keystone

I'm done with our conversation.

I share that trait as well—I need some time to fully generalize my ideas. As I work through this, I’m noticing that some adjustments are necessary, so...

I work in simulation as an engineer, not a computer scientist. I'm new to graph theory and currently only utilizing the basics. Even though the concep...

Someone could say the same thing about the epsilon-delta formulation of a limit, which was introduced to give calculus a more rigorous foundation. Aft...

A real number corresponds to a specific subgraph within a potential structure. In the 1D case, this is represented by a potential curve and the two po...

An actual curve in 1D is unique in that it is fully defined by its endpoints. However, in 2D and higher dimensions, a curve is determined not only by ...

Each indivisible object, whether potential, pseudo, or actual, is represented as a vertex within a structure, regardless of its dimensionality. This a...

@"fishfry": Would you be open to re-engaging with me on this topic (that we discussed months back)? I believe the post quoted here will give you somet...

Good point. I've needed to learn this lesson too many times. Since the functions I'm working with all converge, I don't believe the supremum is necess...

Yeah, my view leans heavily on algorithms. Yes. I have since edited the post to clarify this. Suitable for what? Good point. I have since edited the p...

EDIT: LOOKING BACK THIS POST HAS ERRORS. IF YOU CARE TO RESPOND PLEASE LET ME KNOW AND I'LL FIX. THANKS. : While working on my response, I realized it...

Natural number arithmetic does not involve infinities, yet natural numbers are inseparably tied to {\aleph_0}. In a similar vein, I argue that real ca...

I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed th...

I view transfinite cardinal and ordinal numbers as crucial for understanding the nature of infinity, and, as you know, they can be manipulated through...

For the moment, please treat 1/1 and 1.0repeating as distinct objects. Without bringing in the SB tree, let me just say that the former is a fraction ...

I agree. However, the main point of my post was to clarify that I'm not working with Cauchy sequences themselves, but with the algorithm used to const...

Apologies for the typo. Also, I initially used -1/0 to represent negative infinity because that’s how it appears in the Stern-Brocot tree, but since w...

First, I'd like to point out that this part (Part 2) takes some liberties with actual infinities for explanatory purposes (and to keep my individual p...

Actually, I think you're the sinkhole. You seem to enjoy destructive conversations.

I'm working with standard finite graphs, nothing unorthodox about my use of them. As such, I don't need to produce an original definition of them. If ...

It's ironic that you got cold right after I went back, carefully studied, and addressed your comments on topology. That feels harsh, but I suppose I s...

You raised a single issue with my response, which I immediately clarified-specifically, that by "1D drawable," I simply meant a 1D analogue of the est...

Perhaps I'll head in this direction and see what you think... I don’t have much experience with logic yet, but from what I know, my perspective seems ...

Instead of discussing 2D continua and area, let’s simplify by returning to 1D continua and length. Length is not a property of an infinite collection ...

I understand that you prefer not to lead the conversation, but I want to sincerely thank you for asking thoughtful questions that have helped me bette...

Thank you for taking the time to try to understand the figure and for sharing what didn’t make sense. I now realize that I skipped too many steps and ...

Edit: I wrote a response here but I want to reflect further on it so removing...

The edges do not represent objects like lines; rather, they signify connections. When I say that edges can be cut, I mean that these connections can b...

No, k-vertices cannot be partitioned. The only objects in these graphs that can be 'cut' are the edges.

I am doing quite the opposite. K-continua can only be partitioned at the edges connecting the vertices. As such, k-continuum 1 cannot be transformed i...

k-continuum 1 k-continuum 1 is represented by the 3-vertex graph: a - <a b> - b, where: vertices a and b represent k-points. vertex <a b> represents a...

I share your sentiment on all accounts - from the hopes to the doubts. Thank you.

There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks...

It's not you, I should have provided a description of the structure I was referring to. I intend to reply to you but I likely will not find time for s...

What I termed '1D drawable' is intended to be the exact 1D analogue to a planar graph, nothing more. I'm presenting my ideas informally, but certainly...

Let's have another take at this. In this take, I will not mention numbers. I also will not use 'partition' anymore as it suggests that indivisible obj...

I was wrong to mention numbers/intervals without properly establishing the notion of a continua. I'm going to leave that message there, but please all...

The link you provided is fascinating, especially since calculus and graphs are central to my perspective. However, I believe the connection stops ther...

I will use the Euclidean line instead of the plane, since we're currently focused on 1D. I'm interested in distinguishing between objects and operatio...

Finite - correct Undirected - correct Loopless - correct (doesn't this mean that no k-vertex is connected to itself?) CONNECTED: I've realized that I ...

I'm going to respond in two posts. This post covers topics not directly related to my ideas and my next post will cover topics more directly related t...

I thought I indirectly addressed this when I said I was going to go back to your earlier messages on topology and respond to them. But if you're looki...

I think the S-B tree is just one particularly pleasant way to cut a continuum. ...we never did get to calculus in the last thread. I don't know whethe...

I see you've already followed up on this. I haven't read it yet as I really want to spend sufficient time digesting it and responding. As I mentioned ...

Let me restate the examples I mentioned: Naïve infinite set theory is thought to be about actually infinite sets when I think it is really about poten...

That seems to be the way with a lot of things. I'm certainly hoping to finally reach such moments of clarity in logic, and eventually topology.

For example: That set theory is about infinite sets. That Cauchy sequences are infinite sequences. That reals are numbers in the same sense that ratio...

Me too. I often think about giving this up and just spending my free time like most people do. I'm truly open to being pushed in either direction. It'...

fundamental objects: indivisible wholes that serve as the basis for constructing composite objects (which are composed of other fundamental objects) t...

Consider the following as an ostensive definition. Is this not basic? https://i.imgur.com/LMImtl4.png https://i.imgur.com/kD3om8e.png https://i.imgur....