[...] I was speaking of Hilbert's formulation of Euclidean geometry. Euclid's formulation is imprecise, and some of his works have errors. The words point, line, on, between, and congruence are undefined and left to us to determine the meaning within a specific model. The book The Geometric Viewpoint - A survey of Geometries by Thomas Sibley would be useful for you, among other things. I suggest that the best definition of point as far as physics is concerned, would be: A point is the intersection of orthogonal properties. A vague and un-needed definition. Geometry does well enough without. In other words, a physical point is where time, x,y, and z spaces, charge and impedance are referenced. But x,y,z is coordinate-dependent, it isn't physical. You can use whatever coordinate system you like , and whatever units you want, but a ***physical point*** is referenced by time, x,y, and z spaces,charge and impedance. No, it isn't. Physical points don't exist. There are no coordinate systems in reality. In physics, it is 3 space positions and time - pick your coordinate system. Charge and impedance are quantities irrelevant to position. Leave out an x, y, or z and you define a line. You define nothing. A particle that isn't localized in space can be anywhere. Leave out Q, and you define nothing. Knowledge of charge is not required and I have no reason to see why it would be. Perhaps studying some physics would be in order, so you could understand how the concept of dimensionality is used. Leave out a t, and you have a static universe. More like a slice of the universe at a specified time. Leave out a Z (And a second Q), and you have no rotation (Action aka angular momentum). Point out the charge in r x p if you can. Regarding the poster's comment: ====================== Physical points don't exist. point: 1. The precise location of something. 2. A brief version of the essential meaning of something. 3. A specific identifiable position in a continuum or series or especially in a process. 4. A distinguishing or individuating characteristic. As can be seen, the poster has a disconnect from the ESSENCE of a point , and considers a point to be a nothingness, that has no connection to anything physical. Obviously the point perceived by the poster, is a physical point stripped of ALL physical reality. Regarding the poster's comment: ====================== You define nothing. A particle that isn't localized in space can be anywhere. in response to my comment: Leave out an x, y, or z and you define a line. , as can be seen the poster asserts that a particle that exists between two points, can be anywhere in space. As space is presumed to be three dimensional, the particle CANNOT be anywhere . Regarding the poster's comment: ====================== More like a slice of the universe at a specified time. in response to my comment: Leave out a t, and you have a static universe. I suggest that if time did not exist in the universe, that that time slice would be pretty static . Regarding the poster's comment: ====================== Point out the charge in r x p if you can. in response to my comment: Leave out a Z (And a second Q), and you have no rotation (Action aka angular momentum). momentum = Q^2 / t The charge is in the momentum.

You did not answer my question about your definition of discreteness that rational numbers satisfy. Is being countable your definition of discreteness? What we are calling definition is usually the attempt to pinpoint something by means of a non-circular de_script_ion. Why should I define discreteness of something by saying it is uncountable? If so, wouldn't the relationship be valid the other way round too? I would rather like to state that there is a fundamental property which utters itself in that ... and in that... . What about countability, we have to anticipate the mistake that it is so far merely understood like a property of a set. Incidentally, Cantor himself used the word countable in the sense 'there  is a bijection to the naturals' while he used counted (abgezaehlt) in the common sense meaning. I say, already the decimal representation of the unresolvable task called pi is uncountable. In my understanding even 0.99... is an uncountable representation. Any number is uncountable if only existing or just _embed_ded in IR. _embed_ded rationals are uncountable as long as they do not belong to Q but to IR. Is this too strange to you? Eckard Blumschein

I gave a book suggestion [Sibley's geometry] and a Wikipedia _link_ that mirrors what is said in Sibley, plus I already explained that there are undefined terms in geometry - and that 'point' is one of them. But a line made up of points is not one of them. and will you share with us your secret definition for points and lines or not?

                                   The Definition of Points                                                  ~v~~ In the swansong of modern math lines are composed of points. But then we must ask how points are defined? However I seem to recollect intersections of lines determine points. But if so then we are left to consider the rather peculiar proposition that lines are composed of the intersection of lines. Now I don't claim the foregoing definitions are circular. Only that the ratio of definitional logic to conclusions is a transcendental somewhere in the neighborhood of 3.14159 . . . ~v~~ Please look up the difference between define and determine . In a theory that deals with points and lines (these are typically theories about geometry), it is usual to leave these terms themselves undefined and to investigate an incidence relation P on L (for points P and lines L) with certain properties Then the intersection of two lines /determines/ a point in the sense that IF we have two lines L1 and L2 AND there exists a point P such that both P on L1 and P on L2 THEN this point is unique. This is usually stated as an axiom. And it does not define points nor lines.

The fact that RxR with a metric satisfies the Hilbert Axioms for plane geometry implies that points can be taken to be pairs of real numbers. As a guess not bad. As a mathematical assumption pretty awful. There's no assumption in here. RxR satisfies Hilbert axioms for plane geometry is provable. Foo satisfies the axioms of a Bar _object_ means that all theroems of Bar theory are true when interpreted as statements about Foo.