AG, your reasoning assumes that because each countable 4D ball shrinks arbitrarily close to zero, the entire universe must shrink as well. This misinterprets how infinity works in both set theory and general relativity.
Shrinking finite regions doesn’t imply a finite universe. Each of your 4D balls represents a finite spacetime region, but an infinite number of shrinking finite regions does not make the total universe finite. Even if every individual region shrinks, an infinite set of them still covers all of space, preserving its infinite extent. The universe can remain infinite despite local contraction. Imagine an infinite 1D line divided into shrinking segments. Each segment gets smaller, but since there are infinitely many, the total length remains infinite. The same applies in higher dimensions: even as each 4D ball shrinks, the universe as a whole remains infinite because there is no bound on the number of shrinking regions. General relativity allows an infinite universe to contract everywhere without requiring a finite total volume. This is why an infinite universe can undergo a Big Bang—density increases everywhere without demanding a global contraction. Your argument assumes a globally shrinking boundary, implying that these 4D balls define the total size of the universe. But in an infinite universe, no such global boundary exists. There is no edge where "shrinking" causes the entire structure to collapse into a finite size. An infinite universe remains infinite while every finite region contracts. Your logic would only apply if the universe were globally finite from the start. Since an infinite universe has no fixed size to contract, space simply becomes denser everywhere as you go back in time. Quentin All those moments will be lost in time, like tears in rain. (Roy Batty/Rutger Hauer) Le jeu. 20 mars 2025, 12:53, Alan Grayson <[email protected]> a écrit : > > > On Thursday, March 20, 2025 at 2:12:29 AM UTC-6 Alan Grayson wrote: > > On Wednesday, March 19, 2025 at 11:49:50 PM UTC-6 Brent Meeker wrote: > > > > On 3/19/2025 10:09 PM, Alan Grayson wrote: > > > > On Wednesday, March 19, 2025 at 10:50:41 PM UTC-6 Brent Meeker wrote: > > > > On 3/19/2025 9:14 PM, Alan Grayson wrote: > > > > On Wednesday, March 19, 2025 at 3:28:40 PM UTC-6 Brent Meeker wrote: > > > > On 3/19/2025 4:56 AM, Alan Grayson wrote: > > > > On Wednesday, March 19, 2025 at 5:40:48 AM UTC-6 John Clark wrote: > > On Wed, Mar 19, 2025 at 4:30 AM Alan Grayson <[email protected]> wrote: > > *> If the universe is infinite in spatial extent, and we run the clock > backward, is all the mass/energy of the observable region confined to a > tiny or zero volume?* > > > *The short answer is nobody knows what will happen if you run the clock > back to zero, and the mystery remains regardless of if the universe is > finite or infinite. Nobody knows what will happen when things get super > small because our two best physical theories, Quantum Mechanics and General > Relativity, disagree with each other. Most believe that something will > prevent a zero volume from ever occurring, but nobody knows what that > "something" is. * > > *John K Clark See what's on my new list at Extropolis > <https://groups.google.com/g/extropolis>* > > > Maybe it's a 5th force. What I'd like to know is this; assuming an > infinite spatial universe and that it gets very very small as we run the > clock backward, the observable regions shrinks, but what happens to the > unobservable region? Quentin claimed to have an answer, but I can't recall > what it was. AG > > All theories treat the unobservable regions as being similar to the > observable (what else could you justify?). So every finite region, > observable or not shrinks to zero. > > Brent > > > *But if every finite subset of an infinite set strinks to zero, in the > case the assumed infinite set is the spatial extent of the universe, won't > the infinite spatial set of the universe also shrink to zero (which is what > Quentin denies)? AG* > > > > > *No. Brent* > > > But, as I've shown, this contradicts basic set theory. AG > > > Basic set theory has no metric. Shrink to zero in meaningless for a set. > > Brent > > > "No" isn't an argument. It's just a claim. My argument is based on set > theory and topology. If an infinite set can be contained in a countable set > of finite sets, and if they represent spacetime, and each shrinks to zero, > then so will the original infinite set. But maybe the infinite set of > spacetime points cannot be contained in a countable set, in which case we'd > have to use the Axiom of Choice. But I'm not sure if the infinite set of > spacetime points can be covered or contained in an uncountable set created > by applying the Axiom of Choice. In any event, you need an argument to > establish your claim. AG > > > The entire universe can be covered with a countable set of 4 dimensional > balls, each centered at integer clock readings, with unit radii, each ball > includes its boundary. No need in this model for applying the Axiom of > Choice. Each ball is infinite in the number of events it contains, and each > is closed since it contains its boundary, so we can consider each ball as a > finite region of spacetime. As time runs backward, each ball shrinks as > close to zero as desired, and ISTM that the entire universe shrinks with > it. How can the universe remain infinite in spatial extent in this > situation? 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