User talk:Prime.mover

New approach to natural number section
I'm sure you've noticed my new approach to getting the natural numbers section to a higher level. I hope you like the direction I'm going in :). As always, feel free to inform me about any remarks or other concerns that you might have.

If you want to keep an eye on the progress, just take a look at the items on User:Lord_Farin/Sandbox/NN Refactoring that have been marked done (you can also add items you think relevant to the list). &mdash; Lord_Farin (talk) 20:37, 22 October 2014 (UTC)


 * I'm keeping an eye, no worries ...


 * I'm going to let you get on with it, I'm happy with this approach. It's a project I found was beyond my powers of extended concentration. Meanwhile, I'm going to continue trying to make head or tail of Euclid's books on number theory. --prime mover (talk) 20:44, 22 October 2014 (UTC)


 * You've probably noticed that I haven't been doing much; I have been occupied by other things. Fortunately, my explicit planning list helps to keep track of where I was and what needs to be done, so that we can still progress in the end. &mdash; Lord_Farin (talk) 16:40, 7 February 2015 (UTC)


 * The Warner work was among the first work I did for and hence has the least corroboration from other sources. I have it in mind to completely revisit the entire thread in due course. Flag up any possible changes to the flow by means of source review templates and I will make sure I get to them when I re-approach this work.


 * At the moment I have my head down completing this first pass through Euclid, which I am going to concentrate on while I am still motivated. After a while I will need to take a break, but while I'm still making progress I'm sticking with that. After that I have no firm plans. It may depend on what "the people" want. --prime mover (talk) 17:36, 7 February 2015 (UTC)


 * I consider the big Warner revisiting due, at least regarding the NOS. Just so you know. I'm fine with postponing until you get around to it (I haven't been able to find a digital copy, so I can't do it myself). &mdash; Lord_Farin (talk) 21:13, 25 February 2015 (UTC)


 * As I say, leave me be to finish off Euclid (tantalisingly close now, nearly the end of book XI, only two more and some bits and pieces left to do) and then I'll revisit Warner.


 * A medical condition (and utter boredom with parallelepipeds) leaves me unable to concentrate for long periods at the current moment, so work is progressing more slowly than it ought. --prime mover (talk) 21:27, 25 February 2015 (UTC)

Feedback for Proofs
I noticed that the "Maximum Modulus Principle" has no proof, and I am working on one in my sandbox right now. But I am not comfortable posting it yet, and have a few questions I'd like to ask first (primarily looking to see if certain results are already proved on the site, and I just can't find them). Where is the best place to get feedback on works in progress before posting them? The main talk page? The questions page?

By the way, my two issues are:


 * I'm trying to find if what I've called the "Mean Value Property" (in my sandbox proof) is already proved on the site.
 * I know that my reference to "Absolute Value of Complex Integral" doesn't quite apply in this situation, but a very similar principle does. I'm wondering if the more general measure-theoretic version is proved somewhere on the site.

-- Ovenhouse (talk)


 * Feedback on specific works in progress can either be got by posting up on the talk page of the actual page in question, or if you believe there is a wider application for the question, then on the main talk page. Note that all edits can be seen on the "Recent changes" page, so everything that is posted up anywhere is visible by anyone who regularly trawls through this.


 * As far as I know, the "Mean Value Property" has not been done yet. I agree, it merits its own page. Note that every statement used needs to be backed up by a link to a demonstration of the truth of that statement; in particular there are references to integration by substitution and change of variables, neither of which have been covered in the complex domain on . Then of course we have the question of what further statements do these rest on? It's tortoises all the way down.


 * The more general result / similar principle to the "Absolute Value of Complex Integral" page do not as far as I know exist. It would be a worthwhile exercise to put them in place.


 * There has been embarrassingly little work done on complex analysis on this site. I have several times made a start but (such is my nature) I have each time become bogged down in the minutiae of the fundamentals to the extent that I have not yet got as far as complex calculus. Apart from me, the only other person to have done any work on the area was Anghel, who did a considerable amount of excellent work on Jordan curves.  However, the basics of complex calculus are still basically untouched.


 * A new approach from someone new would be welcome. If there are existing results, then (apart from the basic stuff and the aforementioned Jordan curve work) it is incomplete and does not form a coherent body of knowledge. If you are of a mind to flesh it out, then your efforts will be greatly appreciated. If results become duplicated, and as a result some redundancy creeps in, then no matter, we can merge existing pages into any new work that is being done.


 * Please note that there is a stringently-applied house style (there are reasons for this which have been discussed in some detail -- and at some heat -- elsewhere), so please do not be put off by a rash of "tidy" and "refactor" and "MissingLinks" templates appearing all over everything you do :-) -- these are more reminders for us janitors to work on than for you to be constantly going back and reworking stuff. --prime mover (talk) 10:57, 30 December 2014 (UTC)


 * If you are in doubt or cannot find a specific result, do not hesitate to use the MissingLinks template, describing what link is missing. Someone else might find it, or write it up. In this way, at least the reference is recorded. &mdash; Lord_Farin (talk) 12:46, 30 December 2014 (UTC)

James S. Kraft
I deleted the self-redirect Mathematicians/James S. Kraft tonight. It was, however, referred to from other pages. Can you reinstate it with the information that ought to be there? &mdash; Lord_Farin (talk) 20:27, 1 March 2015 (UTC)
 * Fixed. When this sort of thing happens, it's often the case that it ought to be a redirect to the appropriate page in the "Writers" category, in this case: Mathematicians/Writers/James S. Kraft. No worries. --prime mover (talk) 21:04, 1 March 2015 (UTC)

Minimal Infinite Successor Set vs. Natural Numbers
Yesterday, I started on the next task on the list, Definition:Minimal Infinite Successor Set. I posted up Definition:Natural Numbers as Elements of Minimal Infinite Successor Set. Regarding this double nature of the MISS, it would make sense to separate the MISS approached from the context of set theory, and its interpretation as $\N$.

In particular, it would be better for clarity in naming pages to have "Natural Number" only refer to the abstract properties of $\N$ as described by Peano's Axioms (and, to a lesser extent, the NOS). Currently, however, the term also refers to what is more rightly called "Element of MISS". True, this obfuscation is ubiquitous in set theory and related areas, but I still think there's a case to be made for keeping the two separate on. Are you fine with keeping the name "Element of MISS" or would you prefer something else? We can also opt for the minimum of results necessary to prove the equivalence with "Finite Ordinal" and use that name for all other results. There are probably sources defining $\N$ as the collection of finite ordinals, which would be easier to cover if we choose the latter option. I'm curious to hear your thoughts. &mdash; Lord_Farin (talk) 18:59, 5 March 2015 (UTC)


 * The optimum approach from my point of view is also the most work! All of the following are done:


 * a) We have an object called "the set of natural numbers", which is your go-to page for any invocation of "natural numbers" which are more-or-less intuitively understood by people, along with a set of conditions which they fulfil (which can be defined as their "axioms"): they have commutative, associative and well-defined addition and multiplication; the latter distributes over the former; they are countably infinite (yes I know, this is how countable infinity is defined, etc.) And whatever other axioms they have.


 * b) We have an object called a "naturally ordered semigroup", which fulfils certain conditions, and can be shown to fulfil all the axioms of the natural numbers, and that the natural numbers are a NOS, and the natural numbers and a NOS are unique and can be identified as the same thing (the basic approach of Warner, and as it has been implemented on.


 * c) We define the MISS and do the same for it as b) (and we demonstrate that the MISS is derived from the ZF axioms).


 * d) We show that Peano's Axioms also give rise to the MISS.


 * e) We demonstrate that Peano's Axioms, the MISS and NOS all are interderivable.


 * Consequently I see the abstract properties of $\N$ being proven just once, but under whatever structure. The fact that they hold for the other structures is a consequence of the other structures being isomorphic. If the fact that these properties hold is a prerequisite for the proof of those isomorphisms, then yes, they can be proved for those structures.


 * But ultimately it depends on whatever we can find / derive as proofs. If we find (published somewhere) these properties proven using the one structure, then we can show that proof.


 * Hope this hasn't confused too badly. I'm distracted at the moment by a business trip tomorrow for which I am ill-prepared. :-( The good news is that it is in the delightful Luxembourg. :-) The bad news is that I fly in and out on the same day :-( --prime mover (talk) 19:58, 5 March 2015 (UTC)


 * Well, this is more or less what I see as well. However, unless you mean Peano's Axioms by (a), there's not a good way of doing that (we cannot prove things about an intuitive concept with desired rigour). Rather, we can only postulate axiomatisations and constructions which seem reasonable and prove them equivalent (the painstaking process of which I have been intermittently working on for months now).


 * I have carefully proved that NOS and MISS form Peano Structures, and the uniqueness of PA and NOS (well-definedness of MISS upcoming). A proof that PA gives rise to NOS is also due, but still missing at this point (to finish the equivalence of PA and NOS proper).


 * There is room for replacing (or rather, augmenting) the very formal treatment of the operations on $\N$ with intuitive descriptions, and including excerpts of those in the intuitive section of Definition:Natural Numbers.


 * I have given it some more thought and will proceed to split the set-theoretic meaning of "natural number" to "finite ordinal", prove "element of MISS $\iff$ finite ordinal" and reduce the occurrences of "element of MISS" to the minimum.


 * That all said and agreed, I wish you best of luck on your trip. Too bad you won't stay on mainland Europe for too long, it's great :). &mdash; Lord_Farin (talk) 20:54, 5 March 2015 (UTC)


 * I'll leave it completely up to you to accomplish what you believe needs to be done to sort out the Peano Structure approach. My personal view is: as it is so important historically, and as it provides such a usefully accessible entry point for the beginner in axiomatics and the foundations of mathematics (and because I like it so much), it would be good to place it prominently in the dissertation.


 * When all is complete I will go through the various works I have access to (Deskins, Warner, Halmos etc.) and review their contributions in light of what we have -- and the cycle will repeat. :-)


 * I took a deliberate career decision mid last year to specifically not have to work abroad so much as I had been doing -- I'm okay with travelling, but my wife's health is poor and I need to be here for her. Last year I spent considerable time in Luxembourg, and would move there given half a chance. But if I want to keep certain aspects of my life together, that won't be immediately possible ... I love Europe, I always have. --prime mover (talk) 21:04, 5 March 2015 (UTC)

Theorems about multi-defined concepts
I ran into Natural Numbers are Transitive Sets in the ongoing rewriting operation. We already have Equivalence of Definitions of Minimal Infinite Successor Set, so we already know that the $\N$ in Halmos' eyes is equal to the set of finite ordinals. Since ordinals are by definition transitive, there is not really a theorem needing proof.

I'm bringing this up generically: What should we do in these cases? Post up the proof as if the equivalence hadn't been demonstrated? Include a trivial a fortiori reasoning as proof 1? &mdash; Lord_Farin (talk) 10:40, 12 April 2015 (UTC)


 * This result is part of Halmos's presentation of the natural numbers. He starts by setting up the numbers in the usual way: $n^+ = n \cup \{n\}$, introduces the Peano axioms, and then uses this result as part of his proof that $\N$ is a Peano set. That is:


 * "The proof of (V) is not trivial; it depends upon a couple of auxiliary propositions. ... (i) no natural number is a subset of any of its elements, and (ii) every element of a natural number is a subset of it. Sometimes a set with the property that it includes ($\subset$) everything that it contains ($\in$) is called a transitive set. More precisely, to say that $E$ is transitive means that if $x \in y$ and $y \in E$, then $x \in E$. ... In this language, (ii) says that every natural number is transitive."


 * In my mind, this result precedes Equivalence of Definitions of Minimal Infinite Successor Set, coming from the direction that the construction of $\N$ via Peano and Zermelo/Fraenkel. In fact, at this stage of the development of numbers, Halmos has not even defined "ordinal" -- he does not get that far till chapter 18. I believe I stopped at 17, having been unable to get past differences of opinion as to how this material was presented (as usual I am far too easily distracted and derailed by trouble with detail) and I started work on something else instead.


 * I believe there is a point to establishing this basic result, because in the context it is made, it needs to be made clear on the road towards proving that $\N$ is a set of ordinals. --prime mover (talk) 11:36, 12 April 2015 (UTC)


 * Fair enough, I'll keep it in. I got an idea, you will get the e-mails about my actions :). &mdash; Lord_Farin (talk) 11:40, 12 April 2015 (UTC)


 * Cf. Element of Minimal Infinite Successor Set is Transitive Set. I hope you like it. &mdash; Lord_Farin (talk) 11:46, 12 April 2015 (UTC)


 * I think so -- I'm going to revisit Halmos once I've done Warner and make sure it all hangs together. The crucial line is the one in Chapter 11: "A natural number is, by definition, an element of the minimal successor set $\omega$. This definition is the rigorous counterpart of the intuitive description according to which they consist of $0, 1, 2, 3$ "and so on." Incidentally, the symbol we are using for the set of natural numbes ($\omega$) has a plurality of the votes of the writers on the subject, but nothing like a clear majority. In this book that symbol will be used systematically and exclusively in the sense defined above."


 * Thus if we replace all references to "natural numbers" in the results and definitions on this thread by "minimal infinite successor set" as is in progress, we will then know that we are specifically building up the required knowledge about MISS and only after that has been established do we then "equate" them with the Natural Numbers by playing the "equivalence of definitions" card. --prime mover (talk) 11:54, 12 April 2015 (UTC)

Ok, I will add an entry to your worklist (towards the bottom of the NN refactoring page) for going through Halmos. You can add anything coming out of that as a sublist under that item.

Relatedly, have you finished source-reviewing Warner? Probably not. I will move that one to your task list as well. &mdash; Lord_Farin (talk) 11:57, 12 April 2015 (UTC)


 * No I haven't, not yet. I'm still bogged down on rationalising the material around index laws. --prime mover (talk) 11:58, 12 April 2015 (UTC)

Closure notation
Presumably missed in the plethora of other changes: Definition talk:Upper Closure. &mdash; Lord_Farin (talk) 15:47, 13 April 2015 (UTC)