You meant to say the Cantor set has the cardinality of the reals, didn't you? It's the standard example of an uncountable set of measure zero.
1=0.999... infinities and box of chocolates..Phliosophy of Math...
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You meant to say the Cantor set has the cardinality of the reals, didn't you? It's the standard example of an uncountable set of measure zero.
Hmm -- that is one weird set of points. Nowhere dense or continuous but of cardinality of the continuum. I have updated that paragraph. Thanks.
It is easy to go wrong when you reason by analogy rather than by axioms, and I think that's why I got that wrong earlier.
http://en.wikipedia.org/wiki/Continuum_(set_theory)
It is easy to go wrong when you reason by analogy rather than by axioms, and I think that's why I got that wrong earlier.
http://en.wikipedia.org/wiki/Continuum_(set_theory)
Constituent of a line is the "infinitesimal non-zero segment of a line" bounded by two points.
The question is wrong-headed and thus indicates you have no idea what I mean by number and don't explain what you mean by number.
Is 420 = 419.99? Just a little levity from real life. Colorado highway officials changed the 420 mile marker to 419.99 because people kept stealing the 420 sign.
http://abcnews.go.com/Weird/wireStory/mile-marker-420-41999-thwart-thieves-21495213
http://abcnews.go.com/Weird/wireStory/mile-marker-420-41999-thwart-thieves-21495213
How can a question be wrong? Answer to my question at the post #604 would be either
1) A number has a constituent.
Or,
2) A number does not have a constituent.
Or,
3) People might not have thought about the constituent aspect for a number.
Or,
4) I may be the first person to ask such a question.
For this discussion, let us consider the real number whose value represents a quantity along a continuous line.
Numbers, like points, don't have constituents. Just because $$1 + 1 = 2$$ doesn't make 1 a constituent of 2. Neither $$2 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots$$ nor $$2 = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} + \frac{1}{15} + \frac{1}{230} + \frac{1}{57960}$$ means that 2 has those numbers as constituents.
Hmmm, a very thought provoking question there, handsa.
All I can think of offhand is that a decimal 'number string' may have both 'zero' (ie value-empty place(s) ) or 'non-zero' (ie, value-occupied place(s) ), each 'place representing the POTENTIAL 'value constituent' which may appear (or not) in each 'place' involved in 'constructing' that string, and, as that string may represent A POINT, then by extension one maybe could say that the number string constituent place values (zeros and non-zeros) are the 'constituents' of that 'point' the string represents? Just a quick thought/perspective, no more than that! What this 'take' on it may represent in maths/physics construct/reality functions/processes etc is definitely up for discussion! In general, I observe that any state/point is the outcome/resultant of the various 'constituent' factors/processes that lead to such state/point properties/location etc etc. That's all I have time to ponder /comment at this time, mate!
Sorry I don't have time to pursue this interesting question/discussion further, handsa. Have to go. Very busy. Thanks for the intriguing thought/question! Cheers.
The above is your own conclusion or you have some reference.
Considering analytic geometry, a number can be represented by a 'segment of a line'. If this 'segment of a line' can have some constituent; why not the number itself also be having some constituent?
Any constituent should satisfy the 'definition of constituent'.
All mathematical textbooks on Number Theory, Set Theory and Analysis strongly support this conclusion by defining numbers without even suggesting a place for constituents in such a role. You cannot argue with a definition, at best you may propose an alternate definition but you have to recognize what the mainstream definitions have been for over 100 years before you can know enough to propose an alternate. If you do propose an alternate, Richard Hamming and others will require you to do the heavy lifting to demonstrate that you have made a contribution.
A segment of a line does not represent a number in Analytic Geometry. A length does. Segments have positions and orientations in addition to their property of having lengths. 1027 line segments may all have the same length. That's 1027 different segments and just one number. Likewise a set of 3 items has 3 members, but the number 3 applies equally well to any set which can be put in a one-to-one relationship with that set and doesn't depend on any particular set or it's members.
My definition of constituent of an object (atom, line, Lego sculpture) means you can identify specific parts of a different nature whose removal changes the thing in a fundamental way. Removal of an electron changes an atom into an ion. Removal of a point from a line turns it into two rays (or turns a line segment into two line segments). Removal of a single Lego from a sculpture changes it into a different sculpture and may be repeated until one reaches the null-sculpture of zero Legos. You cannot "identify" or "remove" $$\frac{1}{230}$$ from $$2$$ in the same way because $$\frac{1}{230}$$ is also number and thus doesn't have a different nature than $$2$$, $$\frac{1}{230}$$ cannot be identified as a specific component of $$2$$ as the latter is in many ways more fundamental than the former and does not admit a unique decomposition into labelled pieces $$\frac{459}{230} + \frac{1}{230} = 2 = \frac{5}{3} + \frac{1}{3}$$ and since subtraction is identical to addition of the additive inverse number, subtraction isn't removal for numbers in the same sense as for lines or Legos.
(If you start with $$2$$ and subtract $$\frac{3}{7}$$ or $$\sqrt{2}$$ multiple times, you reach no natural stopping point as you do with Legos. An irrational number like $$\sqrt{2}$$ or $$\pi$$ is an even more nonsensical thing to claim that it has identifiable constituents.)
OK.
So an electron is a constituent of an atom.
Removal of a point from a line segment, does not affect the length of the line segment. Because, by definition a point is dimension-less(ie its radius is zero). Radius is zero means its diameter is also zero. That means length of a point in any direction is zero. So, by removing a point only 'zero length' is being removed from the line segment. Thus the length of the line segment remains unaffected.
So, by removing a point, a line segment can not turn into two line segments.
Removal of the endpoint of a closed line segment does not change the length but changes it from a closed line segment (with nameable end points) to a half-open line segment. But if you remove a point from the interior of the line segment, you get two half-open line segments.
Length isn't the only property of a line segment; it also has zero thickness. When lines cross, we can also say a line cuts a line at a point. If we remove that point we make that cut literally and the line falls apart into two, just as a disc or rectangular area is cut by a line that passes through its interior.
Length is just one property of a line segment.
Actually the length is partitioned between the two resulting line segments.
This conclusion does not follow from the assumptions.
Are you assuming a point as something which has some dimension(infinitesimal length)?
If these two segments are added, will it restore the original length?
--http://en.wikipedia.org/wiki/Half-line#RayGiven a line and any point A on it, we may consider A as decomposing this line into two parts. Each such part is called a ray (or half-line) and the point A is called its initial point. The point A is considered to be a member of the ray.[5] Intuitively, a ray consists of those points on a line passing through A and proceeding indefinitely, starting at A, in one direction only along the line. However, in order to use this concept of a ray in proofs a more precise definition is required.
Given distinct points A and B, they determine a unique ray with initial point A. As two points define a unique line, this ray consists of all the points between A and B (including A and B) and all the points C on the line through A and B such that B is between A and C.[6] This is, at times, also expressed as the set of all points C such that A is not between B and C.[7] A point D, on the line determined by A and B but not in the ray with initial point A determined by B, will determine another ray with initial point A. With respect to the AB ray, the AD ray is called the opposite ray.
Ray
Thus, we would say that two different points, A and B, define a line and a decomposition of this line into the disjoint union of an open segment (A, B) and two rays, BC and AD (the point D is not drawn in the diagram, but is to the left of A on the line AB). These are not opposite rays since they have different initial points.
The definition of a ray depends upon the notion of betweenness for points on a line. It follows that rays exist only for geometries for which this notion exists, typically Euclidean geometry or affine geometry over an ordered field.
BTW, in mathematics I think the term "component" is used instead of "constituent", the latter term more often used in politics.
So is a point a component of a line? It must be if you can remove a point from a line. Likewise a line segment must be a component of the line it's part of.
A point has no dimensions, but it can have coordinates.hansda said:
The sum of the two segments will be equal to the original length (because the removed point has no length). But the set of points in the union of disjoint segments won't be the same set because one point is missing.
The only way to restore the original line is to include the missing point.
Thanks for the link.
It does not matter which term you use, as long as its definition is correct. I think rpenner's definition for the term in post #611 is ok.
But the length of the line remains unaffected.
YES. Thats true.
Co-ordinates define the location of a point, where infinite number of points can be placed.
Correct.
If you add 8 points in a location and remove 5 points from that location, how many points you should count at that location for the purpose of set.
"But" is incorrect. x - 0 = x, so of course the total length is not expected to change. Therefore you have no reason to write "but" as you have no alternate expectation. Even if you remove the point from somewhere inbetween the endpoints, the total length is not expected to change. x - 0 = a + b. Length is just one property of a line segment. Removing a point breaks the property of continuity at that point. If I have two lines that meet at a point and I removed that point from one of the lines, then the untouched line no longer intersects either of the broken halves of the line. If I remove an end point from a closed line segment, I get a half-open line segment which has a different (topological) nature.
A point's only property is location. You cannot "add points to a location" as point and location are synonymous (in Euclidean geometry and related topics where there is no concept of movement). Same location means same point.
No, a coordinate defines the location of a single point. A line or line segment has an infinite number of locations, each defining a single point.hansda said:
yes .. that makes more sense. A point doesn't exist until it is located would be a better way of looking IMO
but how can you remove a point that doesn't actually exist? [ as it is zero in dimension ]
Whether zero dimensional or 1/infinity diameter the intersection of two lines would still be happening...??