User talk:Lord Farin

Template for Gentzen proofs?
Since we have a template for non-Gentzen tableau proofs, it might be worth constructing one for the new Gentzen ones you're constructing at the moment. It should save some considerable effort and page space. What say? --prime mover (talk) 15:05, 2 April 2014 (UTC)


 * I don't expect many proofs in this style, since even Ben-Ari tries to move on to different proof systems quickly. It is not a proof system that is very common or popular (other Gentzen systems are, however). I deem it wise to not create a template specifically for this proof style.


 * Nonetheless, given that we can expect more and more proof systems in the future, it might be a good idea to generalise Template:BeginTableau to arbitrary proof systems, and provide a generic template which can act as a semantic hull around the MediaWiki syntax, just like Template:Eqn and Template:Axiom do. This will make it easier to adapt to future needs. &mdash; Lord_Farin (talk) 17:36, 2 April 2014 (UTC)


 * More challenging ... I'm not immediately planning on doing all that much more in this area (I only raised it in the first place so as to have something to plant ZFC in back in 2008 and it all got a bit unwieldy) so for the moment I'll leave this until I have another manic episode. --prime mover (talk) 19:28, 2 April 2014 (UTC)

Boolean Algebra
I have been investigating the "Wanted Pages" list and clearing out dead links to superseded pages. One such is Category:Boolean Algebra, where there is still a link from the Definition:Algebra page. Is it appropriate to replace this with Definition:Boolean Algebra in your opinion, or to remove it from that particular list in which it appears? --prime mover (talk) 08:23, 2 May 2014 (UTC)
 * ... or indeed replace with Category:Boolean Algebras? --prime mover (talk) 08:27, 2 May 2014 (UTC)


 * I think it'd be best to remove it. A link to Definition:Boolean Algebra could be added to the "Algebra (Abstract Algebra)" subpage. &mdash; Lord_Farin (talk) 21:11, 2 May 2014 (UTC)

Big refactoring job on Warner thread
I see the good work you're doing on the refactoring the Naturally Ordered Semigroup stuff I did way back when (some of the earliest stuff I posted up). It's going to be a while before I get to reviewing the citation flow (and I don't really want to make a start until I'm fairly sure you're finished) so request the pages flagged up for deletion be retained till after that's been done.

I want to get as much of this tedious Euclid topped off before I go any further with anything else, so bear with me for the moment. In the meantime, good work! --prime mover (talk) 22:38, 19 January 2015 (UTC)


 * Thanks. It's satisfactory to put everything I have prepared into place, and it's going much smoother than I anticipated. I'll give you a note when I think I'm finished. I'll write up new proofs in the process so as to connect with the other axiomatisations, but that shouldn't hurt Warner too much. Don't expect things to be perfect the first time around, but I should be able to put up a correct architecture that we can build upon. &mdash; Lord_Farin (talk) 22:44, 19 January 2015 (UTC)

Templates
I would rather that you hadn't deleted all those templates. I know they were sitting around not doing much, but I had plans for some of them -- particularly "Authorship" and the various "Syllogism" templates. Not sure where I was going with BookRef but I know it started as a good idea. Could we have these back, or do you have good reasons for keeping them deleted? --prime mover (talk) 21:13, 1 March 2015 (UTC)


 * You can have them back, but since some of them had been sitting idle for more than a year, I figured any plan for using them would amount to starting from scratch anyway, so a round of cleaning wouldn't be bad. But sure, restore what you'd like. Perhaps a good idea, though, to keep this kind of early-stage idea in your sandbox &mdash; makes it that much clearer that it's something you plan working on some day, or at least the call to drop it is all yours. &mdash; Lord_Farin (talk) 21:57, 1 March 2015 (UTC)


 * Okay good call, I'll get on with doing that. Be aware though, a year is a short time in ProofWikiLand ... --prime mover (talk) 22:15, 1 March 2015 (UTC)

Lots of source reviews
Yes, as you see I am splitting down composite pages into their simple component parts -- which leaves a considerable amount of redirecting the source flow.

If you find you no longer have the patience for this minute level of detail, then as far as I'm concerned it's no problem to leave the source flow merely landing on the parent page (particularly in the case of De Morgan).

Have fun. --prime mover (talk) 23:10, 24 March 2015 (UTC)


 * I've taken out most, but the last few require me to dig up five sources. This is beyond my patience now. I will get to it eventually. &mdash; Lord_Farin (talk) 18:30, 25 March 2015 (UTC)


 * All is done. &mdash; Lord_Farin (talk) 22:40, 26 March 2015 (UTC)


 * Good job. I am myself working methodically through Warner. I should reach the Natural Numbers in a week or so -- if I don't get sidetracked by something that needs fixing or that got missed and needs elaboration. --prime mover (talk) 22:56, 26 March 2015 (UTC)

Question about Definition:Smallest/Ordered Set
Back in December 2013 you placed an "expand" template onto Definition:Smallest/Ordered Set saying:
 * Some links to this page are actually referring to the smallest element of a subset. This has to be defined, and the links changed to the resulting subpage.

I've looked at this, and raised such a subpage Definition:Smallest/Ordered Set/Subset, but I'm unclear as to how well this serves us.

After all, a subset is a set, and has the same properties as a set, and so the smallest element of a subset is defined in the same as the smallest element of a general set -- whether that set is a superset of it or not.

Hence a page containing something like "the smallest element of $T$ where $T \subseteq S$" is equally well served by Definition:Smallest/Ordered Set as it is to a separate page defining the concept of "smallest" in the specific context where the set in question happens to be the subset of an ordered superset. None of the works I have raise it in a separate definition (however I see you have access to the Munkres and Birkhoff works which I don't, so while I can't check them, maybe you can).

The only circumstance I can think of where the concept may be worth separating out and emphasising is the one where $S$ is partially-ordered while $T$ is totally ordered -- but we have that covered by the page Definition:Chain (Set Theory).

Can you take a look at this and add some thoughts? I should have asked the question at the time, but back in Dec. 2013 I was freezing my nuts off in $-10^\circ \mathrm{C}$ blizzards, and the brain was less well disposed to think about it. :-) --prime mover (talk) 08:34, 5 April 2015 (UTC)


 * The most apparent definition arises in the context of a woset. While it is ultimately the same definition, a reader may fail to recognise that the ordered set one is talking about is actually the subset.


 * One might add that Munkres explicitly defines this on subsets. It is also relevant in the intuitive coupling of largest element and supremum.


 * I'm pretty sure, though, that Munkres triggered me to create the expand template. Does that address your concerns in an adequate manner? &mdash; Lord_Farin (talk) 21:55, 6 April 2015 (UTC)


 * Okay, given that I'm unfamiliar with the Munkres, has the treatment I have offered on the subpage I added consistent with his treatment? --prime mover (talk) 22:01, 6 April 2015 (UTC)


 * Most definitely, although he doesn't bother with the formalism of writing $\preceq \restriction_T$. &mdash; Lord_Farin (talk) 22:39, 6 April 2015 (UTC)

mathbin, mathrel, mathop
Recommend we go through and change "mathop" to "mathbin" and "mathrel" as appropriate, as we progress? I confess it's not a job I fancy doing in one go. :-) I picked up on the differences when I went through the work on TeX codes at the weekend, so I'm fairly well up to speed on it and I appreciate that it "ought" to be done, but it's not something I can nerve myself up to.


 * Gradual seems like just fine to me. Or "as noticed". &mdash; Lord_Farin (talk) 21:56, 7 April 2015 (UTC)

While I'm here, I wonder whether the "mathbin" entries on Inverse of Order Isomorphism is Order Isomorphism should actually be "mathrel" as $\preceq_1$ etc. is technically a relation not an operation. What say? --prime mover (talk) 18:13, 7 April 2015 (UTC)


 * It was a subconscious thought after changing that page (among others) that this was actually wrong that triggered me bringing this up in the first place. I have fixed it.


 * Further investigation ... in places where we have  and so on, we can in fact just remove the mathop altogether, and just have , because it appears that MathJax may have fixed the bug whereby subscripted relations were not being treated as relations.


 * Check out:

So not only is mathop now demonstrably different from and inferior to mathrel, mathrel is not even needed.

IIRC the reason we started using mathop in the first place was because it was the only one of these modifiers that had actually been implemented on MathJax adequately at the time. The new version of MathJax seems to have resolved all of this. --prime mover (talk) 06:44, 8 April 2015 (UTC)


 * Awesomeness (TM). &mdash; Lord_Farin (talk) 14:59, 8 April 2015 (UTC)


 * ... although another reason we started using mathop was to put some spacing between summation limits. Neither mathrel nor mathbin do the job here, only mathop does what we want it to:








 * --prime mover (talk) 14:58, 8 April 2015 (UTC)
 * --prime mover (talk) 14:58, 8 April 2015 (UTC)


 * Hmm... Some cursory search and testing have convinced me that continuing to use \mathop is the way to go, although it is not an elegant solution. There is simply no other option I know of, except manually fiddling around with spacing. &mdash; Lord_Farin (talk) 15:18, 8 April 2015 (UTC)

NOS
I have finally reached the Naturally Ordered Semigroup chapter of Warner (having hurried through the chapters on Ordering without completing all the outstanding work -- I'll return to that later). Note that I am checking the pages off on your User:Lord Farin/Sandbox/NN Refactoring‎ page as I go, particularly when I delete stuff that has a delete mark against it. You may want to follow along behind and check I'm not breaking stuff. --prime mover (talk) 21:00, 10 April 2015 (UTC)


 * Thanks. Looking good. If I now can muster the energy to work through the off-line list I have compiled for MISS, we might actually get somewhere with this project. &mdash; Lord_Farin (talk) 08:37, 11 April 2015 (UTC)

Families
Advice needed on where to go with this one.

Munkres is the only author I've seen who defines the terminology of indexing sets using definitions which are intuitively clear:


 * a) An Definition:Indexing Function is the actual mapping from $x: I \to S$
 * b) A Definition:Indexed Family is the set of elements of $S$, together with the mapping itself.

Every other source I've seen calls the mapping itself the "family", despite giving it the intuitively-suggestive notation $\langle x_i \rangle$ (or whatever notation used).

Consequently I have decided to go with Munkres's definition, despite the fact that he appears to be in a minority.

Or do you see this as: everybody else has failed to define the concept precisely enough, and has made a mistake by failing adequately to distinguish between the mapping itself and its codomain?

It is also interesting that Munkres is the only one who specifies that the mapping itself needs to be a surjection -- a point which I still have yet to address in the definitions.

As a practical matter, however, I have changed the links on Halmos and Givant so as to allow it to point to Definition:Indexing Function instead of Definition:Indexed Family. You may wish to review this in order to make sure the flow accurately reflects the specific treatment of Halmos and Givant here: do they define a "family" as the mapping, or the mapping together with its codomain?

I also need to consider whether to include the "also defined as" on Definition:Indexing Set/Family on a separate page and then transclude it onto other pages in the same way as Notation and Note on Terminology.

Are you able to cast an eye over this area? I just need a second opinion on it. Thanks. --prime mover (talk) 09:09, 25 April 2015 (UTC)


 * There are basically two points of view at work here which are conceptually quite different. The majority of writers takes the indexing set $I$ as the starting point; from this point of view, it is more natural to define the concept in the way they do (e.g. Halmos/Givant do it in this fashion), as simply the function $x: I \to S$.


 * On the other hand, we have Munkres (and it's inconceivable he is alone). The viewpoint now is radically different: We start with a set $S$ (in Munkres' case, a set of sets), which we want to index. In this view, it makes sense to bring $S$ along explicitly, and to require an indexing function to be surjective -- after all, it was not merely a subset of $S$ that was to be indexed.


 * With this in mind, I think it is not warranted to choose between the two. Rather, we should aim for an exposition that somehow manages to unify these treatments. I'm sorry to burden you with this task, but ultimately I think this is the best way to go. &mdash; Lord_Farin (talk) 09:53, 25 April 2015 (UTC)


 * Are you happy with the approach which I have just taken? --prime mover (talk) 21:49, 26 April 2015 (UTC)


 * Except for the note I left on one of the talk pages, the content looks good. Regarding presentation, it is maybe a bit awkward to have Indexing Set as the master page. Perhaps Indexed Family ought to be the parent page, with all related concepts as subpages; in this way, subpages will describe portions of the entire thing, which is the Indexed Family. (But I understand if you don't want to embark on that journey right now.) &mdash; Lord_Farin (talk) 08:08, 27 April 2015 (UTC)


 * You're right, it does appear as thought "indexed family" is the main concept, and all the other parts of it should perhaps be subpages. It's not quite as big a job as it looks on the surface, though, because our redirect strategy is a good one. --prime mover (talk) 15:31, 27 April 2015 (UTC)


 * All instances of Definition:Family have now been reviewed and replaced by a link to the appropriate instance of "indexed family", be it "of sets" or "of subsets" or neither. Time for a break; Definition:Family can now become a disambiguation page as suggested in a conv with Joel on Definition talk:Indexing Set/Function a while back. --prime mover (talk) 20:03, 27 April 2015 (UTC)

Abstract Algebra and all that malarkey
I'm getting terminally bogged down in the Abstract Algebra work from Warner, so I am going to take a break from it and work on something else for a bit. I have a yearning to start something new, or continue from where I left off, as this maintenance and refactoring is starting to give me nightmares.

If you have any inspiration as to how to clean up the mess that is the recursive definition of the general $n$th power of an element that I abandoned so cruelly last week, then please feel free to succour and nurture it back to health. --prime mover (talk) 16:04, 3 May 2015 (UTC)

Problems
Are there any problems with the site? It seems like someone has gone nuts on adding deletion requests EmperorZelos (talk) 07:47, 16 May 2015 (UTC)


 * Yes, there are some problems regarding that. I'm sorry. I will see if I have the time today to sort that out for you. &mdash; Lord_Farin (talk) 08:35, 16 May 2015 (UTC)
 * Is it anything alarming as in the site itself is threatened or is it just someone gone rogue? EmperorZelos (talk) 08:36, 16 May 2015 (UTC)


 * Rest assured, I won't let the site be threatened in this way :). Thanks for caring, though :). &mdash; Lord_Farin (talk) 08:38, 16 May 2015 (UTC)
 * Great :) I love this place, I wish there was a book like this where one could just find proofs upon proofs upon proofs. EmperorZelos (talk) 08:39, 16 May 2015 (UTC)

For what it's worth, all extraneous flags for deletion have been reverted. &mdash; Lord_Farin (talk) 13:44, 16 May 2015 (UTC)

Catalan's Identity
Are you suggesting that I do not use $a$ $b$ $c$ $d$? Sorry if I misinterpreted your message. --kc_kennylau (talk) 21:45, 21 May 2015 (UTC)


 * Yes. &mdash; Lord_Farin (talk) 16:17, 22 May 2015 (UTC)

Properties of Fibonacci Numbers
Can we move User:kc_kennylau/Sandbox to the main namespace now? --kc_kennylau (talk) 01:03, 24 May 2015 (UTC)


 * Done. Nice work :). &mdash; Lord_Farin (talk) 10:02, 24 May 2015 (UTC)


 * Thanks. I was wondering why you did not use the "move" function. --kc_kennylau (talk) 10:03, 24 May 2015 (UTC)


 * Oh, I presume I could've. But since your sandbox will probably be used for other purposes as well, I thought it nice to leave something there, instead of it being a redirect or outright deleted. &mdash; Lord_Farin (talk) 10:06, 24 May 2015 (UTC)

CBS
I'm not sure if this is the correct way to answer you. My proof of CBS is essentially the same. (If you read both versions you will notice the similarities.) -- Inconsistency


 * Conversations are easier to follow if they take place on a single page. I've moved this part of the conversation to your talk, where we will continue our discussion. &mdash; Lord_Farin (talk) 16:34, 2 June 2015 (UTC)

Eqn template: puzzling
So one of the rules of the game is: do not have adjacent  or   in an Eqn template, or it will break.

So I'm puzzled at how this:

manages all right.

I think the fact that the  and   come in pairs very close together may have something to do with it -- but had you noticed this?

It's on page Consecutive Fibonacci Numbers are Coprime if you're interested. --prime mover (talk) 21:24, 2 June 2015 (UTC)


 * It's definitely a fluke, and should be expected to break at any moment. Thanks for spotting, you're sharp :). &mdash; Lord_Farin (talk) 21:26, 2 June 2015 (UTC)

More on the eqn template
Check out Inverse Cosecant is Odd Function.

Note how the $\pi$ on one line is overlapped by the $2$ on the line above.

This is the source of the comment I made about adding some top and bottom padding to the eqn template.

I tried experimenting with this but could not get it to work.

Are you able to do some magic here? --prime mover (talk) 22:22, 5 June 2015 (UTC)


 * I've added 3px bottom and top margin. If you feel it should be 2 or 4 (5 is a bit too much I think, it adds up to 10 total in that case), just change the 3px occurrences to whatever you deem suitable. But 3px works fine for me. &mdash; Lord_Farin (talk) 22:49, 5 June 2015 (UTC)


 * That looks perfect. Good job. --prime mover (talk) 04:54, 6 June 2015 (UTC)

Help needed with eqn-intertext
The current  is all very well, but it puts all text into the first column, which then expands to fill it and compromises the appeal of the page.

Is there a way of making the text spread over all the columns of the table, perhaps using a "merge columns" technique? A good example of a page where it is to be made use of is Primitive of p x + q over Root of a x + b. --prime mover (talk) 06:17, 11 June 2015 (UTC)


 * Sorry for not minding my own business, but I solved it. --kc_kennylau (talk) 06:29, 11 June 2015 (UTC)


 * Brilliant! I knew it was simple, I just didn't care to look up how to do it ... :-) --prime mover (talk) 06:38, 11 June 2015 (UTC)

Where is the style guide?
You referred to "a long-established style guide" but you didn't link to it. Nor do I see anything appearing to be a link to it among the many links in the column on the left.

Could you tell me where it is? Michael Hardy (talk) 22:55, 24 July 2015 (UTC)


 * Help:FAQ has an entry on what you ask and is available in the left menu. I see that you found Help:Editing/House Style on your own. &mdash; Lord_Farin (talk) 09:21, 25 July 2015 (UTC)

Jacobson's treatment of Peano axioms
treats the Peano structure in considerable detail, but uses a $1$-based approach, not $0$-based.

How do you envisage this being treated?

a) State "Some treatments define the unique non-successor element as $1$ instead of $0$", and leave it at that

b) Provide a full definition of each addition, multiplication, etc. in the 1-based schema, and separately demonstrate the commutativity / associativity / distributivity laws?

c) A combination of the two?

Since I have the book and have been tasked with the source review, I suppose it devolves to me to do whichever approach is decided. --prime mover (talk) 07:13, 22 September 2015 (UTC)


 * I feel we can get away with a). Therefore, I think it is more valuable to dedicate our limited resources to other areas of PW. In an ideal world, b) would be preferable, but due to the declining importance of the 1-based approach this does not have priority for me at the moment. (A similar thing will go for meticulously proving the equivalence of the defined operations in the various axiomatisations -- I'm going to postpone it.)


 * If in due time we find a good number of sources on the 1-based approach, which furthermore make the top of the list of books to be covered, we can reconsider.


 * Until then, a) suffices in my book. I feel that once multiplication and the ordinal entanglement are revisited, the section about $\N$ is sufficiently coherent to serve as a stable basis for further development of other areas of PW. &mdash; Lord_Farin (talk) 15:15, 22 September 2015 (UTC)


 * I implemented a template "Non-Successor Element 1" which states the case and has been used in various places and in particular Definition:Addition/Peano Structure and Definition:Multiplication/Natural Numbers -- the latter of which now has a muce clearer boundary between $0$-based and $1$-based.

Further number theory axiom question
This page: Axiom:Axiom Schema for 1-Based Natural Numbers is what I renamed Axiom:Axiomatization of 1-Based Natural Numbers, as there were already some pages linking to that directly. It fixed a load of redlinks.

What's the plan here? Leave it at that? --prime mover (talk) 19:23, 22 September 2015 (UTC)


 * The plan was to rename it because it's not a "schema", it's a finite set of axioms. Apparently I somehow forgot to address the pages linking to it. &mdash; Lord_Farin (talk) 20:06, 22 September 2015 (UTC)


 * In general I've been quite tired from this refactoring endeavour, so I'm patching up the final pieces that really need to be done and then I'll move somewhere else. &mdash; Lord_Farin (talk) 20:06, 22 September 2015 (UTC)


 * All those are done except for a few links from your backup pages etc. I haven't done Definition:1-Based Natural Numbers yet as it's not so contentious. --prime mover (talk) 21:16, 22 September 2015 (UTC)


 * The last thing on my list is to address multiplication. I'll try to make time these days, it'll feel good to be done. &mdash; Lord_Farin (talk) 15:31, 23 September 2015 (UTC)

Recreating links to moved pages
There are a lot of pages in your backup area which were moved from their main pages without leaving a redirect. Trouble is, these have left some redlinks, in particular the source flow from Well-Ordering Principle which had not had all its links changed.

I have gone through and replaced a few of the redirects, but I realise the problem is more widespread as there are many pages in your backup with no redirects to them, and I am worried we are going to lose the flow (specially around the Induction proofs).

My preferred technique is never to move without leaving a redirect, and never delete the redirect until all links via the redirect have been repointed. I use "What links here" to check, and then go through and change each one methodically.

Hope this hasn't put you out too badly, as I say there were some redlinks to non-redirected moved pages and it wasn't clear to me exactly where they were supposed to go. I worked it out in the end, but I want to make sure everything is linking before we clear out redirects. --prime mover (talk) 22:23, 22 September 2015 (UTC)


 * I know that the rule is to have redirects. However, I treat the backup area as special. Namely, it is a backup area. Everything there is a snapshot from before an important change, so that if need be, we can easily restore everything by removing the appropriate prefix from the page title. I don't care for redlinks on these pages; they shouldn't be visited anyway, and in theory should be regularly purged.


 * At times, I furthermore take the liberty to spawn redlinks on talk pages, where in my head, they indicate that the section they're in can safely be regarded as obsolete and can in principle be deleted.


 * For all other links, I always try to make sure they are changed appropriately.


 * With this procedure I preserve the structural integrity of the site with minimal effort. I'm not inclined to change it. But perhaps it would've been good to make these assumptions more explicit, as they apparently led to some confusion on your part -- sorry for that. &mdash; Lord_Farin (talk) 15:31, 23 September 2015 (UTC)


 * Point is that not all the live pages had been amended so that the links had all been updated, so the redlinks (well, one, anyway) were on live pages not on backup pages. Hence and so. I still prefer not to delete a redirect till all the pages that link to it have been addressed. Even backup pages, because at least if you've got a redirect to a backup page, you have a chance at working out where the link ought to go to, otherwise one is lost. --prime mover (talk) 16:18, 23 September 2015 (UTC)


 * There's a valid point there. I'll consider it next time. I'll also try not to screw up the redirecting of live links. &mdash; Lord_Farin (talk) 16:48, 23 September 2015 (UTC)

Injection refactoring
The Injection page now has 6 definitions, all complete with equivalence proofs. This of course means source review. There are 6 outstanding works, some of which I believe you have. Feel free to address it.

Annoyingly, none of my source works use Definition 2, but such is the effort of restructuring this mammoth page again I don't feel like revisiting it at the moment. --prime mover (talk) 21:52, 25 September 2015 (UTC)


 * It's done. And you get to be lucky, because Takeuti/Zaring uses Def 2. Good work. &mdash; Lord_Farin (talk) 10:45, 26 September 2015 (UTC)


 * Wonderful. Another little corner of the universe tidied up. --prime mover (talk) 11:59, 26 September 2015 (UTC)

More refactoring
I am working through the various works I have which treat natural numbers by axiomatics -- that's mainly my Halmos, my Deskins and my Jacobson, so as to clear my backlog of Source Reviews and tidy up this area according to what you left open for me to attend to.

Please note I am leaving you any source works I don't have access to, on the grounds that you might (and you seem to). These will trickle into your inbox during the course of time.

There's an outstanding review of the Howson on Definition:Natural Numbers which you may be able to sort out. I intended links on this page to refer to places in the source works which contain nothing much more complicated than "The natural numbers are the ordinary counting numbers: $1, 2, 3, ...$ at whatever levels of sophistication this goes into, usually the first of five lines containing a simple list of one-line definitions of $\N, \Z, \Q, \R, \C$. Anything deeper goes into the subpages.

I have a job of work to do on Halmos, which seems to have got particularly fragmented. --prime mover (talk) 18:48, 26 September 2015 (UTC)

Deletion of Natural Numbers form Naturally Ordered Semigroup
I've been tidying up some of my Source Review tasks, and have come upon Natural Numbers form Naturally Ordered Semigroup.

There are a number of pages dependent upon this, from when the definition of Natural Numbers was based on that analysis. Without this page, there is no longer an explicit page defining the link between the Naturally Ordered Semigroup and the Natural Numbers. There is a definite place for a page that links up the derivations that state:
 * a) The Natural Numbers fulfil the requirements for being a Natural Ordered Semigroup
 * b) The Naturally Ordered Semigroup is unique under isomorphism

thereby allowing us to identify the natural numbers with the NOS.

We already have a transcluded link on the Natural Numbers page which defines the $\N$ as an instance of an NOS, but I am afraid that if we are not careful we will (under the guise of tidying up) remove some important links in the chain of reasoning that allow us to define $\N$ from the NOS axioms. It's not a mainstream derivation, but I believe it has some benefit, if nothing because Seth Warner is so delightfully thorough in everything he does, and his construction is a paradigm of such attention to detail.

So, in short, can we either leave this, or make an effort to restructure all the pages which depend on it so as to allow alternative reasoning (e.g. Peano-based) for their justification? There's a huge pile of pages (just do "What links here" from it, you'll see the extent of the task) which I'm not in a position to address right now, I'm supposed to be packing to leave for Romania in a couple of hours. --prime mover (talk) 10:41, 11 October 2015 (UTC)


 * You make a fair point. But, and this will remain my point of view, there is no such thing as "the natural numbers" in existence with the adequate mathematical precision. Therefore, the NOS was invented as an adequate or at least plausible axiomatisation of $\N$.


 * Now of course the axiomatisation may be strengthened if it's proven equivalent to some other prevalent and so far highly successful treatises of $\N$ in a precise way. But it cannot and will not ever replace the intuition everyone has on $\N$.


 * Therefore I think the only viable option is to rephrase all the linking pages. It needn't be hard. We only need to say "consider $\N$ defined as a NOS" instead of referring to said theorem. I'll try to find the time. &mdash; Lord_Farin (talk) 15:13, 11 October 2015 (UTC)


 * I still think there is a disconnect -- we have "consider $\N$ defined as a NOS" but do we have a page establishing why we can do this? My original thinking behind this page was purely so there is a direct link to a page which says "... and we can use the NOS as a model for the natural numbers because ..."


 * You're right, it all boils down to an "intuitive" understanding of what makes $\N$ what it is -- but exactly how to establish this I'm not sure, beyond a vague "... and this is a model for $\N$ because it's infinite and it's got addition, multiplication and ordering and they act like ..."


 * Do we need a separate page again for the "axioms" of what define natural numbers from an "intuitive" perspective?


 * If this isn't all done by the time I get back, I'll continue giving it some thought, but at the moment (on 3 hours sleep and inadequate coffee) I'm not able to summon enough grey cells. :-) --prime mover (talk) 05:10, 12 October 2015 (UTC)


 * There is no reason why we can do that beyond proving a lot of results with NOS that we expect to hold for $\N$, as well as not managing to prove anything that is false for $\N$ (the latter being critical to the axiomatisation's success). There is no such list of "axioms" beyond the vague sentence you wrote down. Anything more is simply philosophy of maths, which is admittedly beyond the scope of this site.


 * It's admittedly a leap of faith. But it's all we've got, because there cannot be a rigorous jump from our intuition to the safe confines of mathematical rigour -- such would prove all instances of "our intuition" are equivalent which is contradicted by the existence of open problems about the natural numbers. Sad but true. &mdash; Lord_Farin (talk) 16:05, 12 October 2015 (UTC)