Talk:Main Page

Well-orderings: sets and classes
I'm looking again at the properties of well-orderings and order isomorphisms in the context of class theory, still using Smullyan and Fitting as my mentor.

During the course of this I am finding they derive some results which take certain truths of bijections and order isomorphisms for granted, but they themselves do not prove these results (e.g. Composite of Order Isomorphisms is Order Isomorphism and Inverse of Order Isomorphism is Order Isomorphism and so on) are fairly trivial and which we have proved in the context of sets not classes.

It is at this stage where I am tempted to expand the scope of these latter results and rework them in the context of class theory not just set theory.

I am not sure at this stage whether to make them class-theoretical results only, and just add "they apply to sets too" or implement a separate class-theory version only ("This result carries over to classes as well:" etc.).

Bear with me while this is all a bit of a mess still. --prime mover (talk) 05:28, 6 July 2022 (UTC)


 * The inexhaustible prime mover... -- 23:00, 6 July 2022 (UTC)


 * I confess I am getting a little tired, staring at it and not making any sense of it. --prime mover (talk) 23:12, 6 July 2022 (UTC)

Another stage implementing class theoretical approach
Fundamental results in well-order theory in the implementation of the class-theoretical approach require that all the basic algebra of set composition (composition, identity and inverses basically) in the context of ordered sets need to have their class-theoretical versions.

Work is to take place in due course, after I've had a break from it for a short while. --prime mover (talk) 21:33, 9 July 2022 (UTC)

Shortcut City
In order to drastically simplify the entire area, I am currently experimenting with leaving a lot of the basic definitions alone, but merely removing the stipulation that the domains and ranges of the mappings and relations in question are in fact sets.

Hence we start with "Let $f: S \to T$ be a mapping" and leaving it up to the context for the reader to determine for themself what the nature of the domain and range actually are. To those raised on set theory only, they will understand that $S$ and $T$ are sets because it is implicit in the definition. To those raised on class theory, the class-theoretical definition is found under the "class-theoretical definition" subpage of the definition page for mapping and relation.

I have a further idea to streamline the whole area: use the term collection consistently when referring to a collection which may be either-a-class-or-a-set-and-the-definition-is-identical-for-either, and somewhere on the page on which this appears, transclude a boilerplate page explaining 's philosophy on the matter: "On, the term collection is specifically and unilaterally defined as being ..." and so on, explaining why we do that. Then we never have to worry about splitting the pages up into "class-theoretical version" and "set-theoretical version" but merely rely upon the context from which the definition page is linked.

(And the weather is going to be glorious today and I have household chores to do before going to a piano concert this afternoon so I'm not planning on spending all day at the keyboard today, despite the fact that I have awoken bright and shiny -- so I don't plan on doing all or any of this today.) --prime mover (talk) 08:06, 10 July 2022 (UTC)


 * I saw the curse of class theory on here years ago. It's so interesting that it might finally see its defeat.  01:09, 17 July 2022 (UTC)

Sanity check
Just checking the following theorem isn't on the site (can't find it): let $X$, $Y$, $Z$ be topological spaces. Give $X \times Y$ the product topology. Let $f : X \times Y \to Z$ be continuous. Then for $x \in X$, the map $y \mapsto \map f {x, y}$ is continuous as a map $Y \to Z$. We called this the $x$-vertical section in measure theory. Basically hinges on the continuity of $y \mapsto \tuple {x, y}$. Caliburn (talk) 10:17, 19 July 2022 (UTC)

How to describe exactly the theorem I want to find
I am now trying to find the proof of this theorem: $a^s>1$ when $a>1$ and $s>0$, but so far I haven't found it. I am not a native English speaker, how can I accurately describe the theorem I am trying to find in natural language? --AuroraAeon (talk) 02:25, 23 August 2022 (UTC)


 * It is possible that theorem does not exist on . This is because it wasn't included in the source work upon which the bulk of the basics of real analysis was built. --prime mover (talk) 05:34, 23 August 2022 (UTC)


 * The best I found is this: Power Function on Base Greater than One is Strictly Increasing, which states $a^s > a^0 = 1$ when $a > 1$ and $s > 0$.
 * For what it's worth I used the following on Google: "site:proofwiki.org power greater one". &mdash; Lord_Farin (talk) 06:05, 23 August 2022 (UTC)

Notation Confusion
You see, there's a lot of pages in the site like this, which writes stuffs like "Let $a\in \mathbb{C}$ be a complex number.

Is it really necessary?--AuroraAeon (talk) 00:28, 25 August 2022 (UTC)


 * 100%. No confusion about what $a$ is. Convenient reinforcement of housestyle, too. The page Definition:Complex Numbers notes alternative symbols exist in the literature:
 * "Variants on $\C$ are often seen, for example $\mathbf C$, $\CC$ and $\mathfrak C$, or even just $C$."
 * Constantly being explicit and saying $\C$ means "complex numbers" increases styling uniformity.
 * Not necessary in a single book with a few authors. But, a smart investment for a $40,000+$ page wiki.  01:28, 25 August 2022 (UTC)


 * What he said.
 * Let me add: we don't write "Let $a\in \mathbb{C}$ be a complex number." We write "Let $a \in \C$ be a complex number."
 * Note the difference. It's not particularly subtle. --prime mover (talk) 06:00, 25 August 2022 (UTC)


 * I got it. Where can I obtain a systemic knowledge of the common-used housestyle in ?--AuroraAeon (talk) 06:39, 25 August 2022 (UTC)
 * Found it here, but thank you anyway.
 * By the way, I notice that the archive of this page hasn't been updated for a long time.--AuroraAeon (talk) 06:47, 25 August 2022 (UTC)


 * Yes it's been a busy year, but as there are a number of discussions which may not have been resolved I'm not ready to archive it yet. --prime mover (talk) 09:03, 25 August 2022 (UTC)

Mathematicians pages - include family?
I'm throwing this open to debate, since this came up:

Is there any value to adding a "family" section to the Mathematician pages? That is, a section that gives the mother and father and perhaps also children of the mathematician?

My immediate reaction to this is "no, it's too much work and too tangential" but then on some of the pages we do have mentions of family -- specifically when those family members are also featured in our database.

So is it worth setting up another formal section to the Mathematician pages titled "Family" (or "Lineage" or whatever)?

Before I settle the question with a big fat "no" I want to see what others think. --prime mover (talk) 09:01, 25 August 2022 (UTC)


 * No thank you. I would not forbid mentions of family, but only put them there when they could be of interest, i.e. when the family member can stand on their own. Otherwise it's just fluff. /2c &mdash; Lord_Farin (talk) 11:49, 25 August 2022 (UTC)

New simple proofs for Logarithm Tends to Infinity and Logarithm Tends to Negative Infinity
Hi

ln e^a = a

let a = ∞, e^∞ = ∞ ln ∞ = ∞

let a = -∞, e^-∞ = 0+ ln 0+ = -∞

Also these proofs not found on Wikipedia https://en.wikipedia.org/wiki/List_of_limits.

Order Notation - resolving the various confusions
Having become mentally and psychologically blocked on class theory and ordinal theory, I've found myself addressing Definition:Order Notation.

In an attempt to find something definitive, I have dug out my cobwebbed old with an eye towards the fact that they cover everything in some detail, and appear to be consistent in their approach.

However, their approach is different from that taken by the various definitions I've seen, and I'm about to take a view on whether they merit genuinely different and parallel definitions, or whether we can implement an "also defined as" approach.

The CLR approach defines, for example, $\map \OO {\map g n}$ as the set of functions: $\set {\map f n: \exists c, n_0 \in \N_{>0}: \forall n > n_0: 0 \le \map f n \le c \map g n}$.

(This of course presupposes the fact that $f$ and $g$ are positive, and $f$ and $g$ are sequences rather than real functions, etc. so there will need to be adjustment of this to match the reality of what is what in pure mathematical contexts.)

As a result I may need to make several run-ups to this topic and maybe backtrack a fair bit.

Please bear with me while I have the scaffolding up. And please, if you see something that you know is obviously wrong in this area, put it up in a talk page rather than editing the page in question, as in that way we avoid edit conflicts and wikiwars. --prime mover (talk) 07:38, 27 August 2022 (UTC)

Proof Rules refactoring executed
Hi all, the refactoring which had been prepared at User:Lord_Farin/Sandbox/Proof Rules has been executed.

The net effect is that our proof rules and rules of inference have been made ready to accommodate more than propositional logic, in particular also predicate logic. And that it is now much easier to build a new proof system on top of these rules in a coherent manner. This opens up further and more thorough coverage of predicate logic and propositional logic. My focus will be on PredLog going forward.

This mainly means that there are new and rewritten pages Definition:Natural Deduction and Definition:Rule of Inference. The old pages have been saved in my backup realm User:Lord_Farin/Backup.

What follows is a number of source review activities related to this change. I have access to a number of them, but not all.

The whole list is on the preparation page User:Lord_Farin/Sandbox/Proof Rules at the bottom. Any help appreciated. &mdash; Lord_Farin (talk) 10:31, 2 September 2022 (UTC)


 * I have a number of the works in question -- if you don't have them I probably do, so assign them to me. I can always reassign them if I can't do them for any reason. --prime mover (talk) 12:02, 2 September 2022 (UTC)


 * I did the works I could locate and assigned the rest to you. &mdash; Lord_Farin (talk) 09:28, 4 September 2022 (UTC)


 * Yes okay it's on my radar, but it's not a trivial exercise and I will need to get myself back into the mindflow of each of the books to do a proper job. Been distracted with irritating trivia. --prime mover (talk) 09:30, 4 September 2022 (UTC)

Philosophical quibbling over "undefined"
Show of hands here: is it reasonable to accept the tacit implication that if $a = b = 0$ then $a$ and $b$ are not coprime?

If $a = b = 0$ then GCD of $a$ and $b$ is not defined.

Hence by implication $\gcd \set {a, b}$ is not equal to $1$.

Are there philosophical constructs in which a function "being undefined" can also have the same truth value as the function actually having a specific value at the undefined point?

I am afraid I have wasted enough time on this utterly pointless argument, and I'd like either a little backup or to be told explicitly that I don't know what I'm talking about. --prime mover (talk) 11:06, 4 September 2022 (UTC)


 * The definition is fine, especially now that Coprime Integers cannot Both be Zero exists. A pinch of accuracy would be gained by saying "$\gcd\set{a,b}$ exists and is equal to $1$". This mostly serves as a safeguard against simplistic application (compare limit and integral identities) and is not strictly necessary. &mdash; Lord_Farin (talk) 11:51, 4 September 2022 (UTC)


 * I hate the fact that this is even considered worth doing, but clearly we seem to need something of the sort. --prime mover (talk) 12:09, 4 September 2022 (UTC)


 * There are two positions possible. We could argue that if one part of an expression is undefined, then the entire expression is undefined.
 * We could also argue that a statement like $m \perp n$ is shorthand for: $\exists \tuple{ m,n } \in \set {\tuple {a, b}: a \: \mathrm{coprime \: with} \: b}$, the truth set of the relation. Then $n \perp 0 \iff n \in \set{ 1, -1 }$ is a true statement, as $\tuple{ 0,0 }$ does not lie in the truth set, and this would be my position. - I believe that was the original problematic phrasing. --Anghel (talk) 13:11, 4 September 2022 (UTC)


 * I'm still not sure there never was a problem. I wonder what the consequences are if we just carry on as we were. --prime mover (talk) 21:55, 4 September 2022 (UTC)

Extension to the template TFAE
Is there any chance that the template TFAE can be extended to accept an axiom: page as an alternative to a definition: page? Otherwise a definition: redirect to the axiom: page is needed. --Leigh.Samphier (talk) 13:28, 15 September 2022 (UTC)


 * Yes of course, good call. I can get on and do that in due course. --prime mover (talk) 13:55, 15 September 2022 (UTC)

Disambiguation of Quotient Mapping
The concept of a Definition:Quotient Mapping is fairly well case-hardened into as the mapping from a set to the set of images of the equivalence classes under a given equivalence.

In topology, we have a definition Definition:Quotient Mapping (Topology) which is (at least on the surface) different in concept from a Definition:Quotient Mapping.

To add to the confusion, we have a Definition:Quotient Topology which uses the Definition:Quotient Mapping in its definition.

It also seems as though Definition:Quotient Mapping (Topology) is closely tied to the Definition:Quotient Topology, a.k.a. / closely related to the Definition:Identification Topology which uses the Definition:Quotient Mapping in its definition.

Is it advisable at this stage to rename Definition:Quotient Mapping to e.g. Definition:Canonical Surjection (there's already a redirect) to allay confusion -- or is there a way of knitting Definition:Quotient Mapping (Topology) and Definition:Quotient Mapping so that the latter can be immediately seen as being a special case of the former?

Advice needed to a) reduce confusion and/or b) improve cohesion. --prime mover (talk) 07:55, 19 September 2022 (UTC)


 * Don't know if this helps.


 * If $f:\struct{S_1, \tau_1} \to \struct{S_2, \tau_2}$ is a Definition:Quotient Mapping (Topology), then there is one and only one homeomorphism  $r: \struct{S_1/\RR, \tau_\RR} \to \struct{S_2, \tau_2}$ such that:


 * $f = r \circ q_{\RR}$


 * where:


 * $(1) \quad \RR = \set{\tuple{x, y} : x, y \in S_1: \map f x = \map f y}$ is an Definition:Equivalence Relation


 * $(2) \quad \tau_\RR$ is the Definition:Quotient Topology induced by $\RR$


 * $(3) \quad q_\RR: S_1 \to S_1 / \RR$ is the Definition:Quotient Mapping


 * --Leigh.Samphier (talk) 11:06, 19 September 2022 (UTC)


 * I have just uploaded a proof for Leigh Samphier's theorem: Quotient Mapping Induces Homeomorphism between Quotient Space and Image (where I renamed $r$ as $\tilde f$). The homeomorphism $r: S_1 / \RR_f \to S_2$ will be defined so:


 * $\map f s = \map {r}{\eqclass s { \RR_f} }$


 * If we squint our eyes enough to remove $r$ from this equation, and ignore the fact that the image set of $f$ is $S_2$ rather than $S_1 / \RR_f$ (which makes sense, as $r$ is a homeomorphism ), we can pretend that $f$ is a quotient mapping in the standard definition. --Anghel (talk) 20:41, 19 September 2022 (UTC)


 * To address the disambiguation question. The definition of Definition:Quotient Mapping (Topology) for a quotient map is ubiquitous.


 * The definition Definition:Quotient Mapping is variously called one of canonnical/natural surjection/projection and rarely called quotient map. I personally dislike the use of natural as it most often indicates an informal left to the reader notion. I dislike the use of projection for sets as this seems inappropriate, but is appropriate in algebra settings where the question being asked is whether an algebra is the direct product of a subalgrebra and the quotient of the algebra by the subalgebra ($A = S \times A/S$).


 * That only leaves canonical surjection, which I have reservations for its use as it isn't suggestive of the involvement of an Equivalence Relation. But nothing else comes up. --Leigh.Samphier (talk) 09:43, 20 September 2022 (UTC)


 * After further thinking. For the record, I'm happier with canonical surjection than natural blah blah -- but now I think about it, I really don't like abandoning the "quotient" term which underpins the understanding, and is common to the whole gamut of quotient structures.


 * As the Definition:Quotient Mapping (Topology) is indeed a construct also (indirectly) created from an equivalence relation, and therefore is the "same sort of thing" as a more general Definition:Quotient Mapping, I now think we should leave the names alone (they are consistent and logical) and if necessary add an explanatory note transcluded from a subpage to explain the situation.


 * You could add a theorem that showed that a Definition:Quotient Mapping where the set and quotient set are endowed with discrete topologies is indeed a Definition:Quotient Mapping (Topology). I can't decide if this is illuminating and completes the loop or is plain clumsy. --Leigh.Samphier (talk) 10:36, 20 September 2022 (UTC)


 * I wonder whether we should perhaps be making more of the "explanatory note" style pages. I note that sometimes "Note" and "Remarks" sections (which I like to make try to steer clear of) could / should be evolved into "/Explanatory Note" pages, as we have done -- which will help to make it more obvious what is central to the information on the page and what is peripheral, discursive and explanatory.


 * Evolution is a continuous process. --prime mover (talk) 10:05, 20 September 2022 (UTC)


 * There is definitely a place for explanatory notes, and they need to be formal in the sense that they are a sequence of statements where each statement begins by referencing an existing theorem/definition and restates the conclusion/defining conditions of the theorem/definition. There should be no statements that are not referencing an existing theorem/definition, stubs should be avoided. With a correctly structured sequence the explanation should follow. --Leigh.Samphier (talk) 10:36, 20 September 2022 (UTC)


 * I have just put up a theorem Continuous Surjection Induces Continuous Bijection from Quotient Space, which uses both Definition:Quotient Mapping and Definition:Quotient Mapping (Topology) in its statement. I ended up using Definition:Canonical Surjection instead of Definition:Quotient Mapping to avoid confusing the readers, just for this solitary case. I would like to know if anybody has a better solution.


 * For the record, I also like the name Definition:Quotient Mapping better than any of the suggested alternatives. --Anghel (talk) 22:42, 22 September 2022 (UTC)


 * A sensible approach. --prime mover (talk) 05:51, 23 September 2022 (UTC)