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.
Kevin Clancy
Karma: 13
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: . Could this be made more explicit at the end of the “other way” section? e.g. “What we’ve really shown here is that ”.
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 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 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 as I’ve said, that would be a circular definition.
If you want to learn more about rigorous foundations for real numbers and related topics, I think that the book Calculus by Michael Spivak is a very approachable and well respected introduction to the topic.
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.
Alexei This page includes a conditional example that only shows up for people who know real analysis. I’m imagining that someone who reads this may have real analysis familiarity, but has not marked themselves as such on Arbital. I’m not sure what the solution to this is. Maybe it is to automatically add a section at the bottom saying that there is additional content for people who know the subjects that are mentioned in conditionals.
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.