After a rather interesting discussion in #math on freenode, I'm perplexed by the fact that there is no way to construct lines from points. For example, I initially thought of constructing a line between 0 and 1 by repeatedly dividing into two smaller halves, eg: 0,1 0,0.5,1 0,0.25,0.5,0.75,1 ..etc But obviously this approach will miss the irrationals and cover only rationals of the form (1/2)^n thus, missing all others with such as (1/3)^n, (1/5)^n, (1/m)^n where m is prime. So, in the hypothetical case where I do cover all the points arising from (1/2)^n (1/3)^n (1/m)^n ... I would STILL miss the irrationals.
It's hard to answer without knowing what precisely you mean by "constructing a line from points". If you mean "enumerate the set of points lying on a line segment as a countable sequence", then that can't happen by a counting argument: that set is uncountable.
Yeah that's basically what I meant, is there an uncountable argument that can be used to construct R (without resorting to Dedekind cuts)? I can easily construct rationals by a countable argument (since there exists an injective function from Q to N)
One of the ways to go from N to R is to construct infinite sequences of naturals to represent reals. You don't miss any numbers doing this. Or more conventionally you can use Dedekind cuts or Cauchy sequences of rationals.
The problem is that a line does not comprise points because points have no dimension. No two points form a line segment until you connect them with a line. No matter how small the inteval is between two points on a line, the points merely define the end points of the line segment between them but comprise no part of the line segment itself. To construct the line segment one would have to place a straight edge between the two points. This can not be done mathematically even with infinite series, the job belongs with the technician who does not worry about infintesimals.
A line is what its defined to be. And when studying plane geometry, a line is most typically defined as a set of points. This is adequate, because the geometry is provided by Euclidean geometry. If you wanted to talk about a line on its own, you would use Euclidean line geometry -- a structure typically consisting of nothing more than points and a betweenness relation. I really suggest that you learn some topology before you start making brazen assertions about dealing with points in spaces. (And as an interesting note, please pay attention to the fact specifying a line in the Euclidean plane is, in fact, strictly less information than specifying a pair of points in the Euclidean plane)
Yes, also a line can be defined by two points but the line is not those points. You can also define a line by the equation y = Ax + B which is in effect defining a line by an infinite number of points. But a line has a dimension while points have no dimension. An infinite number of zeros add to nothing which is why you have to rely on more than points such as a betweenness relation. There are of course many ways to define things by use of elements which have nothing in common with the thing defined.
your question about how to construct the line from simple operations of division is one of the oldest questions in mathematics originating in the time of the ancient Greeks. the Greeks, particularly the Pythagoreans, imagined that God constructed space through a sequence of geometric constructions using a straight edge and a compass. Space was the product of an unending geometric construction. Your example of constructing new points by finding mid-points is one possible straight edge and compass construction but there are many others and it was not known until Gauss what they all were. At first the Pythagoreans thought that all points of space were simple divisions of whole distances. For them all of space was just ratios of whole straight edge lengths. All numbers were rational and in fact the word rational means ratio. When they discovered the Pythagorean Theorem they immediately wanted to know which ratio equals the square root of 2 which they constructed as the hypotenuse of a right triangle whose sides have length 1. When they proved the the square root of 2 is not a ratio, it stunned their concept of God and space and they kept this result a secret from outsiders. Now a new questions came up. That was what are all of the lengths that can be constructed using a straight edge and compass. This was a question about what space actually was when viewed as a geometrical construct. Gauss finally solved this age old question by proving that all constructable points in space are solutions of a certain type of polynomial equation. From this it was immediately clear that not all points were constructable because many numbers were solutions of polynomial equations different from those that represented straght edge and compass constructions, for instance cube roots. I believe that there were attempts to generalize the idea of geometric construction and am not sure when people realized that the entire program must fail because even if you construct for ever you will end up with only a countable infinity of points. I know that the debate over whether infinity even existed raged into the late nineteenth century and that Cantor's construction of transfinite numbers was violently attacked. Dedekind told Cantor that infinity did not exist and when Cantor rebutted that there are infinitely many integers, Dedekind responded, "Yes, but the integers were created by God." Cantor conceived of real numbers as limits of infinite successions of converging numbers, what we today call Cauchy sequences. To him these limits could only exist as what he called "actual infinities" as points that occur at the end after all possible points in the infinite sequence have been traversed. This idea of an actual infinity flew in the face of the ancient and then still accepted belief that infinite successions could not exist and that the idea of infinity was merely the idea of a finite succession that can always be extended further - such as counting integers. This rejection of infinity goes back to the Greeks and arguments against the possibility of infinite sequences led to many conclusions in philosophy and theology. Zeno's paradox uses the impossibility of infinity to prove that Achilles can never catch a tortoise that is given a head start in a race and therefore that all motion is an illusion. Aristotle directly rejected the idea of infinity as vacuous and said that all it really was was the idea of a finite succession that can always be extended one step further. Acquinas proved the existence of God by arguing that if there was not a first cause then there would be an infinite regress of causes, a clear impossibility. So God exists as the first cause. Cantor destroyed this attitude by showing that the numbers must exist as actual infinities. And just to seal the point he demonstrated an inductive method of generating infinities successively in order of size. His inductive method was much more powerful than Aristotle and Zeno's method and represents a new way of reasoning not known to his predecessors. Today we tend to forget these profound historical thoughts and take mathematical objects as givens as if they were never discovered and as though they never affected the way we think. Your naive question about constructing the reals is really a profound question that has led to much of modern mathematics and which helped shape all of intellectual history. . .
What precisely do you mean by adding infinitely many things? And why should that have anything at all to do with the notion of dimension?
Prove it. (Hint: you cannot -- there exists a model of Euclidean geometry where the lines are literally pairs of points)
Not in my context, but what do you mean by model? A model could be anything you wanted it to be. My earlier post was based upon what I remembered from my math teacher back in the late 50's to the 1960's. I am not sure what grade I was in.
If you really do not know what a model for an axiom system is, then this discussion is way over your head! (Much of it is over my head!) A "model" is a specific assignment of meanings to the "undefined terms" in the system such that all axioms are true. If there exist a statement "A" in the vocabulary of the system that is true in one model for the system but false in another, then "A" cannot be proved or disproved in that system. Hurkyl's point is that it is possible to assign specific meaning to "point", "line", etc. in Euclidean geometry such that a line is defined as exactly two points. Therefore, the statement that a line is not two points cannot be proved in Euclidean geometry. If that is not the case in your "context", then your context is not that of Euclidean geometry.
A model just means a specific set of objects that are mapped onto the terms in the axiom system so that these objects obey the axioms. Instead of a continuum as you are thinking of, Euclidean geometry, according to Hurky, has a model where lines are just pairs of points. I am not sure what this model really looks like. But I think you are interested in whether one can construct the continuum - not just any model of the line - from points, using only simple geometric constructions. This age old question is completely reasonable and occupied the minds of mathematicians throughout history. I suspect that ruler and straight edge constructions were enhanced with other constructive techniques in attempts to perform difficult constructions such as the trisection of an angle. I seem to remember that Descartes tried some novel constructions using curves. While Gauss showed that ruler and compass constructions could never produce the continuum, the question of how the continuum came into existence remained. Cantor, I think, believed that the continuum arose through completion of Cauchy sequences and he modeled the process of completion as an act of inductive thought, by God no doubt, that defined ''actual infinities" as ideas that represent the entire process of succession. Another way of saying this is that an actual infinity is an idea in which the entire infinite succession of rational numbers in the Cauchy sequence is a predicate. It is interesting that this philosophical and theological line of thought led to real mathematics. Before our realistic modern attitude that theology and science are separate domains of inquiry, mathematicians and physicists thought they were revealing God's laws. The idea that the universe is fundamentally similar to the mind of God is found in Kepler, Riemann, and many others, (probably Bach and Beethoven). It is an age old idea that God is that of which all things are predicated. This abstruse theological idea might have been the inspiration for Cantor's actual infinities. After all, an actual infinity is that of which an infinite succession is predicated. In modern physics we do not worry about these ideas of creation. We just take any model that explains data. Not so historically and certain key advances were made though this fusion of mathematics, physics, and theology. Another example is the Ptolemaic system which was rejected intellectually long before Copernicus because people felt that the centers of the circles in epicycles were arbitrary and therefore could not represent Divine geometry. In modern times we would not have rejected the Ptolemaic system because it provides an accurate model of planetary orbits. In fact, the mathematical technique of epicycles is similar to the modern method of Fourier series.
Right, and that's sort of the key point. Euclidean geometry, as typically presented today, is a theory that talks about "points", "lines", "incidence", "betweenness", and "congruence". It doesn't tell you anything about points or lines 'really are' and it doesn't really care -- you don't need any of that to be able to study geometry. Of course, sometimes we have specific applications in mind -- e.g. maybe we want to study equations in two real variables (which I'll call x and y). Then we might say that points "really are" pairs of real numbers, and that lines "really are" certain sets of points. Or, maybe we'd prefer to say that lines "really are" equations of the form ax + by = c. Or maybe we'd prefer to say that lines "really are" pairs of distinct points. Or maybe we'd prefer something else entirely. It makes no difference what the semantics are: Euclidean geometry is the same whether we say that lines are certain sets of points or certain equations or something else entirely. In fact, in practice we often exploit this fact -- rather than binding ourselves to one and only one meaning, we instead use whatever is most convenient at the time. We might make one calculation where we use "line" to mean a pair of points, and then turn right around and use "line" to mean a kind of equation in the very next statement. Defining a "line" as a certain set of points (where a point is in the line in the set-theoretic sense if and only if the point lies on the line in the geometric senes) happens to be one of the more convenient meanings -- note that we also need to talk about "rays", "circles", "discs", "triangles", "parabolas", and all sorts of other things. Expressing them as sets of points allows us to use set theory to describe how they relate -- it would be very cumbersome to do otherwise! And except for some rather unusual ideas, shapes in the Euclidean plane are completely determined by the set of points lying on them, so nothing is "lost" by expressing shapes as sets of points. Geometrically, it really looks like a Euclidean plane.
What then is meant by the statement that line AB passes through point C where point C is neither point A or point B? To say that a line is just two points is akin to saying since the boundaries of Utah are all that is needed to define that state then Salt Lake City is not part of Utah. You may also note that the boundaries of a state are not part of that state even though they define it.
It means that the incidence predicate "C lies on [tex]\overline{AB}[/tex]" is true. The incidence predicate is one of the fundamental undefined terms of Euclidean geometry (in its usual formulation), and there is no reason it should coincide with the set-theoretic relation [itex]\in[/itex]. In particular the "locus of points lying on [tex]\overline{AB}[/tex]" consists of many more points than just A and B.
In fact, if we have a model where lines are pairs of points, there's no formal reason either of those points should even lie on that line! (However, in the particular model I was imagining, the two points did, in fact, lie on the line)
One particular practical application of "lines are pairs of points" semantics comes from trying to study compass-and-straightedge constructions. Here, it is most natural to define a line as the pair of points used to 'construct' it. (And different pairs can define the same line) Similarly, a circle is most naturally defined as an ordered pair of points (one names the origin, the other gives the radius) -- and is a natural example where a shape is defined by a pair of points, but one of those points doesn't even lie on it!
Then I think your wording is inaccurate and confusing. You are not defining a line AS a pair of two points, but in fact you are defining a line BY a pair of points. PS in my model a line may be defined by any two separate points that lie on the line (or points not on the line if you add other defining elements) but a line and points are two entirely distinct concepts. This does not conflict with the geometry that was taught to me back in grade school.