Re Why real numbers and points can't model continuum

Re: Why real numbers and points can't model continuum Re: Why real numbers and points can't model continuum Source: http://sci.tech−archive.net/Archive/sci.math/2007−11/msg04140.html • From: Randy Poe • Date: Wed, 21 Nov 2007 06:00:15 −0800 (PST) On Nov 21, 4:06 am, Venkat Reddy wrote: On Nov 20, 8:27 pm, Randy Poe wrote: On Nov 20, 10:04 am, Venkat Reddy wrote: On Nov 20, 7:17 pm, Phil Carmody wrote: Venkat Reddy writes: Are you finding my reasoning incorrect or just ruling it out because my wording is not rigorous? Re: Why real numbers and points can't model continuum 1 Re: Why real numbers and points can't model continuum I'm not finding your reasoning, period. I suggest you come out of the "set" mindset and imagine about lines and intersections without using sets for a while and then read my post again. It should help you. C'mon, you can do it. Only after you supply some axioms that usefully replace my set theory ones. Euclid's axioms, postulates and Hilbert's axioms of geometry do not seem to use any sets. Do they help you? http://en.wikipedia.org/wiki/Hilbert's_axioms "Hilbert's axioms do not constitute a first−order theory because the axioms in group V cannot be expressed in first− order logic. Therefore Hilbert's axioms, unlike Tarski's, implicitly DRAW ON SET THEORY..." Lines? They're sets of points. Intersections? They're sets too. If intersections are sets, are they empty sets or a single element sets? Re: Why real numbers and points can't model continuum 2 Re: Why real numbers and points can't model continuum Intersection of two sets means the set of elements which are contained in both. An intersection can contain any number of elements. The intersection of two distinct lines may be either empty or contain a single element. Wikipedia (http://en.wikipedia.org/wiki/Line−line_intersection) says it is empty set. No, it says it is either an empty set, or a single point, or an entire line (they are the same line). If empty set, then does it mean there is no point at which lines intersect? Yes, if the set of intersection points is empty, then there are no intersection points. Ok, I stand corrected in interpreting that wikipedia article. But if the intersection of two non−parallel lines is a set containing a single point, then do other resulting line segments also contain this point? Resulting in what way? Some do, some don't. Line segments can intersect in zero, one, or infinitely many points. In other words, is the folowing info enough for you decide if the line segments will contain the intersection point or not? Line x=2 cuts the line y=3 at (2,3). Re: Why real numbers and points can't model continuum 3 Re: Why real numbers and points can't model continuum That is equivalent to saying "the point {2,3} is contained in both the line x=2 and the line y=3". Geometriically speaking, I don't need any more extra info to distribute the point (2,3) among the line segments. The undefined operation in this post (there's always one): "distributing a point among line segments". There's only one point. It isn't distributed. It is a member of both lines, just like the number 2 is a member of "the set of prime numbers" and "the set of numbers > 0" and "the set of even numbers" and "the set of even positive numbers" and ... I don't need to "distribute" 2 among those sets. But in the case of sets, I don't think this distribution is is defined, without requiring an explicit specification. That's right. As I said, this "distribution" is not defined. Neither in the case of sets nor lines which we can think of as sets. It's both meaningless and unnecessary. Whatever helps you think of 2 as a member of all those sets while being a single unique value, that same model holds for points which are intersections of lines. − Randy . Re: Why real numbers and points can't model continuum 4