NTuft

ℕ , the set of all natural numbers: https://infogalactic.com/info/Natural_number#Notation ℝ , the set of all real numbers: https://infogalactic.com/info/Real_number They're irrational? I do not know if they enjoy such being ascribed as such... Perhaps, also, it's been proven that i' = aleph_1, while the ℕ = aleph_0, and the ℝ ≥ aleph_2

Let aleph_0 = N, the set of Natural numbers If we take the set of square roots of prime numbers as a subset of the real numbers, we can say that this subset is smaller than all reals, R. This subset, although smaller than all Reals, can have each member of the set equate to the set N, 1:1, as the decimals keep extending infinitely to match the increasing Naturals. Hence, this set is of aleph (cardinality) greater than N, aleph_0, but less than the cardinality of all reals, R, because it is a subset therefrom; hence, the cardinality of R is at least aleph=2.

{emphasis added} Hello again. I'm no expert either. I don't read what you'd quoted to lead to that following statement. From: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Gillman544-553.pdf Two Classical Surprises Concerning the Axiom of Choice and the Continuum Hypothesis by Leonard Gillman So it may be independent, but, as I think @studiot was alluding to more generally, we have to work from an accepted basis to get anywhere. So, @joigus, bringing in ZF axioms is most pertinent. Since the discussion is under set theory, here's an overview which I don't think is too tedious: from: https://www.math.fsu.edu/~hoeij/spring2020/Axioms.pdf So I think it needs to be formulated under this rubric (A1-A8 or A1-A9) and with understanding of cardinality/ordinality (i.e. using what's agreed upon and the relevent well-defined terms). Thanks I think that this is the purpose of set theory: numbers are what we commonly refer to as the collection of objects in the Natural/Real set. Once the objects (numbers) are defined, operations can be defined to act upon them, certain inputs giving certain outputs.

So, if we take the set of square roots of prime numbers, each number in that set can match the set of "Natural" numbers by extending the decimal points. As the "Natural" numbers get large a further decimal 0-9 matches each n, n+1, ...; 1:1. The ordinality of i' is the same as the ordinality of "infinity", and there is a degree of aleph=2 to the subset i' before it reaches anything including all the "Reals". So, between the sets "infinity", i', various Real sets, and the Natural set, there are at least two degrees, hence the aleph=2, hence Cantor's conjecture is towards the negative. Does that make sense? P.s. yada, yada, cardinality.

@joigus , @studiot , Does this serve to "Disproving" Cantor's hypothesis? So, if we take the set of square roots of prime numbers, each number in that set can match the set of "Natural" numbers by extending the decimal points. As the "Natural" numbers get large a further decimal 0-9 matches each n, n+1, ...; 1:1. The ordinality of i' is the same as the ordinality of "infinity", and there is a degree of aleph=2 to the subset i' before it reaches anything including all the "Reals". So, between the sets "infinity", i', various Real sets, and the Natural set, there are at least two degrees, hence the aleph=2, hence Cantor's conjecture is towards the negative. Does that make sense? Thanks

@studiot I do need to study number theory, and I woud like to hear exposition from you on what you find interesting, or if you had a resource to recommend to, "get up to speed", I would read into it. I was using it only in the sense that I could arbitrarily set rules for a number theory on my own terms, and I doubt now anything I've posited provides value other than what is enumerated or established by number theory already developed. To change it somewhat, I'd say that the first derivative from infinity should include the set of all square roots of numbers. Then, integers can be teased out from the sq. rts of their squares, and the leftover inoperable numbers are the primes and semi-primes that have prime factors still needing the ^(1/2) exponentiation. More or less, I had the hope that this idea of deriving from infinity was a "bolt from a dark cloud", but it is little else than a fever dream or pipe dream that now I can see has little applicability to any actual problems. @joigus, Yes, I suppose so -- I had chosen 'i' as my term to define, even though it is already defined as a constant, which is confusing. Furthermore, I was actually ignorant that in complex numbers the term i is actually operated on! From some wiki reading: cyclic values: i-3 = i i-2 = -1 i-1 = -i i0 = 1 i1=i i2= -1 i3=-i i4=1 i5=i i6=-1 Now, what you're showing I think is a partial differential equation? My math is limited -- hence why in my discussion I was talking about trying to derive "z" (formerly "i") in terms of something else, "the field". This might not make any sense whatsoever. So, again, I'm in need of education. That said, from above cyclic results, it looks to me that the values that come from i when it is operated on are quite limited, and maybe this set of results could be expanded? The cyclic values of 1,-1,i,-i are what graph onto the complex plane as a magnitude(length?), or argument (direction)? If the magnitudes/arguments were expanded to include the set of larger square roots of primes would this be impossible or without basis? To my mind, this would simply expand the idea of degree of magnitude in complex number graphing. However, my understanding of complex numbers and complex graphing is admittedly lacking. I am quite ignorant as I keep repeating and likely in need of a Physics 201 or 301 course. I'm going to continue my study of Quantum Chemistry as I think it incorporates a lot of what I want to learn. I'm also looking at some books on basics covering General Relativity and more modern theories (e.g. Hyperspace by Kaku) from a reading list I saw on another board. I have enough insight I think to see I'm low man on the totem pole in this discussion, so I appreciate your fellows' input, and if you had some reading material on number theory or physics that you'd recommend I would be glad to have those suggestions. I can see you're willing to engage and that alone I can appreciate. Thanks noted^

Studiot, thanks for the assessment. Yes, including or redifining i (i2=(-1)) is like an ad hoc hitchhiker, I don't think it makes sense. Need something other than "i". I also appreciate the number theory run up, I think you get what I was trying. Then from last, & from joigus, I agree, this is tangled up puree, as I can't now see any physical correlates with which this helps. I suppose the hope was the math would have physical correlates, but of course that should show itself. That said, just what Studiot also mentioned, is what I had in mind. To start it from "the wrong end". I don't take it as granted that that is wrong -- that I would say is the math conjecture here.

Thank you, joigus. I know I needed help on the calculus involved here, and I doubted most of all that that part was feasible or correctly formulated. I understand a derivative to be a measure of infinitesimal change at a point. It is mentioned in the discourse that I had an idea that the imaginary unit i could be redefined to encompass all sq. rt.s of primes, instead of just i = (-1)^(1/2). An interjection here -- if you could point me somewhere to learn how to insert the MatLab looking equations I'd appreciate it. Now, getting back to the first, if i is not only i=(-1)^(1/2) , but is instead defined as I posit, could that make it a "variable", or moving target; a differentiable quantity? I accept that the basis may be off and this all may be "math crankery". Furthermore, I would add that subsets of the number set of sq. rt.s of primes could be aggregated into the "integer" multipliers that are used to obtain allowable energy states in the Bohr model of allowed energies. Because this is following from the simplest elucidation, that of the spectral lines of Hydrogen, the more complicated sets of numbers could be conceptualized as "variable", because the size of the central, positive charge in the nucleus, and therefore by extension the base electron cloud (whatever the combination may be) would comprise a differently constituted subset of primes as encompassing the allowed energy excitation states, and hopefully also, somehow, describing the kinetic energy component in motion from the higher-order derivative.

I take it you consider infinity to be the constant? Do you suppose that that may be a way of working with a concept that facilitates maths operations built up on a number theory from below it? I want to consider chemistry as built out of only positive and negative charges, a la electrostatics. I want to consider physics as having gravitation equivalent to magnetism (ergo electrogravitational force) and explained by central points of gravitation/positive charge at nucleus, with negative charges in orbit; again, really just "electrostatics", or maybe that is not the right term. More targets now available.

Summary: Idea of deriving down to i''' from infinity, potentially establishing number theory from set of sq.rt. of primes, i'. Then, (*highly speculative*) i'' as Kinetic Energy, i'" as Potential Energy from electrostatics for new Hamiltonian in quantum mechanics? In need of critique on mathematics/calculus, in particular. Derivations done with respect to Field "shape", encompassing decreasing dimensions with each derivative for conceptualization, see discussion. Hello, Although this hinges on mathematics I will make this only just a discussion with cursory equations. I would encourage criticism, as I suspect my math is completely off-base. That said, I think the idea might have some merit, and could perhaps be improved upon by the forum discussion. I propose to take the first derivative from infinity (i) with respect to the field shape or conformer that exists. I call this first derivative i' (di/dAc [Ac, Actual conformer]) , and define that the set of departure, or domain, or number set, that emerges is: the set of square roots of prime numbers. I think of this as logically coherent as the set i' could potentially also be infinite, and I do not think any operations can be performed on these numbers, save for further derivation. IMO, the field shape or conformer here is of the nature of anti-DeSitter saddle conformer (negative curvature), but that is the physics and not mathemathics. That said, to discuss the physics, I think there is a need for 4D here a la Quaternion mathematics. See the Euler identity [wiki: ht tps://en.wikipedia.org/wiki/Euler's_identity]: under Generalizations section: it seems Quaternion exponentiation transforms i (imaginary number, i = sq. rt. (-1)...) found within e^i*pi, into i= (1/sq.rt.(3))* ( i+/-j+/-k)... Back to maths, the second derivative from the set i' yields i'' as of type (di''/dAc) = (1/2)*(1/ i' ^(1/2)) ; this "function" (I do not know if it is a function) appears to me to have a limit approaching 0. I think of it as this i'' derivation has now opened "top-down" number theory by opening 1 & 2 for operations as 1/2, alongside the set of i' which are now inhabiting the denominator. This 1/2 coefficient could in some way relate to Kinetic Energy, as I'm hopeful this theory is addressing only just Field shape & Energy / information. To continue with Field discussion, I theorize about the field here as being mappable by hyperbolic geometry on the Poincare disk. There again is the constant negative curvature, and the apparent 2D shape map has perhaps an infinitesimal extension to 3D being understood from the nature of what's going on. Infinitesimal sections here integrate back to more complex (and I do mean perhaps real + imaginary by complex) sections of field math above. To third derivative: di'''/dAc = [- (1/4)*(1/ (i')^(3/2)]. So, that there is a -1/4 coefficient, which looks to me to dovetail with the Potential Energy derivations from electrostatics that give rise to Bohr's frequency condition. Which is arrived at by applying Coulomb's law I think, and eventually first complexifies to delta E = [{mass(electron)*e^4 / 8e.(permittivity)^2*h^2} {(1/n1^2)-(1/n2^2)}] where n2>n1, which then simplifies to delta E = hv = KE + PE. That, of course, E = (n)hv, is our same result from Planck on blackbody radiation and Einstein on the photoelectric effect. That is then going in to the Hamiltonian's of quantum mechanics. However, I'm hoping to reset the basis of i, the imaginary unit, as a derivative from infinity encompassing all sq. rt. of primes in stead of just i = (-1)^(1/2). Last paragraph largely follows from initial review of Donald A. McQuarrie's, Quantum Chemistry, 2nd Ed., early chapters. Now, to touch base with field again, here at this i''' derivative level, the field again is simplified and I think can be descibed from a Polar graphing plot only really using theta (angle) and r (radian length) to map vectors of forces [or, Quaternions applied throughout has explanatory potential; I know no Quaternion math], which I think of as electrogravitational force from a circle-dot centralization [see the visualization of a wave function under Preliminaries on the wiki for Schrödinger equation: ht tps://en.wikipedia.org/wiki/Schrödinger_equation]. Apparent 2D vector plot is perhaps really 3D, with infinitesimal extension in 3rd because as soon as a movement or extension takes place, a distortion takes place, and a conformer or shape to space-time begins to take shape. Some derivations teased out for assessment: i' = number set of all sq. rt.s of prime no.s i'' = (1/2)*i(-1/2) i''' = -(1/4)*i(-3/2) i'v = (3/8)*i(-5/2) iv = -(15/16)*i(-7/2) ivi = (105/32)*i(-9/2) ivii = -(945/65)*i(-11/2) ... So, please, please, fire away at the bad math (or physics or chemistry) here -- I know it is in need of more work. There is poorly formalized notation here, and I am not sure I am deriving a continuous function, or that one can derive a number set in relation to a field shape or conformer. But, to me, this mathematics related back to the physics/chemistry makes some logical sense, which I think and hope is providential to better understandings. Thank you, NTuft