Question for the math folk nit interval [0,1] contains infinitely many points. Is that correct? If so, how many points do you think it should have? — fishfry
Something discrete. Yet "discrete space" is impossible if it is to remain space.
And modern physics does not posit infinite divisibility. In fact in physics, space is divisible down to the Planck length, equal to around 1.616255...×10−351.616255...×10−35 meters. Below this distance, our physics breaks down and we cannot sensibly speak of what goes on or how space is. There's a Planck time as well, a minimum time interval below which our physics breaks down and can't be applied. — fishfry
This is a commonly held sophistry. As i demonstrated on this thread, everything in this world is made of infinite parts and I BELIEVE the conclusion is that everything is finite and infinite in the
exact same respect. That last part is what I was trying to explore
Oh darn, I sandbagged myself again. People always bring up Banach-Tarski, and I say, "B-T is at heart a simply syntactic phenomenon that I could describe in a page of exposition if anyone was interested," and they invariably have no interest. One of these days someone's going to say, "I'd like to see that" and I'll do it. But I see once again that you name-checked B-T but don't actually have an interest in it. And I got hopeful, only to be disappointed again. I am telling you that the heart of B-T is simple and surprising and perfectly clear, but nobody wants to hear about it. I pointed you at the references but you had no questions. One of these days ... — fishfry
I don't see how anyone with a brain wouldn't want to know how to get two objects out of one without referring to infinities. Such a theorem is incredible and I hope you do codify it into a thesis that others will read and appreciate. I for one am having trouble with it because it's of such a nature which I do not think I will understand it by READING it, as opposed to having it explained in person where I can cross examine every step. Reading it is just to much for me
In math, given a line, you can pick any two points, label one point 0 and the other point 1, and that length is your basic unit. — fishfry
That is arbitrary, as is the Plank length
That falls directly out of calculus. And as andrewk noted in the Gabriel's horn thread, it's analogous to the fact that the infinite series 1/2 + 1/3 + 1/4 + 1/5 ... sums to infinity, yet the series 1/4 + 1/9 + 1/16 + 1/25 + ... has a finite sum. Just a mathematical fact that takes a bit of getting used to, but is undeniably true. — fishfry
Something I need to consider more, thanks.
Well Euclid's axioms are a fine set of basic axioms. And if you drop the parallel postulate and replace it with either zero or many parallels through a point parallel to a given line, you get various flavors of non-Euclidean geometry. In modern times, Tarski's axiomitization of Euclidean geometry is of interest.
https://en.wikipedia.org/wiki/Tarski%27s_axioms — fishfry
"Sir Henry Savile remarked in 1621, there were only two blemishes in Euclid, the theory of parallels and the theory of proportion. It is now known that these are almost the only points in which Euclid is free from blemish. Countless errors are involved in his first eight propositions. That is to say, not only is it doubtful whether his axioms are true, which is a comparatively trivial matter, but it is certain that his propositions do not follow from the axioms which he enunciates."
Bertrand Russell
It becomes very confusing, which is why I was trying to find something basic about space that I could use as "first principles" in a Cartesian fashion