Wikipedia:Reference desk/Archives/Mathematics/2016 September 7

= September 7 =

Model theory:
Let S be an infinite countable set of axioms such that, for every finite natural n - the n-th axiom states that v > n.

Let X be the set $$\Bigl\{x\Bigg|\exists(a\in\mathbb N_1,b\in\mathbb N_0,c\in\mathbb N_0)(ax=bv+c \lor ax+c=bv)\Bigr\}$$.

Does PA $$\cup$$ S have a model satisfying, for every x∈X in the model, that every m,n in the model - which satisfy m+n=x - satisfy m∈X and n∈X?

HOTmag (talk) 06:39, 7 September 2016 (UTC)
 * To be clear, in your definition of X, the N refers to the genuine (finite) naturals? Or do you just mean that a, b, and c have to be elements of the model?
 * If by N you mean the finite naturals, then I think the answer is no. For an element x to be in X as witnessed by a, b, c, either b is zero or it is not.  If b is zero, then x is finite.  If b is nonzero, then x is neither larger than v by an infinite factor, nor smaller than v by an infinite factor.
 * Well, that is, unless a, b, and c are all zero, but that trivializes the definition of X, so I'm going to assume you're excluding that case.
 * So if you look at, say, the square root of v (rounded down), then that is not finite, but it is smaller than v by an infinite factor, so it cannot be in X. --Trovatore (talk) 06:47, 7 September 2016 (UTC)
 * Yes: by $$\mathbb N$$, I mean the set of finite naturals (Thanks to your comments I've just made it clear in my original post). However, I can't see why you think the square root of v (rounded down) must be in the model. If it's not in the model, nor is any non-natural power of v, then my question arises again: Is there such a model satisfying my requirement? HOTmag (talk) 07:34, 7 September 2016 (UTC)
 * It follows from PA that every number has a square root (in the sense that for any m, there is a greatest n such that n2&le;m). --Trovatore (talk) 07:45, 7 September 2016 (UTC)
 * Thanks. HOTmag (talk) 07:50, 7 September 2016 (UTC)

Consequences of a model hypothesis
What are the logical consequences or implications of the hypothesis of equal mass density of the proton and electron?--82.137.11.134 (talk) 13:35, 7 September 2016 (UTC)

($$\frac{m_p}{r_p^3} = \frac{m_e}{r_e^3}$$ where mp, me mass of proton and electron, and rp, re radiuses.)--82.137.12.96 (talk) 10:37, 8 September 2016 (UTC)
 * You would be better off on the science reference desk with questions like this. At most mathematics can calulate the change or if there is some inconsistency if one tweaks some model of nature. In this case the tweaking would have to be severe because currently electrons are not considered to have any internal structure or extent and density is not an applicable concept. The first logical consequence if one can call it that with modern physics is that electrons would have to be composed of other particles. Dmcq (talk) 15:04, 7 September 2016 (UTC)
 * ...which they are, but your point stands that "density" is an inapplicable macroscopic concept. Tigraan Click here to contact me
 * Tigraan, quarks compose hadrons, such as protons and neutrons, but not electrons. Rojomoke (talk) 04:46, 8 September 2016 (UTC)
 * I'm mainly interested (as I've said below) in calculations and the analysis of mathematical reasoning steps of formulae derivation and less in conventional standard assumptions in modern physics about the structure of electrons. You say that it follows the necessity of electron being composed of other particles. How does this follow, which are the steps of reaching this conclusion? Please deploy the reasoning!
 * You also mention the word inconsistency. Is there an inconsistency (with perhaps modern physics assumptions) arising from this working hypothesis of equality of mass density understood as the distribution of mass of the proton over the geometric volume of proton or electron taking into consideration the proton radius as determined by electron scattering. What are the implications of this equality hypothesis for electron radius as reasoning steps?--82.137.14.50 (talk) 22:30, 7 September 2016 (UTC)
 * Take this to the science desk. It is not a maths question. That proton radius comes from it being composed of quarks like an atom has a size because its structure. Dmcq (talk) 09:13, 8 September 2016 (UTC)
 * Could you ignore the quarks structure presupposition and explore the mathematical reasoning based on the initial hypothesis? It has nothing to do with exploring the consequences of the proposed working hypothesis as a mathematical proposition!--82.137.12.96 (talk) 09:31, 8 September 2016 (UTC)
 * PS: I don't quite understand why do you insist in saying that this not a math question when I said the main interest is in mathematical formulae derivation and analysis!? Have I not expressed clearly the mathematical aspect of this posted question? Is it somehow due to the fact that I haven't attached a tex formula to the initial statement of the question?--82.137.12.96 (talk) 09:40, 8 September 2016 (UTC)

Your question is awfully short on detail. If it referred to a standard mathematical calculation that happens to take place in physical models, people here could go to the physics textbooks and find the details of that calculation; but since you use concepts widely out of the main path ("mass density" for microscopic particles), I (for one) have no idea what you have in mind. What "mathematical formulae" are you referring to? Tigraan Click here to contact me 09:49, 8 September 2016 (UTC)
 * Yes, I provide some details. It seems that context details of this question involve another related question: what are the mathematical bases of formulation of the concept of quarks? As I understand, the quarks have come into existence starting from analysis of electron scattering experiments cross-section results on protons involving some scattering formula based on Rutherford scattering equation modified by Marshall Rosenbluth. (It is possible that, to my (limited) knowledge, calculation details not be found in some standard textbooks, being considered an advanced topic, and mostly primary research articles address the calculation details).
 * The initial formula that I should have added is about rendering of words regarding the equality of mass (of proton and electron) distributed on each particle volume calculated based on radiuses of these particles. From this (geometric-like) equality one can get an estimation of the (charge) radius of electron knowing the determined radius of the proton based on electron scattering. I'll insert the equation.--82.137.12.96 (talk) 10:21, 8 September 2016 (UTC)
 * The formula associated to initial hypothesis to be analyzed is:
 * $$\frac{m_p}{r_p^3} = \frac{m_e}{r_e^3}$$ where mp, me mass of proton and electron, and rp, re radiuses. A common factor 4/3π that multiplies the cube of radiuses disapears from both sides of equality. --82.137.12.96 (talk) 10:28, 8 September 2016 (UTC)
 * For additional context about the concept of densities applied to microscopic particles the page electron magnetic moment could be considered to analyze reasoning connections.--82.137.12.96 (talk) 11:06, 8 September 2016 (UTC)

Change of orbital parameters of Moon
What is the change of the orbital motion of Moon during its gradual distancing from Earth? Does this distancing occur by a spiral trajectory? What is the time needed for the Moon to completely exit Earth orbit and how is it calculated? Thanks.--82.137.13.100 (talk) 14:56, 7 September 2016 (UTC)
 * Please put such questions on the science reference desk. The moon would never escape the earth and anyway they're both due to be burnt up by the sun in another five thousand million years or so. Have a look at the article about the moon. Dmcq (talk) 15:08, 7 September 2016 (UTC)


 * I support Dmcq's suggestion to ask at Reference desk/Science, but maybe Lunar_Laser_Ranging_experiment could help? Tigraan Click here to contact me 15:15, 7 September 2016 (UTC)
 * I see there in the suggested link it says that the Moon is spiraling away with an anomalously high speed. One aspect is certain that the Moon is spiraling. I'm more interested in the equations of spiraling and their derivation within the frame of celestial mechanics as a mathematical discipline, that's way I've posted the request here and not at Science Refdesk. Of course the aspect related to the physical causes of spiraling can be asked at Science Refdesk to get a complementary view.--82.137.14.50 (talk) 21:59, 7 September 2016 (UTC)
 * As a rule of thumb, if the question is phrased in terms of physics and the real world, it should be in WP:RD/S. If it's phrased in terms of mathematical equations, it should be here on WP:RD/Ma (even if those equations were inspired by a physics problem). As it stands the question requires a lot of non-mathematical domain knowledge to answer, and as such this is not the best place for it. If you were able to present the relevant model and only needed our help solving it, it would be a different matter. The people at WP:RD/S should be able to help you model this properly and solve the model - physicists aren't strangers to mathematical derivations, you know. -- Meni Rosenfeld (talk) 11:48, 8 September 2016 (UTC)
 * OK. Then I rephrase the question to focus on mathematical modelling aspects leaving the real Moon aspects to the other Refdesk(Science). To this purspose let's consider a (fictitious astronomical) central body with an orbiting satelite. How is, in this fictionalized example, the time required to get a certain degree of modification change of a given the elliptic orbit (semi-axes a and b) by spiraling calculated? More geometrically, what is the (kinematic) overlap between an ellipse and a spiral?--82.137.12.230 (talk) 18:50, 9 September 2016 (UTC)
 * See Tidal locking. Notice the factor Q in the formula which is poorly known, depends on the particular bodies and changes with time. This is a sign this is a physics question not a maths question. Dmcq (talk) 20:27, 9 September 2016 (UTC)
 * Interesting formula. Any indication of how has it been established or derived? (semitheoretically or experimentally) Could someone access the full text of Icarus (journal) article and extract some info?--82.137.12.230 (talk) 20:43, 9 September 2016 (UTC)
 * Try Googling on suitable search terms. For instance I just stuck in 'Tidally locked satellites' and got which probably gives what you want. Questions on it should be addressed to the science reference desk. Mathematical modelling of questions like this comes under physics. As Meni Rosenfeld said above they will have absolutely no problem with the mathematics of things like this and a lot of far more difficult maths too. Dmcq (talk) 22:27, 9 September 2016 (UTC)


 * See the last paragraph of Orbit of the Moon. -- ToE 00:53, 8 September 2016 (UTC)