The real numbers are by construction, continuous, as any knife cut does in fact define a real number.
This is a fundamental fact about analysis and the real numbers as defined and used for over 100 years.
This is just an assumption.
No, it is not an assumption about physical reality, but a true statement (true by definition) for the real numbers and the Euclidean line. You can't have two lines cross in Euclidean space and fail to have them intersect in a point.
Example from analytic geometry.
$$ A_1 x + B_1 y = C_1 $$ is the equation of a straight line in the x-y plane. $$A_2 x + B_2 y = C_2$$ is the equation of a different straight line. If $$A_1 B_2 - A_2 B_1 \neq 0$$ then it is guaranteed that these two lines are not parallel and thus cross. Thus they intersect in exactly one point. What is that point?
$$\begin{pmatrix} x_0 \\ y_0 \end{pmatrix} = \begin{pmatrix} A_1 & B_1 \\ A_2 & B_2 \end{pmatrix} ^{-1} \begin{pmatrix} C_1 \\ C_2 \end{pmatrix} = \frac{1}{A_1 B_2 - A_2 B_1} \begin{pmatrix} B_2 & -B_1 \\ -A_2 & A_1 \end{pmatrix} \begin{pmatrix} C_1 \\ C_2 \end{pmatrix} = \begin{pmatrix} \frac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1} \\ \frac{- A_2 C_1 + A_1 C_2}{A_1 B_2 - A_2 B_1}\end{pmatrix}$$
And so the point $$(x_0, \, y_0 )$$ is the only simultaneous solution to both equations whenever the two lines are not parallel.
$$A_1 x_0 + B_1 y_0 = A_1 \frac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1} + B_1 \frac{- A_2 C_1 + A_1 C_2}{A_1 B_2 - A_2 B_1} = \frac{A_1 B_2 C_1 - \underline{A_1 B_1 C_2} - A_2 B_1 C_1 + \underline{A_1 B_1 C_2}}{A_1 B_2 - A_2 B_1} = \frac{ A_1 B_2 - A_2 B_1 }{A_1 B_2 - A_2 B_1} C_1 = C_1 \\ A_2 x_0 + B_2 y_0 = A_2 \frac{B_2 C_1 - B_1 C_2}{A_1 B_2 - A_2 B_1} + B_2 \frac{- A_2 C_1 + A_1 C_2}{A_1 B_2 - A_2 B_1} = \frac{\underline{A_2 B_2 C_1} - A_2 B_1 C_2 - \underline{A_2 B_2 C_1} + A_1 B_2 C_2}{A_1 B_2 - A_2 B_1} = \frac{- A_2 B_1 + A_1 B_2 } {A_1 B_2 - A_2 B_1} C_2 = C_2$$
If $$A_1, A_2, B_1, B_2, C_1, C_3$$ are all integers, it does not follow that $$x_0 \; \textrm{and} \; y_0$$ are integers. But if $$A_1, A_2, B_1, B_2, C_1, C_3$$ are all real numbers, then when the lines cross it follows that $$x_0 \; \textrm{and} \; y_0$$ are real numbers. Thus the strength of algebra of real numbers is added to the strengths of geometry without the need to ponder if the intersection point exists.
May be the ancient Greek assumption, as you mentioned earlier.
The Greeks, Romans, English, etc. all assumed the geometry of Euclidean space was the geometry of reality. This does not have to be true, and indeed, in the early twentieth century the presumption of mathematical flatness was found to be flawed. This does not provide evidence that the assumption of continuity was flawed or that the mathematical Euclidean space is not a self-consistent system.
But most important, it is impossible to say that reality is better modeled by a non-continuous mathematical model of space until you have developed a rigorous model of how such mathematics actually works and how (via the principle of correspondence) that no one has yet noticed the lack of continuity in the physical world. 1-D is simple, because the lattice spacing might be too small to notice, but higher dimensional lattices break the property of isotropy that makes circles round.
You claims about mathematics are wrong and your claims about physical reality are not argued from the evidence.
A point on a line corresponds to the mathematical number 0(zero).
You mean the distance between a point and itself is zero. Why limit it to lines?
Every non-zero number corresponds to a segment of line whose length is non-zero.
This correspondence is arbitrary. You need a segment chosen to be of one specific non-zero length before you can establish a correspondence between real numbers and lengths. (Examples, 1 inch, 1 meter, 1 Ångstrom, etc.). Since Euclidean geometry is scale-invariant this choice of unit length is completely arbitrary.
Adding a number is same as adding its corresponding segment of line.
That is the basis of much of analytic geometry when adding just finite numbers of segments together.
A point can be considered as a segment of line, whose length is zero.
That follows.
So, adding a point is same as adding the mathematical number 0(zero).
x + 0 = x.
Adding infinite number of points is same as adding infinite numbers of 0's(zeroes). And the result will be a point or the number 0(zero). We will not get any non-zero finite segment of line or a non-zero number.
There is no such operation as adding an infinite number of points. You add finite numbers of segments. The cardinality of points in a segment of non-zero length is larger than the cardinality of the set of counting numbers, so because you have two different orders of infinity at work here, your argument does not hold water. It has a big hole in it.
There is such a thing as a Cantor dust which is a infinite collection of points with
just the cardinality of the counting numbers the same cardinality as the points of a line segment. It has zero measure. So this is an example of an infinite collection of points
not forming a continuum. You need more
than just points to make a continuum as in the classical construction of a Cantor dust one starts with a line segment and postulate away the middle thirds of of all the segments until almost all points have been removed
and the structure is no longer continuous.
So, it is wrong to assume a point as the constituent of a line.
No, what is wrong is for someone uneducated in a field to make empty pronouncements about the invalidity of a field and expect people to listen to him.
As already advised earlier, I don't have time to spare for prolonged discussion at this time because I am busy finalizing my stuff ready for publication. I only came in to give you a heads up regarding the respective stances in this particular side-discussion between handsa and rpenner, mainly because my upcoming publication will, among many other things, provide that which rpenner has been requesting regarding re-definitions/improvements etc; as well as explaining those subtle things which QQ and handsa have been trying to convey.