My question is this What is the least upper bound ordinal of my linear n-symbol partition ordinal?

I am self-learning person. I create the ordinal sequence that I can't prove it mathematically.

I know the fundamental step by step method of how to increase my ordinal to any higher ordinal but I don't know how to write it down mathematically. The the fundamental step by step is like an axiom which is taken to be true. The problem is as the ordinal get larger, the fundamental step by step will be longer to work up to the higher ordinal. At some point, it will take too long time to do.

I try fix this by looking for the pattern can create some formula to make larger step. But the problem is that the formula is totally based on the observation which isn't proved to be correct. However, I can't find any example that make the formula incorrect at all. What should I do ? Should I keep using it ?

Another problem is that I don't know how to prove those formula because it is the formula in my ordinal which appear to be different from the normal mathematical for me. Also it might be because of that my question is too hard for me or probably beyond my knowledge. If it is really beyond my knowledge, what should I do ? Which knowledge I have to learn it understand my ordinal ?

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    $\begingroup$ In the post you linked, you basically gave out a huge list of examples, but it doesn't seem like you have some formulas that explain what your encoding method is that you made up, or what exactly you're doing in general. If you have not taken a basic course or read a book regarding the basics of proofs, sets, or functions, you might want to start there first before jumping into ordinals. $\endgroup$ Commented Jul 10, 2023 at 18:17
  • $\begingroup$ @Accelerator Ok, I just read the Naive Set Theory by Paul Halmos. But the problem is I don't know how to explain my encoding method mathematically. Base on Naive Set Theory by Paul Halmos, in my system, "0" is empty set. The successor of "0" is "00". The successor of "00" is "000". The successor of "000" is "0000". And so on. Do I explain my encoding method mathematically correctly ?. $\endgroup$ Commented Jul 10, 2023 at 20:07
  • $\begingroup$ Sorry, I don't know much about ordinals to give good feedback on it. Maybe you could say something like, "Let $0$ be the empty set and its successor is $00$. Then the successor of that is $000$, then $0000$, and so on." I don't know how it fits into your post you linked, but I was thinking you could also try to describe in plain English how a list of zeroes maps to its corresponding ordinal given some certain conditions. That's all I know. $\endgroup$ Commented Jul 10, 2023 at 23:00
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  • $\begingroup$ @hardmath Thank you but after I read Naive Set Theory by Paul Halmos. I think I will be able to understand the my question soon. $\endgroup$ Commented Jul 11, 2023 at 17:58
  • $\begingroup$ Interaction will do plenty of good. Even if the question isn't clear enough initially, your willingness to talk will bring others to the table as well, as this meta thread (and Mike Earnest in the main thread) showed. With interaction you will also be able to eventually ask the right questions clearly without needing help beforehand. $\endgroup$ Commented Jul 12, 2023 at 2:40

3 Answers 3


Let me offer some general advice that is not directed at the linked question specifically, but nonetheless may be helpful.

No matter how advanced you are mathematically, there are questions that you will be able to ask which you won't be able to answer. To take an extreme example, a bright primary school student might be able to understand the statement of Fermat's Last Theorem, even though only a tiny fraction of professional mathematicians can even begin to comprehend its proof. Asking questions about your subject material, and trying to the answer them yourself, is a great way to understand it more deeply. However, you must be prepared for the possibility that neither you, nor even your lecturer, can answer them. You may take solace in the fact that even in cases where you can't answer your questions fully, this is often a useful learning experience.

It is undoubtedly frustrating when you realise that it is likely that the answer to one of your questions is too advanced for you. However, so long as it meets the guidelines of the site, I would recommend posting it here anyway, for two reasons:

  1. Well-posed questions that receive good answers are useful to the mathematical community as a whole, not just the particular person who asked the question. (A possible exception to this occurs when your question is a duplicate – please search for duplicates before asking.)
  2. In months or years to come, you might re-discover your question, and realise that you now have the ability to understand its answer. This is a very satisfying experience.

On the other hand, it might seem defeating if you cannot answer any question you have about the material you are learning. One way to mitigate this issue is by picking up a good introductory book on the subject. Often the authors of these books attempt to make the presentation self-contained – so many questions that arise naturally from reading the book are also answered in the book (or at least, they can be answered with the tools introduced in the book).

  • $\begingroup$ Thank you for your advice. However, I have no idea what introductory book that I should looking for. I do read a lot about ordinal on the internet. I know I might not understand them all but I don't see how they can help me to solve my ordinal problem because my ordinal work in different ways. As I work at it, I feel like it is a new kind of ordinal that is different from normal mathematical ordinal. Some of it is same but some of it is different. That is why I have no idea what kind of introductory book that I should looking for. $\endgroup$ Commented Jul 10, 2023 at 12:41
  • $\begingroup$ @Justamanintheworld: Before jumping in and learning about ordinals, I would recommend learning more basic set theory. One very nicely written book about set theory (that does cover ordinals towards the end) is Naive Set Theory by Paul Halmos. I cannot say whether that book will answer your specific question if you read it. I can say that, if you understand the content of the book, then you will be in a much better place to learn the answer to your question. $\endgroup$
    – Joe
    Commented Jul 10, 2023 at 12:46
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    $\begingroup$ @Justamanintheworld: I'm not trying to discourage you, but usually, when beginners to a subject try to "invent" a new concept, this doesn't lead anywhere fruitful. New constructions that are actually useful are invariably formulated by mathematicians with significant expertise in their area. $\endgroup$
    – Joe
    Commented Jul 10, 2023 at 12:48
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    $\begingroup$ @Justamanintheworld: Anyway, learning set theory with Halmos's book might not be the best idea if you haven't taken a University-level course in pure mathematics before. Have you? $\endgroup$
    – Joe
    Commented Jul 10, 2023 at 12:50
  • $\begingroup$ No, I haven't taken any University-level course in pure mathematics before. Should I read Naive Set Theory by Paul Halmos ? Or is there any better choice ? $\endgroup$ Commented Jul 10, 2023 at 12:58
  • $\begingroup$ @Justamanintheworld: Usually, students start off with learning basic analysis or basic linear algebra. What book to recommend really depends on your pre-existing knowledge. For instance, have you learnt calculus before (derivatives, integrals, and so forth)? $\endgroup$
    – Joe
    Commented Jul 10, 2023 at 13:03
  • $\begingroup$ Yes, I have learnt calculus in university. I am not sure if I have learn about basic analysis or basic linear algebra or not. Because I never heard them from any of my instructors. Maybe, they call them with different name. $\endgroup$ Commented Jul 10, 2023 at 13:07
  • $\begingroup$ @Justamanintheworld: Basic analysis studies the properties of the real numbers, and proves many of the theorems you have met before in calculus class (e.g. the chain rule). Linear algebra is the study of vector spaces, linear maps (also known as linear transformations), and matrices. $\endgroup$
    – Joe
    Commented Jul 10, 2023 at 15:37
  • $\begingroup$ Ok, I do see some proves of some theorem of calculus (e.g. the chain rule) in the calculus text book but never have specific class of basic analysis in university. Also same for the linear algebra, I meet some of them but never have specific class of it in university. $\endgroup$ Commented Jul 10, 2023 at 15:51
  • $\begingroup$ @Justamanintheworld: Okay. I think you would be able to read Halmos, but it probably will be challenging. There are definitely parts of it which strike me as being particularly difficult (e.g. the proof of Zorn's lemma), and so it would help if you have someone to guide you if you get stuck. You can also consult Mathematics Stack Exchange! $\endgroup$
    – Joe
    Commented Jul 11, 2023 at 10:59
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    $\begingroup$ @Justamanintheworld: If you don't have a good time with Halmos, then I'd recommend learning a different subject (rather than picking up another set theory book). I estimate that someone who has worked their way through a book such as Michael Spivak's Calculus would be ready for Halmos's book. $\endgroup$
    – Joe
    Commented Jul 11, 2023 at 15:11
  • $\begingroup$ Thank you very much. But I think Halmos is good. I like it and I have good time with it. Now, I think I understand how to define ordinal now but I still don't totally understand about the axiom of substitution. Is it ok to define something like ω*ω = {ω,ω*2,ω*3,ω*4,...} ? $\endgroup$ Commented Jul 11, 2023 at 17:56
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    $\begingroup$ @Just a man in the world: But I think Halmos is good. I like it and I have good time with it. --- As a big-picture overview of certain non-formal/non-axiomatic aspects of ordinal numbers, you might be interested in John Baez's essays on ordinal numbers in the Azimuth blog: Part 1 and Part 2 and Part 3. $\endgroup$ Commented Jul 12, 2023 at 19:43
  • $\begingroup$ @DaveL.Renfro Thank you. In fact, I already real something like that as well. Overall, I feel like there are something that could be more there in normal ordinal system since it hit the limit at Bachmann–Howard ordinal. That is why I create my new ordinal system. Maybe, if I have more knowledge and extent it, it might be able to go beyond the Bachmann–Howard ordinal. $\endgroup$ Commented Jul 14, 2023 at 20:32

There's no restriction on the number of good questions you can ask. I've been in a similar self-learning situation, both in Math SE and other sites, and as long as it's not an emergency, what I do is ask either a simpler question to which I don't know the answer, or one that I do, or usually kind-of do know the answer or can guess, but can't defend it.

I sometimes list my guess right in the question. Often one or two answers are posted with formal answers, helpful insights and links to further reading, especially if I mention in the question that those are particularly welcome.

I use these as stepping-stones to get closer to being able to form my real question clearly enough that it either gets more helpful comments or an answer or two.

As long as your questions are received as acceptable Stack Exchange question, there is absolutely nothing wrong with working a problem through a series of questions and answers, especially if each one links to the previous for background reference.


Having edited the question you linked at length, I would also suggest that learning to express your questions clearly (also known as learning to write academic papers) is a good skill, too.

Academic writing, such as you are attempting in your question is hard work and takes years to learn. I spent about fifty minutes and didn't even edit through your entire question -- and in theory, I at least have some idea what I'm doing!

But I will easily spend fifty hours writing a paper draft -- and that is after studying the topic for a year!

There are those who work on paper-writing full-time. For me, it is just a part-time hobby at the moment. If you are interested in this, I would consider entering academia. Get an undergraduate degree in math, then an MS and PhD. Along the way, you will be working with others who share your passion. And the courses you take will ideally align with it as well. If you can, at the same time, convince someone to pay you for your degrees, all the better!

  • $\begingroup$ I understand. In fact, I have undergraduate degree in physics. However, I found that academia isn't my way. I am just don't like it because I don't want to learn everything about it. I only want to learn only some specific subject that I want to but I can learn other relate subject only when I need them. $\endgroup$ Commented Jul 18, 2023 at 17:23

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