KJW

I don't know what operating system you're using but on my computer, I can select "Print to PDF" as my printer. It's part of the printer options rather than save options.

(ds)² = A(t)² ((c dt)² – (dx)² – (dy)² – (dz)²) For the cosmological redshift, Z, the use of the conformally flat metric simplifies the calculation because the equation of a light-like trajectory in spacetime has the simple form of a straight line. Let the observer be at the origin of the coordinate system (t = 0 ; x = 0 ; y = 0 ; z = 0), and let the emitter of two light-pulses, an infinitesimal interval of time apart, be at x = X > 0 ; y = 0 ; z = 0. Then, the equation of the two light-pulses: t + x/c = 0 ; y = 0 ; z = 0 and: t + x/c – dt = 0 ; y = 0 ; z = 0 Thus, the emitter emitted the two light-pulses at: t = –X/c and: t = –X/c + dt These two light-pulses were observed at: t = 0 and: t = dt Then the cosmological redshift, Z: Z = (ds(t = 0) / ds(t = –X/c)) – 1 = (A(0) c dt / A(–X/c) c dt) – 1 And therefore: Z = (A(0) / A(–X/c)) – 1 Note that for a redshift, Z > 0, A(0) > A(–X/c), and for a blueshift, Z < 0, A(0) < A(–X/c) Also note that the x, y, and z coordinates of the conformally flat metric are the same as for the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, and therefore the cosmological redshift, specified in terms of –X/c is unchanged for the FLRW metric. For example: (ds)² = exp(k t)² ((c dt)² – (dx)² – (dy)² – (dz)²) Z = exp(k X/c) – 1

How are you printing the files? From the LibreOffice program or from File Explorer? Can you print to a PDF file? Can you print PDF files? What does Google say? I would've thought that it is the program sending the data to the printer that handles the file type, not the printer.

I don't recall seeing it. It did occur to me that my question could be addressed by considering a principle that, given that reality is based on a spacetime with a non-flat metric, every quantity that emerges from Ricci Calculus corresponds to some quantity in the physical world. So, for example, the Einstein tensor corresponds to some conserved quantity in the physical world. That it corresponds to the energy-momentum tensor is largely a matter of physics rather than mathematics. However, which physical quantity a given mathematical quantity corresponds to does depend on the precise matching of the mathematical properties of the two quantities. Nevertheless, I'm still curious about the rationale behind choosing the Ricci scalar for the Einstein-Hilbert action. Thank you. 🙂

Another advantage of such algebraic presentation is that it can be applied to uncountably infinite sets.

I have already said that for a finite set, it is obvious that the number of subsets is greater than the number of elements, and that it doesn't require the diagonal argument to prove this. The reason I did apply the diagonal argument was to demonstrate the procedure of the diagonal argument, rather than to actually prove anything. But Cantor's theorem is not just about finite sets. Although Cantor's theorem is true for finite sets, including the empty set, it is for infinite sets that the theorem is intended to be applied. And it is for infinite sets that the diagonal argument demonstrates its power. No. Although it is true for finite sets that a proper subset has fewer elements than the set, it is not true for infinite sets. Indeed, it is a defining property of infinite sets that they can be mapped one-to-one onto proper subsets. As far back as Galileo Galilei, it was demonstrated that the number of square numbers is equal to the number of numbers themselves: [math]1 \longleftrightarrow 1[/math] [math]2 \longleftrightarrow 4[/math] [math]3 \longleftrightarrow 9[/math] [math]4 \longleftrightarrow 16[/math] [math]5 \longleftrightarrow 25[/math] [math]6 \longleftrightarrow 36[/math] [math]7 \longleftrightarrow 49[/math] [math]8 \longleftrightarrow 64[/math] [math]...[/math] [math]n \longleftrightarrow n^2[/math] [math]...[/math] Indeed, one can even form a one-to-one correspondence between [math]n[/math] and [math]2^n[/math] for all [math]n \in \textstyle \mathbb {N}[/math]: [math]1 \longleftrightarrow 2[/math] [math]2 \longleftrightarrow 4[/math] [math]3 \longleftrightarrow 8[/math] [math]4 \longleftrightarrow 16[/math] [math]5 \longleftrightarrow 32[/math] [math]6 \longleftrightarrow 64[/math] [math]7 \longleftrightarrow 128[/math] [math]8 \longleftrightarrow 256[/math] [math]...[/math] [math]n \longleftrightarrow 2^n[/math] [math]...[/math] This allows one to create the following one-to-one correspondence of the natural numbers to subsets of the natural numbers: [math]1 \longleftrightarrow 00000000...[/math] [math]2 \longleftrightarrow 10000000...[/math] [math]3 \longleftrightarrow 01000000...[/math] [math]4 \longleftrightarrow 11000000...[/math] [math]5 \longleftrightarrow 00100000...[/math] [math]6 \longleftrightarrow 10100000...[/math] [math]7 \longleftrightarrow 01100000...[/math] [math]8 \longleftrightarrow 11100000...[/math] [math]...[/math] Note that applying Cantor's diagonal argument to this gives [math]11111111...[/math] as the representation of the subset of the natural numbers that is not in this list. That is, it appears as if the set of the natural numbers itself is the "final" subset of the set of all subsets. But note that the above list is not a list of the set of all subsets but is a list of the set of all finite subsets, and this can be mapped one-to-one onto the set of natural numbers. The subset corresponding to [math]11111111...[/math], the set of natural numbers itself, is not a finite subset so it doesn't belong to the above list of subsets. In other words, Cantor's diagonal argument is only excluding a subset that would be excluded anyway.

[math]1 \longleftrightarrow 1[/math] [math]2 \longleftrightarrow 4[/math] [math]3 \longleftrightarrow 9[/math] [math]4 \longleftrightarrow 16[/math] [math]5 \longleftrightarrow 25[/math] [math]6 \longleftrightarrow 36[/math] [math]7 \longleftrightarrow 49[/math] [math]8 \longleftrightarrow 64[/math] [math]...[/math] [math]n \longleftrightarrow n^2[/math] [math]...[/math] [math]\textstyle \mathbb {N}[/math]

How can it not be about his theorem or the power set? Firstly, Cantor's diagonal argument is about proving his theorem. It makes no sense at all to discuss the diagonal argument without mentioning his theorem. And secondly, his theorem is about the power set in relation to the set. That (not b) is not in the square list is the whole point of the diagonal argument. It shows that the number of possible sequences is greater than the number of rows in the square list. It's only "self-fulfilling" because for a finite list, the theorem is so obviously true. But it is much less obvious for an infinite list. No, the row count u is supposed to be equal to the column count v. The row count is not the number of possible sequences (the number of subsets of the given set). The row count is the number of elements of the given set (equal to the column count). The question is "Is the negation of the diagonal in the square list?" The diagonal argument is that it is not. If you want to extend the list to cover all possible sequences, then it is only natural that the negation of the diagonal is in the extended part of the list. But the diagonal argument is about the "incomplete" square list only.

That's only true in two dimensions. The corresponding quantity in four dimensions (ignoring the numeric factor) was given above: Rijkl Rpqrs δijpqklrs For even dimensions in general, the quantity is: Rijkl Rpqrs ... Ruvwx δijpq...uvklrs...wx I am not aware of any corresponding quantity for odd dimensions.

Yes, as I alluded to in my first post, the Einstein tensor is the only tensor that is linear in the second-order derivatives of the metric tensor. One of the questions that I had considered for a long time was what justified the use of the Ricci scalar in the Einstein-Hilbert action. It turned out that the discovery that every scalar function of the metric tensor and its partial derivatives of any order leads to a conserved quantity was not the answer I was expecting. But the Ricci scalar is the only scalar function (that I'm aware of) that is linear in the second-order derivatives of the metric tensor. That doesn't quite answer my question but at least it is something that distinguishes the Ricci scalar from other scalars. For me, there is the question of why the theory of gravity we have is the one that matches physics. I believe that GR needs to not only conform to physical reality, but also be logical as well.

No. I'm talking about scalars that are distinct from the Ricci scalar and obtained from the Riemann tensor or its covariant derivatives of any order. I should also remark that one such scalar: Rijkl Rpqrs δijpqklrs is the integrand of a topological invariant in four dimensions, and therefore not all the second-degree scalars are independent.

Actually, it's not. In fact, for every scalar function of the metric tensor and its partial derivatives of any order, there is a rank-2 tensor that has identically zero covariant divergence. I call this the general relativity version of Noether's theorem. For the Ricci scalar as the scalar function, the corresponding rank-2 tensor that has identically zero covariant divergence is the Einstein tensor. Two other independent scalar functions are the square of the magnitude of the trace-free part of the Ricci tensor, and the square of the magnitude of the Weyl conformal tensor. There are also various covariant derivatives of curvature tensors from which scalar functions can be derived as the square of the magnitudes. However, the Ricci scalar is probably the only scalar function that is not derived from the square of a magnitude.

Is that what you think I'm doing? No, I am trying point out what I believe your error is. If that seems like I'm repeating what I read, then it would seem that you are not understanding my point. I'm not sure what you mean by this. The objective for me is clear: to explain Cantor's theorem and its proof. There is no doubt in my mind about these, but trying to explain them to you is proving to be a challenge. It's not clear to me that you actually know what it is that is being proven. It appears to me that you have misunderstood my 8 x 8 array. I was hoping that I had explained it, but apparently not well enough. The set of symbols is not {o,1,2,3,4,5,6,7,8}, but simply the binary "this element is in the subset" / "this element is not in the subset" choice. That is, there are not 98 = 43,046,721 possible sequences, but only 28 = 256. In other words, in my 8 x 8 array, you could replace the numbers "1", "2", "3", "4", "5", "6", "7", "8" with "Y" and "o" with "N". I apologise for the confusion. You appear to be treating Cantor's theorem as being about lists and sequences. But it is better to treat Cantor's theorem in terms of a set, its elements, and its subsets. Cantor's theorem then says that for a given set, there are more subsets than elements. The 8 x 8 array is about a set with 8 elements. For this set, there are 256 subsets which is obviously greater than the 8 elements, and thus Cantor's theorem is proven for this set. The diagonal argument wasn't really needed to show this, but I did used it to show how it is applied. In the case where the set is the set of natural numbers, the diagonal argument becomes valuable. You mention "missing sequences" as if their existence is a problem. No, their existence is what is being proven. It's obvious in the case of finite sets, but also true for infinite sets.

That is the power set and not the subject of the 1891 paper analyzed. According to the Wikipedia article "Cantor's theorem", that is precisely what he is proving: I should point out that while I am willing to debate the principles associated with Cantor's theorem, including the diagonal argument, I am unwilling to debate over a translation of the precise wording of an 1891 paper. You presented a paper in which it appears that you have a problem with the relationship between the length of the list and the length of the sequences, indicating that the diagonal argument requires that these two lengths be equal. In the post that you quoted, I showed you why they are equal, but you didn't respond to this. Quite simply, the length of the list is the number of elements in the set, and the length of the sequences is also the number of elements in the set because a subset of a set is made up of elements in the set, and the representation of the subset is a sequence indicating whether or not each element in the set is also in the subset.

Looking at your paper, it seems to me that you misunderstand the theorem being proven by Cantor. Cantor is proving that there does not exist a one-to-one correspondence between a set and the set of all the subsets of that set. It is true for finite sets in a rather trivial manner, and it is for infinite sets that the theorem is truly interesting. But let's consider the case of the finite set {1,2,3,4,5,6,7,8}. Vertically on the left is a list of the eight elements, and to the right of each of the elements is a representation of a subset of that set. If the subset contains a particular element, that element will be shown in its numerical position in the sequence. Otherwise, the numerical position will contain "o": 1=1o345o78 2=1o3o56o8 3=1o3o5o78 4=12oo5678 5=o234o678 6=1o34ooo8 7=12oo567o 8=1oo4oo78 Each subset is unique in the list. Now consider the diagonal: 1o3ooo78 The complement of the diagonal: o2o456oo differs from the first subset in the first element, differs from the second subset in the second element, differs from the third subset in the third element, ... , differs from the eighth subset in the eighth element, and so differs from every subset in the list. The complement is a subset of the eight elements but is not one of the eight subsets in the list. Furthermore, every possible list of eight subsets will have a complement of the diagonal that is missing from that list, proving that there does not exist a one-to-one correspondence between a set and the set of all the subsets of that set.

In the coordinate system of expanding space and time, if one has a light clock based on a pair of mirrors that are at rest in the co-moving frame, then this clock will tick coordinate time and not proper time.

Actually, the clock is co-moving with the cosmological fluid in both expanding space and expanding spacetime. But what @Markus Hanke is saying is that only in the expanding space does the time coordinate correspond to the proper time indicated by a clock co-moving with the cosmological fluid.

No, anyone who understands the proof can see that there are no errors. It's not a difficult proof, one can work through the proof to deduce that it is correct without assuming that it is correct. Calling it "set theory math babble" doesn't bode well for your credibility on the subject. We are talking about the mathematical realm, not the physical realm. You cannot use physical reality to argue anything about mathematics. Doing so also doesn't bode well for your credibility on the subject. I think it is fair to say that in mathematics, everything exists unless it can be mathematically proven not to exist. In the case of infinite sets, we know they mathematically exist because there is no largest natural number, and therefore we have an example of an infinite set, the natural numbers.

You are correct. Actually, on Page 4 of the thread "Cosmological Redshift and metric expansion", I have shown that expanding space and expanding spacetime are interconvertible by a coordinate transformation, and therefore they are actually the same thing and neither is more correct than the other.

Maybe one can get rid of the "2" under the square root by some form of pulley mechanism. The problem with using dimensional analysis to derive formulae is that one can miss out on dimensionless constants, as has happened here.

The list of sequences is infinite, specifically countably infinite because it is a list. It is only displayed as finite because of the limitations of illustrations. In fact, Cantor proved using set-theoretical notation that a set and the set of all its subsets (its power set) cannot be placed into one-to-one correspondence. The diagonal argument is a mere illustration of the proof for a countably infinite set. What Cantor proved is that there is an infinite sequence of distinct infinite cardinalities: natural numbers, power set of natural numbers, power set of power set of natural numbers, power set of power set of power set of natural numbers, etc. I'm not opening that. Please present it in a form that does not require opening a file.

A while ago, I performed a computer simulation of the second law of thermodynamics. It involved a collection of identical entities that could be in one of two states, X or Y, with X transforming to Y and Y transforming to X at a rate governed by some rate constant. The simulation starts with all of the entities in state X, and proceeds to equilibrium. The conclusions that I drew from this simulation are: 1: The second law of thermodynamics is largely mathematical rather than physical because it is able to be simulated. 2: The rate constant is covariant with respect to time reversal. That is, it satisfies the requirement of general relativity. In other words, the thermodynamic arrow of time does not violate the principle of general relativity because the description remains valid even under time reversal. 3: The transition probability is not time reversible, applying only in the forward time direction. However, because the transition probability is based on entity counts, it is necessarily a positive number, unlike the corresponding time reversed value. 4: The transitions X to Y and Y to X in the forward time direction are not the same as the transitions Y to X and X to Y in the reverse time direction. The notion of microscopic reversibility refers to a transition and its reverse in the same time direction, not a transition in forward and reverse time directions. The transitions in the reverse time direction are not governed by transition probability. 5: Causality is governed by the transition probability and only exists in the forward time direction. The reverse time direction appears to be teleological.

Thanks. However, shortly after I posted my question, I realised that you were right, at least for the original Aspect experiment involving polarisation of photons, with non-commutativity having an essential role in the experiment.

Oh, man. If something is it. That is it. A logic that sometimes doesn't allow you to say this and that. That's the essence of Bell''s theorem without a doubt. I've never seen Bell's Theorem described in terms of non-commuting observables. Could you please explain how non-commuting observables relate to Bell's Theorem.

Just in case you don't, an operator [math]L()[/math] is linear if and only if it satisfies: [math]L(\psi + \phi) = L(\psi) + L(\phi)[/math] Linearity is essential to QM because quantum superposition demands it.