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Number theory derivation from infinity; speculations on equations that are derived in terms of the Field


NTuft
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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

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37 minutes ago, NTuft said:

So, please, please, fire away at the bad math (or physics or chemistry) here

I wanted to fire away real bad, but I didn't spot any physics. No chemistry either.

And the only maths is wrong. You can't consider increments of a constant.

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4 minutes ago, joigus said:

I wanted to fire away real bad, but I didn't spot any physics. No chemistry either.

And the only maths is wrong. You can't consider increments of a constant.


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. 

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9 minutes ago, NTuft said:

I take it you consider infinity to be the constant?

Who would consider such a stupid thing?

I mean you do, are doing,

\[ \frac{d}{di}\left(\frac{1}{2}i^{-1/2}\right)=-\frac{1}{4}i^{-3/2} \]

So you seem to be considering the imaginary unit, \( i \), as a constant. variable.

That shows that you don't understand what differentiation is about.

9 minutes ago, NTuft said:

More targets now available. 

No target in sight.

Edited by joigus
correction
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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.  

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6 hours ago, NTuft said:

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.

My problem with what you have written is simple.

As Joigus patiently noted,the quantity commonly denoted by the symbol 'i'  is defined as the square root of minus 1.

5 hours ago, NTuft said:

It is mentioned in the discourse that I had an idea that the imaginary unit i could be redefined

If you really want a quantity that is not the square root of -1 then you should start by saying so, defining what you want your quantity to be, and choosing a suitable symbol for it.

Otherwise you will simply confuse your readers, and lead them to ask what other quantities and terms is the person playing fast and loose with ?

Joigus also noted

6 hours ago, joigus said:

So you seem to be considering the imaginary unit, i , as a constant. variable.

That shows that you don't understand what differentiation is about.

Note he did not say that you cannot differentiate a constant (which of course you can if you know what you are doing)

Reading your response I agree with him that you are attempting to misuse the process of differentiation.

 

 

7 hours ago, NTuft said:

Idea of deriving down to i''' from infinity

For your information, you are starting at 'the wrong end'.

Number theory starts at the beginning and uses 'successor theory'.

It does not (can not) start at infinity for the very good reason that infinity is not an end or a number, it is a shorthand way of saying there is no end.

But there is a beginning.

You start by asserting there is a beginning (ie asserting that there is a number called zero, or a number called 1 depending upon your preference)

And then asserting that every number has a successor.

So there must be a successor to your given first number, say 1.

You call this number 2.

This in turn must have a successor, you call 3.

And so on.

Since this process never ends you have defined every number.

But you have not defined or used 'infinity'.

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I should alsao like to point out that what Mathematicians mean by the word 'Field', particularly in relation to number theory, 

and what Physicists mean are quite different.

You should disentangle the Physics and the Mathematics please.

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3 hours ago, studiot said:

You should disentangle the Physics and the Mathematics please.

I agree. This is not to say that physics and mathematics don't have interesting connections. They do, and they surface from time to time. But in the way the OP is dealing with it, I find it almost impossible to fathom what the proposed connection actually is. Again, I'm missing a clear statement of what is supposed to be the conjecture. It's more like a purée of mathematical, physical, and chemical terms.

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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,

17 hours ago, studiot said:

For your information, you are starting at 'the wrong end'.

Number theory starts at the beginning and uses 'successor theory'.

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.

Edited by NTuft
grammar
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25 minutes ago, NTuft said:

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.

I am going to say +1 for your listening attitude here. This is so much better than many who come to lay out their pet cogitations.

:)

I see you are not fuly convinced about the number theory. Nor have you mentioned infinity again.

That is possibly because there are some interesting twists that appear when you study number theory a bit more.

Some of these would probably help you understand more about numbers and infinity.

Do you wish to discuss this aspect some more ?

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14 hours ago, studiot said:

I am going to say +1 for your listening attitude here. This is so much better than many who come to lay out their pet cogitations.

 

Agreed.

I suppose what you wanna do here, @NTuft, is something like considering a complex variable, \( z \), differentiating by it,

\[ \frac{d}{dz}\left(\frac{1}{2}z^{-1/2}\right)=-\frac{1}{4}z^{-3/2} \]

and then substituting,

\[ \left.\frac{d}{dz}\left(\frac{1}{2}z^{-1/2}\right)\right|_{z=i}=\left.-\frac{1}{4}z^{-3/2}\right|_{z=i}=-\frac{1}{4}i^{-3/2} \]

But be careful; you may be re-discovering complex calculus.

The connection to number theory is something I'm missing.

I'm very skeptic to there being a connection between physics/chemistry and number theory. There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems.

Edited by joigus
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Quote

 

"...
That is possibly because there are some interesting twists that appear when you study number theory a bit more.

Some of these would probably help you understand more about numbers and infinity.

Do you wish to discuss this aspect some more ?"

 

@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,

Quote

"I suppose what you wanna do here,..."

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

  

On 8/15/2021 at 12:52 AM, joigus said:

The connection to number theory is something I'm missing.

I'm very skeptic to there being a connection between physics/chemistry and number theory. There is a connection between physics and complex calculus, provided by harmonic functions in two-dimensional problems.

noted^

Edited by NTuft
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  • 7 months later...
Posted (edited)

@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

Edited by NTuft
re-writes, Clarity
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5 hours ago, NTuft said:

@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

Hi. I'm not an expert, but I can tell you what the present status of Cantor's continuum hypothesis is:

Quote

The continuum hypothesis was advanced by Georg Cantor in 1878,[1] and establishing its truth or falsehood is the first of Hilbert's 23 problems presented in 1900. The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent. This independence was proved in 1963 by Paul Cohen, complementing earlier work by Kurt Gödel in 1940.[2]

(My emphasis.)

https://en.wikipedia.org/wiki/Continuum_hypothesis

So, within the Zermelo-Frenkel set-theory axioms, plus the axiom of choice, you can neither prove nor disprove the continuum hypothesis.

And that's what I know. The possibility of a cardinality sandwiched between aleph_0 and aleph_1 is still conjectural.

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On top of what @joigus said so clearly (+ 1),

You have to delve much deeper into the philosophy of Mathematics beofre you can start using symbols such a = ;  + ;  n+1 and so on.

Consider.

Until you have defined what a number is how can you define addition ?

So how can you give meaning to n + 1 in the definition of a number ?

 

Note in Mathematics defining means loosely 'give meaning to' .

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10 hours ago, joigus said:

The answer to this problem is independent of ZFC, so that either the continuum hypothesis or its negation can be added as an axiom to ZFC set theory, with the resulting theory being consistent if and only if ZFC is consistent.

10 hours ago, joigus said:

So, within the Zermelo-Frenkel set-theory axioms, plus the axiom of choice, you can neither prove nor disprove the continuum hypothesis.

{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

Quote

Kurt Godel proved in 1938 that the General Continuum Hypothesis and the Axiom of Choice are consistent with the usual (Zermelo-Fraenkel) axioms of set theory [4]. Twenty-five years later, Paul Cohen established that the negations of the Continuum Hypothesis and the Axiom of Choice are also consistent with these axioms [3]. Taken together, these results tell us that the Continuum Hypothesis and the Axiom of Choice are independent of the Zermelo-Fraenkel axioms.

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:

Quote

ZFC axioms of set theory
(the axioms of Zermelo, Fraenkel, plus the axiom of Choice)

For details see Wikipedia ”Zermelo-Fraenkel set theory”. Note that the descriptions there are quite formal (They need to be, because the goal is to reduce the rest of math to these axioms. So to avoid circular reasoning, you have to state the axioms without using anything you know from the rest of math! That can be done but it does make the precise statements of the axioms too technical for this class).

Since the formal description may be difficult to read at this point, I typed a more informal description here, and added explanations:

 

A1 Axiom of Extensionality.
This Axiom says that two sets are the same if their elements are the same. You can think of this axiom as defining what a set is.

A2 Axiom of Regularity.
I have never seen a set A for which A is an element of A, but that does not imply that such sets do not exist. Do they exist? How would you construct such a set without using a circular construction? Circular definitions are forbidden in math because if you allow them, then it is very easy to run into contradictions. Now about this axiom. This axiom looks quite technical. It implies that a set A cannot cannot be an element of A. Moreover, it also implies that A cannot be an element of an element of A. Also: A cannot be an element of an element of an element of A, etc.

A3 Axiom schema of specification.
If P(x) is a statement (if you plug in a value for x then P(x) becomes either true or false) and if A is a set, then this axiom says that the elements of A that satisfy P also form a set. We denote that set as {x ∈ A | P(x)}. Read that notation as: The set of all x ∈ A for which P(x) is true. The symbol | means: “for which” (some authors use : instead of |).

A4 Axiom of Pairing .
Says that if x, y are sets then {x, y} is also a set. You may wonder: why do we need this axiom? Isn’t this obviously true? Well, the issue is, it doesn’t matter if people feel that this is obviously true, either way, you can not actually prove this statement. The point of the axioms is to explicitly write down all statements we accept without proof, including those that some might feel are “obvious”. Think of it this way, supposed you’ve played chess for many years. Then you might feel that the rules of chess are obviously true. But you can’t mathematically prove that those rules are true. So then why are the rules of chess true? Because chess organizations have decided that those are the rules they want. Likewise, in the game of math, we have settled on certain rules, and these rules are the axioms and the rules of logic.

A5 Axiom of Union.

If A is a set whose elements are again sets, then the union of those sets is again a set.

A6 Axiom of replacement.

Suppose φ(x, y) is a statement such that for every set x there is precisely one set y for which φ(x, y) is true. Denote that y as f(x). Remark: f looks a lot like a function, but we may not yet call f a function. In math you may only call f a function after specifying the domain AND the co-domain. 1 The axiom says that if we apply f to all elements of some set A, so take f(x) for every x ∈ A, then the axiom says that all those f(x) form a set. We denote that set as {f(x) | x ∈ A}.

A7 Axiom of infinity. Says that there exists an infinite set. Note: It does not matter which infinite set, as long as we have at least one infinite set then the other axioms allow us to construct our favorite infinite set, which is N = {0, 1, 2, 3, . . .}. You may wonder, why do we need an axiom that tells us that an infinite set exists, when we’ve already known about {0, 1, 2, 3, ....} for a long time? The issue here is: How do you define the . . . in the notation {0, 1, 2, 3, . . .} without using circular reasoning? The only thing we can do here is to accept the unprovable statement that there exists at least one infinite set, and then work from there. So wait, any time we need an unproven statement, we can simply call it an axiom? No, because other mathematicians won’t accept additional axioms. Although the current axioms are not proven, they are supported by good evidence. They have been thoroughly put to the test for at least a century. It is outside of the scope of this course to detail the many ways in which the axioms have been investigated, but the upshot is that mathematicians are very confident that the standard axioms (called ZFC), combined with the rules of logic, do not lead to errors. Mathematicians are unlikely to accept more axioms; we do not need more axioms, and we are confident about the ones we have.

A8 Axiom of the Power set.

If A is a set then P(A) is a set.

A9 Axiom of Choice. (AC) Chapter 3 [?] proves that this axiom is equivalent to several other statements:

1) Zorn’s lemma

2) The well-ordering theorem

3) For any cardinals d, e we have d ≤ e or e ≤ d.

4) For any sets A, B, if there is a surjective function from A to B then there is an injective function from B to A.

If you replaced AC by one of these four statements, then ZFC set theory stays the same. The axiom of choice, says that if A is a set whose elements are non-empty sets, then one can pick an element from each of these non-empty sets. This sounds harmless, however, if A is an infinite set, then we have to choose one element from infinitely many sets. How do you make infinitely many choices? Even though we may not be able to do this ourselves, AC says that there does exist a function that makes all these choices. In case you think that this is fishy, that is not unreasonable, but consider this: One can prove (using only A1-A8) that if A1-A9 is contradictory, then A1-A8 must also be contradictory. So if you want to prove that something in A1-A9 is wrong, then you also have to prove that something in A1-A8 is wrong (remember from the comments under A7 that the axioms have been put to the test in many ways).

 

There you have it, a full list of all statements that mathematicians accept without proof. Any statement other than A1-A9 will only be accepted by other mathematicians if it has a proof that only uses the rules of logic and the axioms A1-A9. (The use of other statements is allowed provided that they can be proved from the rules of logic and the axioms.) Proved statements are called theorems, and mathematicians trust them even when they are counter-intuitive (like the existence of infinite sets with different cardinalities!). Conversely, other than A1-A9, mathematicians don’t accept statements that don’t have a proof, no matter how plausible they may sound.

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

9 hours ago, studiot said:

On top of what @joigus said so clearly (+ 1),

You have to delve much deeper into the philosophy of Mathematics beofre you can start using symbols such a = ;  + ;  n+1 and so on.

Consider.

Until you have defined what a number is how can you define addition ?

So how can you give meaning to n + 1 in the definition of a number ?

 

Note in Mathematics defining means loosely 'give meaning to' .

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.

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1 hour ago, NTuft said:

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.

You are going to be a lot more precise to make any headway.

What is "the Natural/Real set" ?

And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

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Posted (edited)
24 minutes ago, studiot said:

You are going to be a lot more precise to make any headway.

What is "the Natural/Real set" ?

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

Quote

And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

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 = aleph_0, and the ℝ ≥ aleph_2 

Edited by NTuft
tooting my own horn
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On 4/19/2022 at 1:54 PM, studiot said:

And what properties do the members of 'the set of square roots of prime numbers enjoy' that have already been proven ?

They are algebraic numbers?

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  • 2 weeks later...
46 minutes ago, NTuft said:

Physical Meaning of Imaginary Unit i

https://shpenkov.com/pdf/ImaginUnitEng.pdf

 

I think this paper is worth the read (9 pgs.), and has some bearing on the relation here between imaginary unit in math, physics of QM, and chemistry.

Thanks

Successfully solving a problem that does not exist to start with. The solution is beautifully clear:

"the mathematical reflection of the dialectical law of affirmation-negation for qualitatively opposite binary judgments about the nature of any object or process."

"the "imaginary" unit i is a symbol for a qualitatively polar opposite essence (property, number, parameter) obeying to the polar opposite (“negative”) algebra of signs relative to the conventional (“positive”) algebra."

In conclusion,

"we believe that the word "imaginary" as not reflecting reality will eventually be removed from mathematics and physics, harmonizing the mathematical structures with the laws of the Universe, and thus expanding the horizons of knowledge."

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Posted (edited)

Communications to Arkadiusz Jadcyzk:

 

8-9-21-UFTwritten-DM.jpg

2ndPageProof-1.jpg

2ndPageProof-2.jpg

 

Special thanks also to Donald A. McQuarrie, George P. Shpenkov, et al.

Edited by NTuft
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Posted (edited)

2x e-mails:

Re: H2+ and the breakdown of the virial theorem <V>/<T> = -2

Hello,

So the average kinetic and potential energy, <T> and <K>, resp., give us <T> + <V> = <E> , and for the 1s hydrogen atom we have <V>/<T>=-2 properly. (pg. 3321)

Then, even at our simplest nucleation, H2+, we have a discrepancy:

[...] paraphrasing pg.5181, the virial theorem starts to move away from -2 with the H2+ molecule, and already needs "optimization" to fit [...]

even though, "...[the virial theorem] is generally true for any system in which the potential energy is coulombic. Equation 7.25 is an example of the virial theorem and is valid for all atoms and molecules." (pg. 3321)[bold added for emphasis]

 

[...] to the issue: could it be that we are miscalculating the Potential Energy because we have reduced what should be a set of values to a constant? It works assuming the 1s electron energy, but something has been lost somewhere where we can't scale up, in "normalization", or in an assumption implicit in Ψ*Ψ to Ψ^2 ; not accounting for what could be hidden in the imaginary unit?

What do you think about why the math breaks down at the simplest nucleation, or getting into a 2 electron system, even? I don't know.
--------------------------

 

 

Re: Addendum on last, cite source. Plus: are we cheating the harmonic oscillator's power series somehow with the "operator method"?

We have : 
"The Schrödinger equation for a harmonic operator [oscillator? it says operator] is


-(hbar^2/2mu)*d''psi/dx'' + (1/2)kx^2psi = E*psi


As we mentioned in Section 5.5, this is a linear differential equation in psi(x), but it does not have constant coefficients. A standard method to solve such equations is to assume that psi(x) can be expressed as a power series in x:

psi(x) =n=0a(sub(n))xn

We substitute this expression into the differential equation and obtain a set of algebraic equations for the a(sub(n)). This method is called the power series method, and is at best tedious. There is an alternative method of solving equation 1 that is based on operator methods, which is not only much easier than the power series method, but is also much more elegant. In this appendix, we shall use operator methods to determine the eigenvalues and eigenfunctions of a harmonic oscillator. Operator methods are used frequently in quantum mechanics and they are well worth assimilating."

/\ = "hat"

"The Hamiltonian operator of a harmonic oscillator is given by

/\H = - (hbar^2)/2mu * d''/dx'' + (1/2)kx^2 =/\P^2/2mu + (k/2/\X^2)

where k=mu*omega^2 is the force constant and /\P and /\X are the momentum and position operators, respectively. If we introduce new operators

/\p = (mu*hbar*omega)^(1/2)/\P    and   /\x = (mu*omega/hbar)^(1/2)/\X

then the Hamiltonian operator can be written as

/\H = [(/\P^2)/2*mu] + (k/2)/\X^2 = [(hbar*omega)/2]*(/\p^2 + /\x^2)
 

Using the fact that [/\P, /\X]= -i * hbar , we have

[/\p, /\x]=/\p/\x - /\x/\p = (1/hbar)*(/\P/\X-/\X/\P = (1/hbar)(-i*hbar) = -1 

We now define the (not necessarily Hermitian) operators /\a+ and /\a_ by


/\a+ = (1/2^(1/2))*(/\x-i/\p)   and /\a_ = (1/2^(1/2))*(/\x+i/\p)

Using these definitions, we have

/\a_/\a+ = (1/2)(/\x+i/\p)(/\x-i/\p) = (1/2)[/\x^2 + i(/\p/\x - /\x/\p) + /\p^2]

= (1/2)(/\x^2+/\p^2 +1) [6]

Similarly,

/\a+/\a_=(1/2)(/\p^2+/\x^2 - 1) [7]

Equations [6] and [7] give us

/\a+/\a_ - /\a_/\a+ = [/\a+, /\a_] = -1 [?...]

Note that the Hamiltonian operator, equation 3, can be written as

/\H=(hbar*omega)*(/\a+/\a_+(1/2)) [9]

To make equation 9 more transparent, we denote the operator /\a+/\a_ by /\v and write equation 9 as

/\H = hbar*omega (/\v + (1/2))" "(pgs. 239-2411)

...
This harmonic oscillator WORKS , but if something got left out in simplification or normalization could that explain why when we try to scale up (continuing to use the simplified Hamiltonian operators methods results) we get inaccurate theoretical vs. experimental energy values?

 @joigus I think there is a "physics" target here, at last.

1McQuarrie, Donald A. (Donald Allan). Quantum Chemisty. 2nd ed., University Science Books, 2008.

 

 

"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" QED unless someone can put a hole in it.

Edited by NTuft
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Posted (edited)
11 hours ago, NTuft said:

even though, "...[the virial theorem] is generally true for any system in which the potential energy is coulombic. Equation 7.25 is an example of the virial theorem and is valid for all atoms and molecules." (pg. 3321)[bold added for emphasis]

I'm not sure what relevance this has on Cantor's hypothesis, but the virial theorem is actually more general: True for any potential that is homogeneous in the spatial coordinates:

\[ V\left(\lambda\boldsymbol{x}\right)=\lambda^{n}V\left(\boldsymbol{x}\right) \]

So ideal gases, harmonic oscillators, and inverse-square systems all have a version of it.  

Sorry. I mistook this thread for another one. Oops!

Edited by joigus
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