KJW

I see it as a contradiction of terms. You can't approach something that is out of reach. Here is an analogy. Did you continue to read what I said after "The term "approaching infinity" is often used in connection to limits."? You seem to have put in a lot of effort to argue against a phrase that is intended to be intuitive. Why can't you approach something that is out of reach? The notion of limits is about approaching something but not actually reaching it.

I think ii and vii are eliminations, the clue being "hot concentrated ethanoic potassium hydroxide solution".

Do you know where the value [math]-\dfrac{1}{12}[/math] comes from? It is the value of [math]\zeta(-1)[/math], where [math]\zeta(z)[/math] is the Riemann zeta function over the complex number field. The Riemann zeta function is defined as: [math]\zeta(z) \buildrel\rm def\over= \dfrac{1}{1^z} + \dfrac{1}{2^z} + \dfrac{1}{3^z} + \dfrac{1}{4^z} + \dfrac{1}{5^z} \> + \> ...[/math] Substituting [math]z = -1[/math] gives the series [math]1 + 2 + 3 + 4 + 5 \> + \> ... \>[/math], but that's not how [math]\zeta(-1) = -\dfrac{1}{12}[/math] is calculated. Calculating [math]\zeta(-1) = -\dfrac{1}{12}[/math] involves a process called "analytic continuation" where the domain of a function is extended as the domain of a continuous function beyond where it would otherwise fail to converge. The series converges for real values [math]> 1[/math], but there is a "pole" at [math]z = 1[/math], and analytic continuation to negative real values from real values [math]> 1[/math] involves going around the pole in the complex number domain. In the case of the Casimir effect, one uses the identity: [math]\zeta(-3) = 1 + 8 + 27 + 64 + 125 \> + \> ... \> = \dfrac{1}{120}[/math] A simple example of a function that is continuous over the entire real number domain, yet its infinite series only converges within a limited domain of the real numbers is: [math]\dfrac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + x^8 - x^{10} \> + \> ... [/math] Although the function is well-defined for [math]|x| \ge 1[/math], the infinite series fails to converge for [math]|x| \ge 1[/math]. But note that for [math]-1 < x < 1[/math], the infinite series converges to the same value as the function. Therefore, one could say that: [math]1 - 4 + 16 - 64 + 256 - 1024 \> + \> ... \> = \dfrac{1}{5}[/math]

[math]f(x)\; {\buildrel\rm def\over=} \;x+1[/math] [math]\buildrel\rm def\over=[/math] [math]\buildrel def \over=[/math]

Did you get your Physics PhD from a home schooling environment or something? There seems to be some rather glaring holes in your physics education... Yeah, I thought that. William.Walker39's argument seems too much like a n00b mistake for it to have come from a PhD physicist.

Again?? You previously asked me a similar question which I answered. The answer to this question is that [math]\aleph_0[/math] is not an element of [math]\mathbb {N}[/math]. [math]\mathbb {N}[/math] contains only finite numbers and [math]\aleph_0[/math] is not a finite number. We've already discussed this, so I suggest you go back to that discussion. The term "approaching infinity" is often used in connection to limits. Suppose a1, a2, …, an, … is a sequence of real numbers. When the limit of the sequence exists, the real number L is the limit of this sequence if and only if for every real number ε > 0, there exists a natural number N such that for all n > N, we have |an − L| < ε. The common notation: [math]\displaystyle \lim _{n \to \infty} a_{n}=L[/math] is read as: "The limit of an as n approaches infinity equals L" or "The limit as n approaches infinity of an equals L". But the formal definition itself doesn't mention "infinity". Indeed, it seems to me that the whole notion of limits is about dealing with the infinite and infinitesimal in a way that remains firmly in the realm of the finite.

[math]\displaystyle \lim _{n \to \infty} a_{n}=L[/math] [math]\lim _{n \to \infty} a_{n}=L[/math]

It concerns the motivation of the author. There have been real world cases of scientists who manipulate data to support their theories. In the justice system, motive is important. There is one important difference that you are overlooking: the mathematical theorem is completely transparent. There is no reliance on trust. I can see from the theorem itself that the motivation of the author is irrelevant, there is no data to manipulate, and this is not the justice system. I disagree. Cantor's theorem is about the existence of different transfinite numbers, based on the mappings of sets of numbers. Have you ever seen the following identity?: [math]1 + 2 + 3 + 4\> +\> ...\> = -\dfrac{1}{12}[/math] This is fairly well-known because it is recognised as one the most bizarre identities in mathematics. It emerges as a result of analytic continuation of the Riemann zeta function. But surely such apparent absurdity has no relevance to physical reality? Actually, a related equally bizarre identity manifests itself in the Casimir effect. I find this quite extraordinary, to be honest. What is its value? The value is [math]\aleph_0[/math]. Why do you think its value should be something else? Where have I said that it does? I think I have indicated that "infinite" is a property. But you also have to recognise that there are different properties that lead to different infinities. In particular, the property of being able to be placed into one-to-one correspondence with the natural numbers, and the property of not being able to be placed into one-to-one correspondence with the natural numbers are two different properties leading to two different infinities.

A fundamental problem with GR is that it is for the most part a local theory. That is, it deals with the various fields at individual points in spacetime. When one changes the coordinate system, the numerical values of the various components of the fields at the various points change according to how the coordinate system changed at those points. And the change in the components at one point is largely independent of the change in the components at another point due to the general nature of coordinate transformations. Thus, for many types of fields, it is impossible to create a total over a region of spacetime, quite simply because changing the coordinate system changes the total in a way that GR does not allow. And because there is no preferred coordinate system in GR, there is no way to say which value of the total over a region of spacetime is the correct value. So, it is impossible to have a total energy-momentum over the entire universe. Furthermore, regions in spacetime are four-dimensional, whereas we tend to think of three-dimensional regions and totals over three-dimensional regions, which also depend on the particular coordinate system.

I heard a joke a few days ago: Women regard men who know a second language as more attractive... unless it's Klingon.

[math] \documentclass{article} \begin{document} Evaluate the sum $\displaystyle\sum\limits_{i=0}^n i^3$. \end{document} [/math] [math]Evaluate the sum $\displaystyle\sum\limits_{i=0}^n i^3$.[/math]

That the union of two disjoint sets, both of which have a cardinality of [math]\aleph_0[/math], has a cardinality of [math]\aleph_0[/math] is logical because we are not dealing with finite sets. The logic of finite sets does not apply to infinite sets. Infinite sets have a logic of their own. If we have two disjoint sets A = {a1, a2, ...,an, ...} and B = {b1, b2, ...,bn, ...}, both of which can be placed into one-to-one correspondence with [math]\mathbb {N}[/math]: [math]1 \longleftrightarrow a_1[/math] [math]2 \longleftrightarrow a_2[/math] [math]...[/math] [math]n \longleftrightarrow a_n[/math] [math]...[/math] and: [math]1 \longleftrightarrow b_1[/math] [math]2 \longleftrightarrow b_2[/math] [math]...[/math] [math]n \longleftrightarrow b_n[/math] [math]...[/math] then we can interleave these two lists to form a list that is also a one-to-one correspondence with [math]\mathbb {N}[/math]: [math]1 \longleftrightarrow a_1[/math] [math]2 \longleftrightarrow b_1[/math] [math]3 \longleftrightarrow a_2[/math] [math]4 \longleftrightarrow b_2[/math] [math]...[/math] [math]2n-1 \longleftrightarrow a_n[/math] [math]2n \longleftrightarrow b_n[/math] [math]...[/math]

So, you have reduced the notion of a soul to mere physiology. The heart obviously exists, but what connection does it have to religious belief? You say the heart moves our body. Are you sure you're not talking about the brain, the nervous system, or muscles?

Earlier in this thread I said that I was "unwilling to debate over a translation of the precise wording of an 1891 paper". I should also have included discussions about the man. The fact of the matter is that the validity of a mathematical theorem is independent of who originated it or how it is expressed in particular writings. Thus, I accept the diagonal argument purely on the basis of its logic, and for you to convince me otherwise, your argument would also have to be purely on the basis of its logic. I don't know about that. It turns out that the mapping from the group of integers [math]\mathbb {Z}[/math] to the group of even integers [math]2\mathbb {Z}[/math] is an endomorphism. This means that properties of the odd numbers map to properties of double of the odd numbers under the endomorphism. And given that this mapping is one-to-one, it is unlikely that the integers [math]\mathbb {Z}[/math] are any richer than the even integers [math]2\mathbb {Z}[/math]. Leopold Kronecker was a subscriber to the philosophy of finitism and was thus in the minority among mathematicians. I am a subscriber to the philosophy of formalism, and regard mathematics as not being rigorous unless it can be derived from first principles (axioms) by mechanical symbol manipulation. The meanings of the symbols are abstract, not necessarily connected to reality, and manifest in the process of manipulation. I do not ask if notions represented by the symbols exist in reality. However, I ask you whether our description of reality is in any way adversely affected by the concept of the transfinite numbers, or the notion that an infinite set can be placed into one-to-one correspondence with proper subsets of that set. In other words, I challenge your notion of existence in mathematics. I don't think any of the reasoning I put forward in this thread even remotely suggests a thinking of a large number as closer to infinity than a small one. I would say the opposite. For example, in the one-to-one mapping of [math]n \longleftrightarrow 2^n[/math], for large [math]n[/math], [math]2^n[/math] is a lot larger than [math]n[/math], yet there was no suggestion that [math]2^n[/math] was any closer to the "end of the list" than [math]n[/math]. How is this E=2D? The mapping: [math]1 \longleftrightarrow 2[/math] [math]2 \longleftrightarrow 4[/math] [math]3 \longleftrightarrow 6[/math] [math]4 \longleftrightarrow 8[/math] [math]...[/math] [math]n \longleftrightarrow 2n[/math] [math]...[/math] is just as true a one-to-one correspondence as the mapping [math]2n-1 \longleftrightarrow 2n[/math]. The cardinality is [math]\aleph_0[/math]. Banach-Tarski paradox

[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.