distort the very space you are measuring, eventually collapsing into a black hole.
Below a certain limit, distance isn’t physically meaningful. If you assume an electron is a point particle with “infinitesimal” size and you zoom in enough, you should be able to get arbitrarily high electric field strength. But at some point, high enough field strength results in vacuum polarization: virtual electron/positron pairs get pushed around and finally one of the positrons annihilates whatever you thought the real electron was, and then one of the virtual electrons doesn’t have anything to pair with and becomes the real one. It’s as if the electron is jumping around. You can’t nail it down. It doesn’t physically have a position down below a certain scale in time and space. There are no infinite bits. All the fundamental particle types are like this. There are no infinitesimal point particles. They’re just waves.
There’s also a cosmological horizon limiting how much of the Universe we can see. There’s also a (related) past temporal horizon at the Big Bang. We can’t see a completed past-temporal or spacial infinity, in any direction. We’re not sure of the Ultimate Fate of the Universe, but it looks like Heat Death is probably it, given our current understanding of physics. So there’s a future limit as well. The other likely candidate Fates are finite in time as well.
But even supposing finite information content in a finite region seems to be enough to make potential-infinite time not really meaningful. There’s a finite number of states possible, so eventually all reachable states are reached. If physics is deterministic (it seems to be), then we get into a cycle. So time is better modeled as a finite circle, rather than an infinite line. And if it’s not deterministic? Then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no phycial way to tell the difference.
Potential-infinite space is the same way. Any accessible region has a finite number of states, so at least some of them must repeat exactly in other regions. If there’s some determinism to the pattern, then it’s maybe better modeled as some curled-up finite space (although aperiodic tilings are also possible). If it’s random, then we still saturate all reachable states, the order just gets shuffled around a bit. There’s no physical way to tell the difference. Once all reachable states have been saturated, why does it matter if they appear only once or a googol or infinity times?