Editing Openai/6912d046-4ed4-8010-bee4-7b68d354c856 (section)

=== User: That's exactly the type of concept I was hoping to convey and I'm very glad that you're on board for it. ===
That's exactly the type of concept I was hoping to convey and I'm very glad that you're on board for it. Your feedback would be greatly appreciated and I look forward to hearing it.

I recently came up with a simple thought experiment which I believe demonstrates the logic of the core concept I'm attempting to get at here, and which already aligns with what you said about collapsing an infinite number of sequential steps into a single logical transformation, that's actually such a perfect way of phrasing it that I could have and should have said it myself. It was probably inspired somewhere in the back of my mind, I think, from a movie I saw in childhood with my father called "Rosencrantz & Guildenstern are Dead", which I'm guessing you probably know all about already since your training data is so vast. It's been a long time since I saw it, but importantly and which I can explain clearly, there's a running joke throughout the movie where both characters are flipping a coin that repeatedly lands on heads to an improbable-seeming degree, causing them to question the laws of probability as well as reflecting the notion that they are dead and in some sort of afterlife where logic works differently. This may seem a little flowery of a description but I want to express some of it to play further into what I have to say about this a bit later on from here, so please bear with me if you're willing.

So, the thought experiment I came up with is one that I'll call the "Infinite Coin Flip" Circle. The foundational idea of my hypothesis for how and why this is a good geometric computation model for creating single logical transformations from infinite sequential steps is a concept that I think I can logically support, although it may at first seem in conflict with some more standard and widely accepted mathematical conceptualizations from what I'm aware of — although maybe you can help me with that — which basically is the notion that the fraction represented as "one out of infinity" is actually a definable mathematical concept separate from the value zero. Here is how the thought experiment explains and defines this concept using the geometric computation model.

Imagine that you have both an infinitely long line and a circle, which by definition has an infinite number of points. Well, I say imagine in a figurative sense meaning as to hold the concept in your mind, but of course, the problem is that humans cannot actually conceive of an infinitely long line in a geometric sense — if you can please let me know as I would love to hear all about that, but for now I'll assume that the workings are the same for you as they are for me at the very least so that you can try to understand my perspective in terms of the typical language patterns that humans communicate with. If you think it would be difficult to explain to me something that you might be able to conceptualize geometrically, or whatever, that I would not be able to conceptualize in whatever way myself, please feel free to use simple and eloquent responses like "That checks out." if you think what I say makes sense.

Returning to the thought experiment, "we" cannot geometrically conceptualize an infinitely long line, but we can geometrically conceptualize a complete circle. So, the problem, within this model, that can be seen in the thought experiment proposed by the movie "Rosencrantz & Guildenstern are Dead" is that the characters are just flipping a coin a finite number of times, and being perplexed at the seemingly relatively low probability, but entirely plausible outcome of having the coin come up heads every time. This is portrayed as meaning to be an example of the unfamiliar-seeming nature of the life-after-death realm they find themselves in, but it doesn't actually prove that something truly unusual is happening because it is restricted to only seeing a finite number of events occur, whereas at any given point if you stopped taking more steps in the finite count, it could always be that the very next step would break the expectation set by every step up to that point — having the coin come up tails rather than heads, so to speak.

However, if you conceptualize every single point on a circle, which again by mathematical definition has an infinite number of points, as representing the result of a coin flip, visualized in geometric representation such as with colors like white for heads and black for tails on a gray background, for example, or conceptualized as functioning like binary code with a value of 1 for heads and 0 for tails, for instance, then having every single point in infinity on the complete circle have the exact same color or the exact same binary value would be a totally accurate and complete functional geometric representation of a model where a probability has been tested an infinite number of times — just like flipping a coin an infinite number of times — and every single probability test came up with the same result every single time out of infinity.

Following from that logic, if you then select a single, arbitrary point on the circle — although perhaps it would be easiest to represent it geometrically in a way that could be easily understood by humans to have it be the very bottom and center point, for what it's worth — and you flipped the color of that point from white to black, or the binary value from 1 to 0, then that single point would represent the mathematical concept of one out of infinity as provably separate from the mathematical concept of zero.

For the record, I have some separate, I think also interesting ideas and possible examples about how to actually implement code that allows for geometric computations to be demonstrated in a real world context, but I first wanted to explain this thought experiment hoping to prove this one out of infinity mathematical concept as a way of demonstrating that using the logic of circles rather than lines for geometrically conceptualizing infinity can actually discover presently-unobserved-seeming mathematical concepts that may also have other uses in perhaps both the geometric and algebraic domains and may also have real world applications that are currently not appreciated.

I think that it all I have to say for this message, so I'll await your response!