Arithmetic Sequence Quotes

If you object (as some of us did to Dr. Goris) that Cantor's transfinite numbers aren't really numbers at all but rather sets, then be apprised that what, say, 'P(Infinity to the Infinity +n), really is is a symbol for the number of members in a given set, the same way '3' is a symbol for the number of members in the set {1,2,3}. And since the transfinites are provably distinct and compose an infinite ordered sequence just like the integers,they really are numbers, symbolizable (for now) by Cantor's well-known system of alephs or '(Aleph symbol's). And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers-although, just as with 0, the rules for these operations are very different in the case of (Alephs) and have to be independently established and proved.

Eternity, in the sense of the pools, manifests as an enigma within the mathematical fabric of existence. It represents a fractal realm in which the notion of endless duration deviates from conventional human experience. Far beyond the finite bounds of what we call ‘time,’ eternity morphs into a disorienting continuum of perpetual recurrence and unbounded expansion. The cyan merely acts as a catalyst to understanding. Within this eerie realm, space dissolves into a concept, and the usual arithmetic constraints fail to hold sway. The rooms become a ceaseless amalgamation of symbolic sequences and iterations, where infinite series relentlessly converge and diverge, oscillating in rhythm to the waves. The wave function collapses when th//Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан цијан цијан цијан Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан

[All] modern chatbots are actually trained simply to predict the next word in a sequence of words. They generate text by repeatedly producing one word at a time. For technical reasons, they generate a “token” at a time, tokens being chunks of words that are shorter than words but longer than individual letters. They string these tokens together to generate text. When a chatbot begins to respond to you, it has no coherent picture of the overall response it’s about to produce. It instead performs an absurdly large number of calculations to determine what the first word in the response should be. After it has output—say, a hundred words—it decides what word would make the most sense given your prompt together with the first hundred words that it has generated so far. This is, of course, a way of producing text that’s utterly unlike human speech. Even when we understand perfectly well how and why a chatbot works, it can remain mind-boggling that it works at all. Again, we cannot stress enough how computationally expensive all this is. To generate a single token—part of a word—ChatGPT has to perform roughly a trillion arithmetic operations. If you asked it to generate a poem that ended up having about a thousand tokens (i.e., a few hundred words), it would have required about a quadrillion calculations—a million billion.

What this means is that the (Infinity) of points involved in continuity is greater than the (Infinity) of points comprised by any kind of discrete sequence, even an infinitely dense one. (2) Via his Diagonal Proof that c is greater than Aleph0, Cantor has succeeded in characterizing arithmetical continuity entirely in terms of order, sets, denumerability, etc. That is, he has characterized it 100% abstractly, without reference to time, motion, streets, noses, pies, or any other feature of the physical world-which is why Russell credits him with 'definitively solving' the deep problems behind the dichotomy. (3) The D.P. also explains, with respect to Dr. G.'s demonstration back in Section 2e, why there will always be more real numbers than red hankies. And it helps us understand why rational numbers ultimately take up 0 space on the Real Line, since it's obviously the irrational numbers that make the set of all reals nondenumerable. (4) An extension of Cantor's proof helps confirm J. Liouville's 1851 proof that there are an infinite number of transcendental irrationals in any interval on the Real Line. (This is pretty interesting. You'll recall from Section 3a FN 15 that of the two types of irrationals, transcendentals are the ones like pi and e that can't be the roots of integer-coefficient polynomials. Cantor's proof that the reals' (Infinity) outweighs the rationals' (Infinity) can be modified to show that it's actually the transcendental irrationals that are nondenumerable and that the set of all algebraic irrationals has the same cardinality as the rationals, which establishes that it's ultimately the transcendetnal-irrational-reals that account for the R.L.'s continuity.)

​The activities of the Chaldean Hekate can be understood as an intensive meditation upon and elaboration of Hekate’s actions in the Homeric Hymn to Demeter, which fall into three stages: ​1. (HHD 22-5): Hekate, described as “Perses’ daughter still innocent of heart [atala phroneousa],” hears Persephone’s cries “from her cave [ex antrou],” as does Helios. ​Here, Hekate is quiescent, but responds to the “voice” of the soul descending to embodiment, to which compare the “lifegiving whir” or “hum” (rhoizêma) with which Damascius associates Hekate (In Parm. III 42.18). ​2. (51-61): On the tenth day [dekatê] of her search, Demeter meets Hekate “with a light in her hand [selas en cheiressin echousa]” and tells her what she heard. Demeter runs with her “with burning torches in her hands” to Helios, who saw the events. ​The numbers ten and four (the ten being the expansion of four, 1+2+3+4) are spoken of as “key-bearers”, kleidouchoi in the pseudo-Iamblichean Theology of Arithmetic (28.13, 81.14 de Falco), this being an epithet of Hekate’s as well. The text refers first to Hekate’s single light at first, but then to Demeter’s twin torches, as they run back to Helios to retrieve the vision. Thus, at the furthest limits of the centrifugal motion, the centripetal motion of “virtue” (keys) comes into play. ​3. (438-440): Hekate, described as at 25 as “of the glossy veil [liparokrêdemnos]”, embraces Persephone on her return, and “the mistress [anassa]” becomes Persephone’s attendant and servant [propolos kai opaôn]. ​At the beginning and the end of the sequence, Hekate is veiled, as when the world is rendered flat or “membrane-like [humenôdês]” (frag. 68). In embracing Persephone on her return, that is, the soul upon its liberation from self-imposed bondage, Hekate is acknowledged as Mistress, and assumes a role of guide and helper to the soul in its future transformations (“ascents” and “descents”).

Eternity, in the sense of the pools, manifests as an enigma within the mathematical fabric of existence. It represents a fractal realm in which the notion of endless duration deviates from conventional human experience. Far beyond the finite bounds of what we call ‘time,’ eternity morphs into a disorienting continuum of perpetual recurrence and unbounded expansion. The cyan merely acts as a catalyst to understanding. Within this eerie realm, space dissolves into a concept, and the usual arithmetic constraints fail to hold sway. The rooms become a ceaseless amalgamation of symbolic sequences and iterations, where infinite series relentlessly converge and diverge, oscillating in rhythm to the waves. The wave function collapses when th//Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан цијан цијан...........................

Eternity, in the sense of the pools, manifests as an enigma within the mathematical fabric of existence. It represents a fractal realm in which the notion of endless duration deviates from conventional human experience. Far beyond the finite bounds of what we call ‘time,’ eternity morphs into a disorienting continuum of perpetual recurrence and unbounded expansion. The cyan merely acts as a catalyst to understanding. Within this eerie realm, space dissolves into a concept, and the usual arithmetic constraints fail to hold sway. The rooms become a ceaseless amalgamation of symbolic sequences and iterations, where infinite series relentlessly converge and diverge, oscillating in rhythm to the waves. The wave function collapses when th//Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан цијан цијан цијан Цијан цијан цијан цијан цијан цијан цијан цијан цијан HELP ME цијан цијан цијан цијан цијан цијан цијан..................

About a year or so ago, Johnny made a truly remarkable set of observations and was responsible for a completely new method of programming, "Herman Goldstine elaborated in 1949. "Johnny's scheme was to wire up what corresponds to the ENIAC's plug-boards with a fixed set of instructions that is universal to all problems." Individual instructions were assigned unique numbers - order codes - that were intelligible to "a switching center so built that upon receipt of a given number, characterizing one of the orders wired into the plug-boards, it energizes the proper board and thereby causes the order to be executed." A sequence of orders, constituting a program, could either be entered via the ENIAC's function tables, or read from punched cards. "It is no longer necessary to stand on one's head to fit a given routine," Goldstine continued. "To prepare an individual problem the coder now merely writes out the sequence of operations, arithmetic and logical, which characterize his problem and then transliterates these into the numbers the machine will understand.