All of Kevin Clancy's Comments + Replies

I think this is an informal presentation of a subject which should only be presented formally. There's already a page called Asymptotic Notation for this topic.

Yes, that's correct. I wonder if it is even a good idea to talk about transitive sets in the transitive relation page, as most people who are interested in transitive relations are not likely to care about transitive sets. When this page is expanded beyond stub status, I hope that it will focus mostly on transitivity, rather than related concepts such transitive sets, posets, and preorders.

There is a page for linearly ordered set. It is called "totally ordered set". This is one of those situations where it would be nice for arbital to have a synonym system.

I see that there is a description of double scaling above. I assume that this is what "the product rule" refers to, but it is never explicitly labeled as such. Maybe giving the paragraph above Reverse a bold heading titled "The Product Rule" would help with this.

I think it's confusing to introduce multi-argument functions before talking about currying. This makes it seem as though multi-argument functions are an intrinsic part of the lambda calculus, rather than just functions that return other functions.

There is already a page about this topic, Join and meet.

I suspect that this is going to be too fast-paced for beginners. They are going to need multiple examples and exercises for each of the concepts introduced.

This is sentence is kind of confusing. It seems like it's trying to say that if we know A = B, then we can substitute A for B (or vice versa) and get.... A = B?

It seems like it should say that if A = B, and A occurs in some equation, then we can substitute B in for A in that equation, and the resulting equation will hold.

This relies on a principle "other way" introduces but, in my opinion, is not explicit enough about: $\frac{n}{m}=n\times \frac{1}{m}$. Could this be made more explicit at the end of the "other way" section? e.g. "What we've really shown here is that $\frac{n}{m}=n\times \frac{1}{m}$".

I think that every metric space is dense in itself. If X is a metric space, then a set E is dense in X whenever every element of X is either a limit point of E or an element of E (or both).

It looks like there is a word missing from this sentence. I'm not sure what it is trying to say.

Correct me if I'm wrong, but isn't it idiosyncratic to define $\le$ as a predicate rather than a relation? I know of at least three books that describe it as a relation: The Joy of Sets by Devlin, Principles of Mathematical Analysis by Rudin, and Introduction to Lattice and Order by Davey and Priestly.

Also, isn't $\le$ called an order rather than a comparison?

I bring this up because I would like there to be consistency between this page and the Partially ordered set page. I think both pages should follow the conventions of mathematics.

The title mentions Cauchy sequences, but the body does not. Doesn't this definition consider classes of non-converging sequences as real numbers?

Thanks Chris. Edit accepted.

I understand what you're saying and I think it's a good point. The problem is that you're developing an algorithm (a non-terminating one) that finds real numbers rather than providing a definition of them. It turns out that providing a definition of real numbers is not a simple as it may at first seem. This presentation is somewhat similar constructive analysis, in which a real number is defined as regularly converging sequence of rational numbers; importantly, constructive analysis does not define real numbers as infinite sums of these sequences, because ... (read more)

Maybe. I haven't done so because the underlying set page describes underlying sets specifically in terms of algebraic structures. I think that a link to that page would therefore just cause confusion.

I have no plans to write about real analysis. I just created this page so that I could use it in conditionals.

I'm pretty tired right now, but this definition seems kind of circular to me. It involves an infinite sum, and infinite sums are defined in terms of limits. But a limit of rational numbers is defined in terms of the set of real numbers. Maybe it would be better to present the definition of real numbers that one would find in a real analysis text.

Alexei I was going to add an examples lens to this page, but I seem to have lost the ability to create lenses. I remember being able to create lenses by placing an orange button in the bottom right corner. That does not currently work, however: the "create lens" icon doesn't show up.

About the citations: what I actually meant was that I want to have a bibliography, so that I can give due credit to any sources that I used to help write this tutorial. I don't want to add a bunch of superscripts into this document.

I can't reproduce it either. Maybe it had something to do with the USB keyboard that I was using.

The editor kept automatically scrolling to top when I was trying to edit this page in Firefox just now.

After another session of using Arbital, I have a few questions and comments.

1.) Is there any mechanism for citations? There probably should be. A lot of what I have written about order theory so far is inspired by Davey and Priestly's Introduction to Lattices and Order. I don't think it's realistic for someone to pull an entire tutorial on a mathematical topic directly out of their brain.

2.) I love the hover-over definition display. It's very convenient for looking up definitions without having to transition to other pages.

3.) It seems that it would be use... (read more)

I intended this to be a wiki page. My plan is to gradually develop it into a full fledged tutorial on order theory (which might take awhile). I see that you have invited me to the mathematics domain. Do I need to join this domain somehow?