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April 22

How are arbitrary hemispheres defined on theWGS84 ellipsoid?

With spheres there's only 1 right answer, there's also formulas that can be used to find points 0.5π radians from another point, is there a formula for the line with half the ellipsoid surface area on each side? Is there a formula for the line where an infinitely far Star of Bethlehem and an anti-Star of Bethlehem at the other end of the ellipsoid normal line would have equal zenith distances ignoring refraction and geoids? Are these lines the same? How far apart can they be? Sagittarian Milky Way (talk) 18:30, 22 April 2024 (UTC)[reply]

Unless an ellipsoid is a sphere (which the WGS 84 reference ellipsoid is not), no portion of it is a mathematical hemisphere. Any plane through its centre divides it though into two equal (congruent) parts. Usually the plane will be a meridional or the equatorial plane. In more general geodetic systems the equator and meridians, although not ellipses, also lie in a plane and can be used for a fairly fair cutting into two parts, which however will normally not be congruent. Calling the two parts "hemispheres", although not correct in a strictly mathematical sense, is nevertheless conventional. --Lambiam 19:11, 22 April 2024 (UTC)[reply]

Yes but that's the easy way out, hemispheres centered on the equator or pole are exactly zero percent of all possible centers. Sagittarian Milky Way (talk) 21:56, 22 April 2024 (UTC)[reply]

Is division by any plane through the centre not general enough?

The sight lines to a point on the celestial sphere and to its celestial opposite are parallel. So are the directions to the respective zeniths from a given place on the ellipsoid and its antipodal place. Therefore the angular distances are the same. --Lambiam 11:03, 23 April 2024 (UTC)[reply]

I don't know how to do the center plane. Either finding points of surface tangency from the point of surface perpendicularity or the point of surface perpendicularity from 2 surface points on the plane. There seems to be a u and a v involved I keep seeing u and v but don't know what that is, or if that's needed when the plane is not arbitrary but has 1 of 3 defining points fixed to the ellipsoid center. I stupidly dropped out before learning u, v and pseudo-delta swirl. Sagittarian Milky Way (talk) 22:31, 23 April 2024 (UTC)[reply]

If they are boldfaced u and v, these variables probably stand for some 3D-vector (

x, y, z

). --Lambiam 16:51, 24 April 2024 (UTC)[reply]

Likely two such vectors orthogonal to each other. —Tamfang (talk) 02:57, 1 May 2024 (UTC)[reply]

I must admit to being intrigued by the idea that the Star of Bethlehem might have been arranged by angels using WGS 84! However as far as I can see there's just ocean at the antipodal point for the birth of Damien Thorn. Pehaps we're safe for a while yet ;) NadVolum (talk) 19:42, 22 April 2024 (UTC)[reply]

If one wanted to start a rival franchise, one could look for holy sites (holy to some cult) in the dark patches of this map. —Tamfang (talk) 02:52, 1 May 2024 (UTC)[reply]

I just want to make orthographic projections, not start a religion. Or equivalently, find star of Bethlehem umbras for anywhere like a lone star over Texas. I'm not sure I didn't make a mistake interpreting the orthographic projection formulae I found. Sagittarian Milky Way (talk) 04:58, 3 May 2024 (UTC)[reply]

I meant a movie series based on some other existing religion's holy site. —Tamfang (talk) 05:15, 4 May 2024 (UTC)[reply]

Ah I see. The Left Behind Antichrist was not born in the southernmost part of the 145°E area of South Pacific archipelago either for some reason (he was born in what is now the EU), and he wasn't the most incompetent sentient on Earth at carpentry, so there should be room for at least a 3rd franchise where he is. There's many Dracula films after all. For some religions an Antichrist-type demon would be extracanonical, I suppose someone could make a film where a boy never follows the middle path, always sleeping and staying awake twice the normal hours for instance, and celibacy then BDSM orgies. Sagittarian Milky Way (talk) 14:54, 4 May 2024 (UTC)[reply]

Or praying to anti-Makkah belly-up in tongues but someone gets the idea to tape-record and play backwards and it's the correct prayers in perfect Arabic pronunciation when he only knows language(s) French Polynesia teaches in school. Someone might try to murder the filmmaker though. Sagittarian Milky Way (talk) 15:11, 4 May 2024 (UTC)[reply]

The anti-Messiah would presumably try to eat as little Kashrut as possible (consuming dairy as close in time as possible if fed beef or poultry), walk at least 1 mile per Shabbat and do all weekend homework then (when it's also Shabbat in Jerusalem), and try to cause Holocaust II. Sagittarian Milky Way (talk) 16:19, 4 May 2024 (UTC)[reply]

The normal on the ellipsoidal through Bethlehem won't go through the center of the Earth and so won't go though the antipodal point. NadVolum (talk) 20:04, 22 April 2024 (UTC)[reply]

It doesn't really matter though cause the idealized celestial sphere/astronomical coordinate system is infinitely far, the lines to the star from anywhere on Earth would be parallel. It would matter for the "ranking all points by distance and picking the nearer half" way as an extremely flattened ellipse could have the (geographic, not geocentric) latitude minus 90 be only a few miles away (plus 90 in the southern hemisphere) Sagittarian Milky Way (talk) 22:15, 22 April 2024 (UTC)[reply]

April 23

Fibonacci numbers and pineapples

The article on Fibonacci numbers mentions the pineapple, but I am unable to discover when, in the literature, this was first discussed. Later in the aforementioned article it states, "In 1830, K. F. Schimper and A. Braun discovered that the parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers." Given that the discussion of pineapples was extremely popular at this time in the early 19th century, one would expect it to be found within that time frame. However, I cannot find anything until the mid to late 20th century, possibly starting with Onderdonk 1970. Does anyone know when Fibonacci numbers were first discussed in reference to the pineapple, and if it was before the 20th century? As it stands, 1830 would fit absolutely perfectly into the pineapple timeline I'm working on, but I can find no supporting evidence for this idea. Viriditas (talk) 23:17, 23 April 2024 (UTC)[reply]

Pineapples are mentioned as an example in a long list of diverse plant species. In other words there's nothing specifically notable about pineapples in relation to Fibonacci numbers. Further on the article also mentions daises and the image shows a chamomile. I think the reason pineapples are mentioned is because someone was able to find a citation for them, and perhaps also because many people can find them in their local supermarket. I remember counting rows on a teasel flower head and coming up with Fibonacci related numbers, but I doubt it's mentioned a lot in the literature. It seems to be a general property of Phyllotaxis, or the way plants grow, though there are exceptions, and the "Repeating spiral" section of that article mentions more about Fibonacci numbers. The article mentions Kepler having pointed out the presence of Fibonacci numbers in nature; that was 400 years ago. I don't know if Kepler ever saw a pineapple though. --RDBury (talk) 02:02, 24 April 2024 (UTC)[reply]

Yes, I’m aware of that. What’s notable in this context is that pineapples were introduced to Europe and it led to a great deal of interest. I’m tying to trace the discussion of pineapples throughout each discipline as it arose within a specific 150 year time frame of interest before cultivation and mechanization led to wider availability of the fruit. One of the reasons so many different disciplines discussed pineapples is because they were considered new, difficult to impossible to grow in cold climates, and didn’t have a previous known history in Europe, giving rise to people in different fields using them as examples in their domain-specific literature. It would be kind of like talking about the Internet in your field of expertise in the 1990s. It was somewhat new and different for the general public and people were trying to apply it to their knowledge base. For example, both Leibniz and Locke wrote about pineapples in the context of philosophy because it was considered unique in taste and unobtainable to the common person due to cost, so it represented an idealized version of an idea that they could use in their work and would attract attention. My post on this topic pertains to pineapple within this timeframe, of which the year 1830 fits. It was at the time, coincidentally, that discussion about Fibonacci numbers and plants arose. My question is whether pineapples were discussed in this context at this time and used as an example, not whether it is of any importance to the math itself. Viriditas (talk) 02:17, 24 April 2024 (UTC)[reply]

The botanists credited with the discovery were Karl Friedrich Schimper and Alexander Braun. Schimper appears to have been the first (in 1830) to describe an observed phyllotactic pattern in term of the Fibonacci sequence (to wit, for the rotational angle between leaves in a stem) and his friend Braun described the next year a Fibonacci pattern in pine cones. I found no evidence these gentlemen or any other 19th-century scientists ever studied the patterns of pineapples. As reported here, the number of spiral rows of fruitlets in pineapples was studied "as early as 1933" by Linford,^[1] however, without referencing the Fibonacci sequence. --Lambiam 16:45, 24 April 2024 (UTC)[reply]

Do you know where I can read their original work? I assume it is in German somewhere? It turns out that the history of the word "pineapple" has a lot of of confusion. "Pineapple" once referred to pine cones in English, while other languages used variations on "ananas" for pineapple, such as German. Viriditas (talk) 20:36, 24 April 2024 (UTC)[reply]

@Lambiam: I just found something interesting. This is unlikely to be true, but there is an implausible chance that Schimper & Braun were mistranslated: "Two names will exist side by side, and only after a time will one gain the upper hand of the other. Thus when the pineapple was introduced into England, it brought with it the name of 'ananas,' erroneously 'anana,' under which last form it is celebrated by Thomson in his Seasons. This name has been nearly or quite superseded by 'pineapple,' manifestly suggested by the likeness of the new fruit to the cone of the pine. It is not a very happy formation; for it is not likeness, but identity, which 'pineapple' suggests, and it gives some excuse to an error, which up to a very late day ran through all German-English and French-English dictionaries; I know not whether even now it has even disappeared. In all of these 'pineapple' is rendered though it signified not the anana, but this cone of the pine; and not very long ago, the Journal des Débats made some uncomplimentary observations on the voracity of the English, who could wind up a Lord Mayor's banquet with fir-cones for dessert." (On the Study of Words, Richard Chenevix Trench, 1893, p. 254.) Viriditas (talk) 20:51, 24 April 2024 (UTC)[reply]

Schimper 1830^[2] and Braun 1831^[3] are independently written articles. I have not looked at Schimper's article, but as described he only discusses the placement of leaves. Braun being an accomplished botanist, he would not have used the term Tannenzapfen (pine cone) as an ambiguous name for the fruit of Ananas comosus, and his article furthermore identifies specific species or at least genera of conifers (Weisstanne = Abies alba; Lerchen = Larix; Rothtanne = Picea abies) whose cones he studied. --Lambiam 05:42, 25 April 2024 (UTC)[reply]

Thank you. This is even more confusing given that pine cones were also called pineapples. Viriditas (talk) 20:24, 26 April 2024 (UTC)[reply]

top of a pine cone

I just uploaded this photo of the top of a pine cone. Can anyone get the Fibonacci numbers from it? I've seen drawings were they show the Fibonacci numbers, but they may be a little idealized, or maybe they had a better example. Bubba73 ^{You talkin' to me?} 06:36, 27 April 2024 (UTC)[reply]

The photo looks good to me. The two numbers in the sequence are 8 (spirals going right), 13 (spirals going left), from 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55... Viriditas (talk) 07:12, 27 April 2024 (UTC)[reply]

It is really hard for me to count around the spirals. Bubba73 ^{You talkin' to me?} 23:55, 27 April 2024 (UTC)[reply]

I understand. What I do to help me focus is to open the image in full screen mode. Then, I place the index finger of my left hand on whatever spiral I designate as #1. Keeping my left finger on the screen on the location of the first spiral, I then take the index finger of my right, and use that as a pointer, so when I eventually end up back at the first position, I don't lose the count, which is how I get 8. Then I do it backwards, resulting in 13. My vision is very poor, so this is the only way I can keep track of the spirals. Viriditas (talk) 00:06, 28 April 2024 (UTC)[reply]

References

^ Linford, M. B., "Fruit quality studies II. Eye number and eye weight". Pineapple Quarterly 3, pp. 185–188 (1933).
^ Schimper, K. F. "Beschreibung des Symphytum Zeyheri und seiner zwei deutschen verwandten der S. bulbosum Schimper und S. tuberosum Jacq.". Magazin für Pharmacie 28, 3–49 (1829); 29, 1–71 (1830).
^ Braun, A. "Vergleichende Untersuchung über die Ordnung der Schuppen an den Tannenzapfen als Einleitung zur Untersuchung der Blattstellung". Nov. Acta Ac. CLC 15, 195–402 (1831).

April 26

duality vs. conjugacy

I noticed that Isbell conjugacy and Isbell duality have the same meaning. So, I would like to know the difference in meaning between duality and conjugacy in mathematics. Also, I found Category:Duality theories, but what is the field of mathematics called Duality theory? However, since Baez (2022) said that the Isbell conjugacy is an adjoints rather than a duality of the category, so I changed the category to Category:Adjoint functors. Thank you, SilverMatsu (talk) 03:54, 26 April 2024 (UTC)[reply]

The original duality occurs in projective geometry, see Duality (projective geometry). At some point people noticed that the axioms of the projective plane where the same (or equivalent) if you swapped the undefined terms "point" and "line". So any theorem in projective geometry can be transformed to a dual theorem by changing the roles of points and lines. The new theorem may simply be a restatement of the original theorem as in Desargues's theorem, but sometimes it's not as in Pappus's hexagon theorem. The result is that you often get two theorems for the price of one proof. You can define a dual category for a given category by reversing the arrows, but category theory was invented long after projective geometry so that's not the original meaning. You can also define the dual curve of a plane curve, the dual space of a vector space, the dual polytope of a polytope, etc. As far as I know there is no all-encompassing "theory of duality", just the custom of using "dual" to describe when mathematical objects seem to occur in pairs in some way. Calling something a dual usually implies that the dual of the dual is in some way identifiable with the original object, but this is not always required. For example the dual of a dual vector space is not identifiable with the original vector space unless it's finite dimensional. Duality does not always exist, for example there doesn't seem to be a useful concept for the dual of a finite group, though you can define one for abelian groups. And sometimes there is a duality that's not called that, for example cohomology can be viewed as the dual of homology. I don't think there is a formal distinction between a "dual" and a "conjugate", but usually a conjugate is the the result of applying an automorphism of order two. For example a complex conjugate is the result of applying the automorphism a+bi → a-bi. Again, this is more of a naming custom than a formal mathematical concept, and there is (apparently) some overlap. I'd say a "conjugate" is usually used when the two objects live inside the same structure, and "dual" is used when you're talking about two different structures. For example the dual of a plane curve lives in the dual of the plane in which the original curve lives. Category theory blurs the distinction between an object and a structure so I can see how the distinction is rather meaningless there. --RDBury (talk) 07:12, 26 April 2024 (UTC)[reply]

Thank you for teaching me so kindly. I'm going to re-read some of the references, keeping in mind what you've taught me. --SilverMatsu (talk) 16:02, 27 April 2024 (UTC)[reply]

Oblate spheroid

I thought of Googling orthographic projection ellipsoid and found these, did I interpret everything right?:

ν=a/(1-e²*(sinΦ)²)^1/2 (why not square root?) x=(ν+h)*cosΦ*cosλ y=(ν+h)*cosΦ*sinλ z=(ν*(1-e²)+h)*sinΦ This seems to be a simple spherical to Cartesian converter with latitudes (Φ) "massaged" so it's not slightly wrong (eccentricity²=0.00669437999014 so not much massaging). Then they convert that to topocentric Cartesian with a matrix I can't solve (now I know why galactic Cartesian's UVW!) but it seems like they also say surface points are U=ν*cosφ*sin(λ-λ_O) V=ν*(sinφ*cosφ_O-cosφ*sinφ_O*cos(λ-λ_O))+e²*(ν_O*sinφ_O-ν*sinφ)cosφ_O where _O means "of the topocentric origin". Did I get that right? If so then I can set an initial guess point at or about 0.25 circumference from the W-axis, use the formulae to find its U and V in "W-axis place"-centered coordinates and the test point is of course √(U²+V²) meters from the W-axis and the part of the ellipsoid with the most meters without being too far from the W-axis-test point plane is the limb of the Earth from infinite distance. The worst-case scenario for how spindly a pie the test points have to be in would be looking at ~the 45th parallel limb with the W-axis in the equator plane. The geocenter depth increases roughly quarter mile from 45.5N to 44.5N so ~56 meters poleward shortens the limb to W-axis line segment by 8 inches which is how much Earth curves in a mile. Azimuth accuracy needed increases "exponentially" with limb coordinate accuracy desired though so 10 miles accuracy would be a lot more than 10x easier than 1 mile. Sagittarian Milky Way (talk) 06:24, 26 April 2024 (UTC)[reply]

April 27

"Distribution diagrams"

I'm trying to show the distribution characteristics of a numerical value in a (finite) population. To do so, I sort the values in ascending order, and then plot the feature values over the position of the value in the sorted sequence, as per the attached example. I'm probably not the first with that idea - is there a standard name for this kind of diagram? And/or is there a better way to visualise such data? --Stephan Schulz (talk) 13:36, 27 April 2024 (UTC)[reply]

If you switch axes (or turn your head sideways) this is the graph of a typical Cumulative distribution function. Perhaps it's better to call it a cumulative frequency instead of a distribution since you're plotting values observed and not the theoretical probability density, but the idea is the same. In particular, your graph resembles the second image shown in the article only turned sideways. The (usual) probability density is simply the derivative of the cumulative distribution function, so if you can estimate the derivative in your diagram that may give a better visual representation. The usual technique is to divide the range in to intervals, and then graph the number of occurrences in each interval. It seems to me that there might be a name in economics for the "sideways" version (enonomists seem to do a lot of things sideways), but I don't know what it would be. --RDBury (talk) 14:41, 27 April 2024 (UTC)[reply]

It is also customary, when plotting a cumulative distribution, to let the (now vertical) axis mark relative values in the range from 0 to 1 (or, equivalently and perhaps more commonly, from 0% to 100%) instead of an absolute ranking like from 1 to 7794 or whatever the sample size may be. --Lambiam 15:06, 27 April 2024 (UTC)[reply]

May 3

Can Carmichael number be Lucas-Carmichael number?

Also, varying the signs, there are four different sequences for similar numbers:

squarefree composite numbers k such that p | k => p-1 | k-1
squarefree composite numbers k such that p | k => p-1 | k+1
squarefree composite numbers k such that p | k => p+1 | k-1
squarefree composite numbers k such that p | k => p+1 | k+1

the 1st sequence is Carmichael numbers, and the 4th sequence is Lucas-Carmichael numbers, but what are the 2nd sequence and the 3rd sequence? Are there any number in at least two of these four sequences? If so, are there any number in at least three of these four sequences? 61.224.150.139 (talk) 05:07, 3 May 2024 (UTC)[reply]

According to Lucas–Carmichael number, it is unknown whether there are any Lucas–Carmichael numbers that are also Carmichael numbers. GalacticShoe (talk) 05:52, 3 May 2024 (UTC)[reply]

2. The sequence is OEIS:A208728, and it starts

15,35,255,455,1295,2703,4355,6479,9215,\ldots

3. The sequence is OEIS:A225711, and it starts

385,2737,6061,6721,17641,24769,25201,\ldots

GalacticShoe (talk) 06:21, 3 May 2024 (UTC)[reply]

Do all numbers in any of these four sequences except 15 and 35 have at least three prime factors? 61.224.150.139 (talk) 06:41, 3 May 2024 (UTC)[reply]

Yes. You can show that:

If $p-1,q-1|pq-1$ , then $p-1|q-1$ and $q-1|p-1$ , implying $p-1=q-1\Rightarrow p=q$ , which is disallowed.
If $p-1,q-1|pq+1$ , then $p-1|q+1$ and $q-1|p+1$ , implying either $p=q-2$ or $p=q+2$ (since they can't be equal.) The rest of this proof is left to the reader since I don't feel like writing it down, but based on the fact that $n|(n+2)^{2}\Leftrightarrow n|4$ , it can be shown that $pq=15,35$ only.
If $p+1,q+1|pq-1$ , then $p+1|q+1$ and $q+1|p+1$ , implying $p+1=q+1\Rightarrow p=q$ , which is disallowed.
If $p+1,q+1|pq+1$ , then $p+1|q-1$ and $q+1|p-1$ , implying $p\leq q-2$ and $p\geq q+2$ , which is not possible.

GalacticShoe (talk) 07:52, 3 May 2024 (UTC)[reply]

Are all of these four sequences infinite? If so, do all of these four sequences contain infinitely many terms with exactly 3 prime factors? Also, do all of these four sequences contain infinitely many terms which are divisible by a given odd prime number? 2402:7500:943:D56F:909B:9877:85C8:AFAA (talk) 02:02, 5 May 2024 (UTC)[reply]

@GalacticShoe: 49.217.60.214 (talk) 05:02, 6 May 2024 (UTC)[reply]

In order for a sequence to be in both 1. and 2., this would require that all prime factors

p|k

satisfy both

p-1|k-1,p-1|k+1\Rightarrow p-1|2\Rightarrow p=2,3

. The only squarefree composite number that is only composed of

2,3

is

6

which can easily be seen to not be in either sequence. Similarly, 3. and 4. would require all prime factors

p|k

to satisfy both

p+1|k-1,p+1|k+1\Rightarrow p+1|2

which does not hold for any primes

p

. Since 1. and 2. cannot coexist, nor can 3. and 4., this means that no number occupies three or more of the four sequences. GalacticShoe (talk) 06:25, 3 May 2024 (UTC)[reply]

Carmichael numbers are the numbers n such that

\lambda (n)

divides n-1, where

\lambda (n)

is the Carmichael lambda function (also called reduced totient function, since it is the reduced form of the Euler totient function) (sequence A002322 in the OEIS), and Lucas-Carmichael numbers should be the numbers n such that

\eta (n)

divides n+1, and this

\eta (n)

should be a reduced form of the Dedekind psi function, use the same reduce rule as the Carmichael lambda function to the Euler totient function (i.e. use the least common multiple in place of the multiplication for

\lambda (mn)

and

\eta (mn)

with coprime m, n), but I cannot even find this function in OEIS (it should start with (start from n=1) 1, 3, 4, 6, 6, 12, 8, 12, 12, 6, 12, 12, 14, 24, 12, 24, 18, 12, 20, 6, 8, 12, 24, 12, …) 2402:7500:943:D56F:909B:9877:85C8:AFAA (talk) 02:13, 5 May 2024 (UTC)[reply]

Factorial & primorial on wikipedia

I know the factorial notation n!. Recently I cam across 5# which I was unfamiliar with. Not knowing its name, (primorial), it proved hard to track down. I searched in wikipedia and Google for "n#" which seemed like the best bet. Both converted it to "n" and reported stuff about the 14th letter of the alphabet. I then thought maybe it is related to factorial, so I looked at wikipedia factorial (n! redirects to factorial on wikipedia, so that works if/when you don't know the term "factorial".)

So is there a way of making n# findable on wikipedia? If so how? -- SGBailey (talk) 21:21, 3 May 2024 (UTC)[reply]

Search for "#" and find Number sign#Mathematics. —Kusma (talk) 22:25, 3 May 2024 (UTC)[reply]

May 5

Origin of notion that there are ב sub 2 many "curves"

(Sorry for awkward heading -- I couldn't get it to put the ב before the 2 because of some strange artifact of RTL rendering.)

I've seen in several places the claim that, as there are $\aleph _{0}$ natural numbers and ${\mathfrak {c}}=2^{\aleph _{0}}$ (sometimes improperly given as $\aleph _{1}$ ) real numbers, there are some greater number of "curves" (sometimes given as f or, again improperly, $\aleph _{2}$ ). Most recently I was reminded of it at our article on George Gamow's (generally excellent) book One Two Three... Infinity.

The usual complaint about these popularizations, a very valid one, is that they uncritically give these cardinalities as aleph numbers in a way that works only if the generalized continuum hypothesis holds. But there's another, quite serious, problem: The claim that there are more "curves" than real numbers is correct only if you have an extremely liberal notion of what constitutes a "curve".

One reasonable notion is that a "curve" is the image of the real line or the unit interval under a continuous function from the reals to Rⁿ (or similar space), but there are only $2^{\aleph _{0}}$ such functions, and therefore the same number of curves.

My best guess is that someone was taking "curve" to mean the graph of an arbitrary function. But these are not typically curves according to any obvious natural-language meaning; they're just scattered points in the plane.

So, question, what's my question? Does anyone know where this idea originated? Was it Gamow, some other popularizer, multiple sources? And what if anything should we do to clean up the text in our One Two Three... Infinity article? I'm thinking an explanatory footnote but ideally I'd want a source directly speaking to the misconception. --Trovatore (talk) 20:58, 5 May 2024 (UTC)[reply]

This Math Stack exchange entry is relevant, but it doesn't seem to cover what you're asking. One problem is that the statement is true by Wikipedia standards; you could cite the book. You would need a reliable source, such as a published article somewhere, to say it was wrong/vague/misleading in order to state that in our article. At the moment the article points out that you'd need the GCH to say what's in the book, but I guess that's supposed to be "common knowledge" (at least among mathies).

This article aims to classify various subsets of the function space 𝐹(ℝ,ℝ) from a constructive-mathematics perspective. The Introduction states: "mathematicians have made numerous attempts to focus on special subsets of this vast vector space (e.g., all real-valued continuous functions [5])", where the cited text is:

Pugh, C.C. Real Mathematical Analysis, 1st ed.; Undergraduate texts in mathematics; Springer Science Business Media: New York, NY, USA, 2002; pp. 223–225.

The latter is available as a pdf here. The article itself denotes this subset as 𝐶(ℝ,ℝ) and concludes in Proposition 4 that 𝐶𝑎𝑟𝑑(𝐶(ℝ,ℝ)) = 𝑐. But this is of course outside the paradise that Cantor created for you. --Lambiam 07:09, 6 May 2024 (UTC)[reply]

The reply given by a fellow Wikipedian to another Math Stack exchange question appears to imply that this also holds within the paradise. --Lambiam 07:23, 6 May 2024 (UTC)[reply]

Cardinality of the continuum § Sets with cardinality of the continuum also lists, without citation, "the set of all continuous functions from

\mathbb {R}

to

\mathbb {R}

". --Lambiam 07:30, 6 May 2024 (UTC)[reply]

Discussed at Stack Exchange. Basically it's because a continuous function from

\mathbb {R}

to

\mathbb {R}

is uniquely determined by its values at rational points. AndrewWTaylor (talk) 16:11, 6 May 2024 (UTC)[reply]

May 6

[1] Linford, M. B., "Fruit quality studies II. Eye number and eye weight". Pineapple Quarterly 3, pp. 185–188 (1933).

[2] Schimper, K. F. "Beschreibung des Symphytum Zeyheri und seiner zwei deutschen verwandten der S. bulbosum Schimper und S. tuberosum Jacq.". Magazin für Pharmacie 28, 3–49 (1829); 29, 1–71 (1830).

[3] Braun, A. "Vergleichende Untersuchung über die Ordnung der Schuppen an den Tannenzapfen als Einleitung zur Untersuchung der Blattstellung". Nov. Acta Ac. CLC 15, 195–402 (1831).

[1]

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