KJW

[math]\textstyle 2\mathbb {Z}[/math] [math]2\textstyle 2\mathbb {Z}[/math] [math]\mathbb {ABCDEFGHIJKLMNOPQRSTUVWXYZ}[/math]

I don't know the details, but Shor's algorithm is a quantum algorithm for finding the prime factors of an integer. This would allow private keys to be determined from public keys much faster than any known classical algorithm.

Earlier in this thread, you indicated that there is only one infinity. Have you changed your mind on that? Anyway, in the case of y=x2 and y=2x, they are the same infinity [math](\aleph_0)[/math]. The thing to note is that for every natural number x, there is one square x2, and one double 2x, and that all the squares and doubles of all the natural numbers are different. That is, the mapping from all the natural numbers to their corresponding squares and doubles is exactly one-to-one. To say that there are not as many squares or evens as natural numbers is to say that there are natural numbers that do not have distinct squares or doubles, which is clearly false. Because a characteristic of infinite sets is that their elements can be mapped one-to-one onto proper subsets, the existence of a set of natural numbers that are not square or even is immaterial.

Do you mean like a neural network?

Thank you. Yes, your labelling the diagonal b was confusing. His negation E of the diagonal D depends on the existence of D. As explained in the previous post, there are no diagonal sequences in the list. They are horizontal as defined by Cantor. But the list shows E1 as equal to D. Thus E1 is an element of the list, so its negation E0 must also be an element of the list. That is a property of the binary set {0, 1.} That E1 is equal to the diagonal is coincidental and not part of the diagonal argument. Note that the given list of sequences is E1, E2, E3,... etc. E0 is not part of the list. E0 is a sequence derived from the given list by negating the diagonal elements avv for v = 1, 2, 3,... etc. The diagonal argument is that E0 cannot be an element of the given list, and therefore the given list must be incomplete with respect to all possible sequences. I don't need to explain this because it is not relevant to the mathematics. And it is a glaring error on your part to think that it is. The symbol 'π' has a value beginning with 3.14159. The symbol 'e' has a value beginning with 2.7182. Perhaps you can tell me what the values of 3.14159 and 2.7182 are. He used ℵ0 to represent a value for 'infinite' sets which have no value. The symbol '0' represents a set with no elements or {}. I find it inconsistent that you use the cardinality of the set {} to specify the value of 0 while at the same time denying my use of the cardinality of the set [math]\textstyle \mathbb {N}[/math] to specify the value of [math]\aleph_0[/math]. Actually, you are correct... any list of things can be counted using the set [math]\textstyle \mathbb {N}[/math]. But it also reveals that you misunderstand the point of the diagonal argument. The point of the diagonal argument is that the set of all the possible sequences cannot be listed. That is, for every possible list of sequences, there will always be sequences that are missing from the list. And for any particular list, the diagonal argument identifies one of the missing sequences. Thus, we have at least two distinct infinities: an infinity that can be listed (countable), and an infinity that cannot be listed (uncountable). I believe that Cantor was referring to an earlier proof that the cardinality of the real numbers is greater than the cardinality of the natural numbers which did not use the diagonal argument. Cantor's theorem is about a set, its elements, and its subsets.

Hmmm. I don't think this is going to end well for Tristan L. Not only restarting a thread closed by a moderator, but actually persisting with the action that led to the original thread being closed.

I don't know what point you are trying to make with this. The diagonal sequence is only used to form the negation of the diagonal sequence, and it is the negation of the diagonal sequence that is not in the list of sequences. Cantor's diagonal argument says nothing about the diagonal sequence itself, and there is no reason why it can't be in the list of sequences. Whether the diagonal sequence is or isn't in the list of sequences is not important to the diagonal argument. Nobody said it was a number. Although I agree about infinity being unbounded, are you sure that is the only property of infinity? Cantor's theorem says otherwise. Indeed, Cantor's theorem implies that there are at least ℵ0 different magnitudes of infinity. ℵ0 is the cardinality of the natural numbers. Perhaps you could tell me what the value of zero is. Congratulations on proving that N is infinite! But even though N is "inexhaustible", there still isn't enough elements in it to cover the set of all subsets of N . We have N as the reference set of cardinality ℵ0 . We can determine if other sets have a cardinality of ℵ0 by determining if it's possible to place their elements into one-to-one correspondence with the elements of N . This may require some ingenuity. For example, one can't simply list the rational numbers Q in numerical order. But one can list them in the order: proving that [math]\textstyle \mathbb {Q}[/math] has the same cardinality as [math]\textstyle \mathbb {N}[/math]. But in the case of the real numbers [math]\textstyle \mathbb {R}[/math], the diagonal argument proves that there does not exist any such list, proving that the cardinality of [math]\textstyle \mathbb {R}[/math] is greater than the cardinality of [math]\textstyle \mathbb {N}[/math]. That is, if one can find a one-to-one correspondence between a given set and [math]\textstyle \mathbb {N}[/math], this proves that the cardinality of the set is [math]\aleph_0[/math], whereas if one can prove that no such one-to-one correspondence exists, then this proves that the cardinality of the set is not [math]\aleph_0[/math].

The mathematical description of the Einstein tensor is actually quite complicated: where: Actually, the above contains somewhat esoteric simplifying notation without which the expression would be even more complicated. The above images were sourced from https://en.wikipedia.org/wiki/Einstein_tensor#Explicit_form

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.