You have done your imagery very well. I will wait and see what comes next. — jgill
I suppose I see some sort of a way to move forward by taking a lattice graph over an area and allowing the number of vertices and edges to increase without bound leading to a countable number of points in the area. — jgill
But this would be inadequate regarding the reals. But you might be able to push into the irrationals some way. — jgill
So far it appears everything you have given is uninteresting from a math perspective. — jgill
I don't think you will get a reaction from anyone but me until you produce a plan moving forward from your images of edges, vertices and surfaces. What is your goal and how do you plan to proceed? — jgill
I don't think you will get a reaction from anyone but me — jgill
1. should be interesting. — jgill
Intuitionism math perhaps. — jgill
You have density, but then continuity is next...I thought you were defining these lines as continuous. Fundamental objects. — jgill
There's an important distinction between handwaving and BS. Handwaving involves vagueness or imprecision, where the core idea might be sound but lacks detail or rigor in its current form. BS, on the other hand, is fundamentally incorrect—an argument that doesn't hold up under scrutiny and lacks substance from the start. — keystone
Indeed, with the very first predicate 'is a continua' still not fully defined, you've piled on a big mess of more of undefined terminology and borrowing of infinitistic objects while you claim to eschew infinitistic mathematics. — TonesInDeepFreeze
1D analogue of the established term "planar diagram" — keystone
Please, give me a chance. — keystone
If your offer to help was sincere — keystone
I don't need to waste my time and energy on you. — TonesInDeepFreeze
It's ironic that you became distant right after I went back, carefully studied, and addressed your comments on topology. — keystone
You need to define "1D analogue of the established term "planar diagram"" in terms that don't presuppose any mathematics that you have not already defined and dervied finitistically and such that it justifies such verbiage as about "embedding in a circle". — TonesInDeepFreeze
You are a sinkhole. — TonesInDeepFreeze
Move on to 2. — jgill
Actually, I think you're the sinkhole. You seem to enjoy destructive conversations. — keystone
I see a mistake in your last figure, typo probably. And I assume -1/0 (meaningless) designates negative infinity, however you define that — jgill
I see nothing of interest so far. — jgill
I've been overlooking the fact that real numbers are typically defined as equivalence classes of Cauchy sequences, not just individual Cauchy sequences. — keystone
Cauchy sequences themselves are infinite sets. — TonesInDeepFreeze
So far I'm not seeing anything beyond a line segment between two points that converge to one. From a continuum to a point. Why should one care about this? — jgill
By moving the focus from the destination to the journey the need for actual infinity vanishes. — keystone
In my recent posts, I have been establishing that real numbers instead describe potential k-curves, which can be thought of as yet to be constructed k-curves which when constructed have the potential to be arbitrarily small (but always retain a non-zero length). — keystone
By "actual infinity" I suppose you mean a kind of number that can be manipulated by arithmetic processes. — jgill
This is either very deep or shallow gobblygook. — jgill
But that program (even in an infinite world*) cannot actually output a set with a cardinality of ℵ0. Potential is important and I feel like it's been forgotten in our Platonist world. — keystone
Elementary calculus does not require "actual" infinities. It gets along quite well with unboundedness, or what you might call potential infinity. — jgill
As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill
I agree that calculus can work quite well with the concepts of unboundedness and potential infinity, but 'actual' infinities are implicitly assumed throughout the standard treatment. — keystone
As I have said before, I have written many papers and notes without ever becoming transfinite. — jgill
Have you written calculus papers/notes that are not (implicitly or explicitly) built upon infinite sets like R? — keystone
I was speaking of ordinal numbers beyond the naturals. Our definitions of "actual" infinities differ. No big deal. — jgill
It might appear that you are moving in the direction of Discrete calculus. — jgill
But go ahead. I am curious. — jgill
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