I can’t disagree with that. It’s a little better, but you’re right, not a lot better.
Isn’t a professor of “Theoretical Science” supposed to ponder what might be even if they cannot prove it? Should there be no such people dreaming this stuff up? Is Paul Steinhardt a woo artist?
Things that I had never considered. The question was, if gravity drops off at the inverse square, is there a point at which the influence of a body reaches zero? The response was, take the sun: it coalesced ~4.5GYa, so, if you could go out 4.5Gly, you would reach a point where you could not observe the sun because its formation is not recorded relatve to where you are: its gravity is zero because it effectively does not yet exist in the local frame of reference.
I see you have never heard of the Total Perspective Vortex. 
I have been back from Frogstar World B for some while.
Not really. Indeed that is for the most part exactly what theoretical science isn’t. “Theoretical” doesn’t mean imaginary or unfounded. It means rigorous science based in hard mathematics. Modern physics at the ragged edge might involve speculative ideas expressed in mathematics (eg String Theory, Loop Quantum Gravity etc) but it isn’t a license to peddle woo.
Sure. But stop pedalling this stuff as if it is science. It isn’t. It is random musings based in the ragged edge of theoretical physics - a place where we know our understanding of physics is incomplete. Incomplete enough that what you dream up may not make enough sense to even be wrong.
This entire topic/thread is speculation and cannot be otherwise.
When you dismiss the Albert Einstein Professor in Science, Director of the Princeton Center for Theoretical Science at Princeton University as pedalling fantasy (“here be dragons”) then I am not sure how this discussion can ever occur to @Chronos or your satisfaction.
In your view, should questions such as the OP’s topic not be asked or discussed because no one can provide rigorous proof?
“peddle”
I know better too. My GF is an editor. She will be very disappointed in me if she sees this.
You have to allow for expansion too. The space between us and a location 4.5 Gly away has expanded since 4.5 GYA, so any light (and gravitational effects) from the formation of the Sun will have travelled further than 4.5 Gly.
In addition to that, all the mass/energy that coalesced into the Sun already existed in this part of the universe, so the gravitational pull from this location will be roughly the same as it was before (although the collapse and ignition events may have been detectable as gravitational waves, assuming very sensitive instruments).
I missed this on first reading: the term Hubble volume caused my eyes to glaze over. But apparently a bog-standard infinite universe implies an infinite number of non-overlapping visible universes. If I understand this properly. So that’s one answer to the OP.
ETA: And yes, that’s one of many multiverse theories, to repost eburacum45’s wiki link.
The matter might not have been the Sun yet, but the matter did exist. In fact, if the collapse into a star happened spherically symmetrically (i.e., a large spherical cloud collapsing straight inward into a small spherically symmetric star), there would be no difference at all in the gravity, anywhere outside of the cloud. Now, real-world stellar formation generally has some asymmetries to it, so there would actually be some changes that would propagate outwards (i.e., a gravitational wave), but that would be an extremely tiny effect, even compared to the already-tiny gravitational force.
There is nothing wrong with such stuff. But when it veers into “probably are” statements, it gets silly.
This is a real problem one sees with a lot of science populism. Real scientists get drawn into unscientific prattling. It isn’t just physicists. Comparing the same scientists publishing in peer reviewed journals versus popular articles on the same topic is illuminating. They are human just like the rest of us. What sells, and gets them repeat invitations back on TV shows and that little bit of limelight isn’t what would pass as science with their peers.
Michio Kaku
I just don’t get it.
Cantor’s arguments about power sets absolutely, positively, most definitely refutes the basic arguments so many people make about “the odds are there has to be a 2nd Earth.”
Nope.
There isn’t enough room, by a higher order of infinity, to ensure a 2nd copy of every possible sizable chunk of the Universe. E.g., an Earth size chunk is phenomenally big enough. And that’s not requiring a Solar System to make Earth … Earth.
Speaking of “tiny odds but non-zero” odds is completely ignoring these different types of infinity. Especially since such an argument would necessarily apply to all possible parts of the Universe.
The power set of any set, including infinite sets, is always bigger. And in the case of infinite sets the power sets are just … too big by a wide degree.
This is not all that hard to understand.
I just don’t get it.
This is not the case. The number of real values between 0 and 1 is not less than the number of all integers, nor is the number of negative integers less than the number of all integers, nor is the number of possible values between 0 and googleplex more than the number of values between 0 and 1.
The mistake being made is that of treating the infinite value as an ordinal value. It is not. It does not behave like a finite ordinal, and the “<”, “=” and “>” operators are not meaningful for infinities. Any given infinite value is both greater than and less than and also equal to any other infinite value, as ordinal comparisons are not meaningful. The comparative relationship between two infinite values is non-binary, rather an infinite range of comparatives.
Actually, yes it is, because we’re talking about cardinals, not ordinals. At least assuming the axiom of choice, any two cardinals satisfy the trichotomy: One is always either <, =, or > the other (and exactly one of these holds). Cantor’s diagonal argument generalizes easily to show that, if S is a set with cardinality |S|, then |P(S)| > |S| is always true, where P is the power set, the set of all subsets of S.
However, that does not apply to the set of finite subsets of S. If S is infinite, then the cardinality of the set of its finite subsets equals the cardinality of S. If it were true that a bounded region of space was determined by a finite amount of information, then you would expect an infinite universe to contain, with probability one, copies of every bounded region.
And another caveat: “with probability one” doesn’t mean that it’s guaranteed to happen…
In fact, it’s greater. Because the number of real values between 0 and 1 has the same cardinality as the power set of the integers. Which, as @ftg says, is always greater.
Your other two examples don’t involve power sets.
But yeah, none of this is relevant to the finite numbers needed to have a match for a section of the Universe.
If a cite is needed:
Here’s Max Tegmark on the ‘crude estimate’ method he used to determine how far away different copies of you may be.
I suspect that is true; there are presumably an infinite number of possible Hubble volumes that don’t happen anywhere, and an infinite number of possible volumes that only happen once, and so on.