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From: Robert Kolker on 20 Apr 2005 16:12 Ross A. Finlayson wrote: > Basically people point to Euclid and notice that geometric points and > lines are defined in terms of each other. If you abstract that (make > that abstract) in a slightly more enlightened way after thousands of > years of research into geometry, the point, and line, and plane, and > circle and sphere, and complex plane and hyperspheres, are all defined, > at a very fundamental level of geometry and as well mathematical logic > and numbers and various set relations. Check out Hilbert's Axioms for geometry. Points, planes and lines are all undefined terms. Hilbert's Axioms Undefined Terms * Points * Lines * Planes * Lie on, contains * Between * Congruent Axioms 1. Axioms of Incidence Postulate I.1. For every two points A, B there exists a line a that contains each of the points A, B. Postulate I.2. For every two points A, B there exists no more than one line that contains each of the points A, B. Postulate I.3. There exists at least two points on a line. There exist at least three points that do not lie on a line. Postulate I.4. For any three points A, B, C that do not lie on the same line there exists a plane ± that contains each of the points A, B, C. For every plane there exists a point which it contains. Postulate I.5. For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C. Postulate I.6. If two points A, B of a line a lie in a plane ± then every point of a lies in the plane ±. Postulate I.7. If two planes ±, ² have a point A in common then they have at least one more point B in common. Postulate I.8. There exist at least four points which do not lie in a plane. 2. Axioms of Order Postulate II.1. If a point B lies between a point A and a point C then the points A, B, C are three distinct points of a line, and B then also lies between C and A. Postulate II.2. For two points A and C, there always exists at least one point B on the line AC such that C lies between A and B. Postulate II.3. Of any three points on a line there exists no more than one that lies between the other two. Postulate II.4. Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC. 3. Axioms of Congruence Postulate III.1. If A, B are two points on a line a, and A' is a point on the same or on another line a' then it is always possible to find a point B' on a given side of the line a' such that AB and A'B' are congruent. Postulate III.2. If a segment A'B' and a segment A"B" are congruent to the same segment AB, then segments A'B' and A"B" are congruent to each other. Postulate III.3. On a line a, let AB and BC be two segments which, except for B, have no points in common. Furthermore, on the same or another line a', let A'B' and B'C' be two segments which, except for B', have no points in common. In that case if ABHA'B' and BCHB'C', then ACHA'C'. Postulate III.4. If ABC is an angle and if B'C' is a ray, then there is exactly one ray B'A' on each "side" of line B'C' such that A'B'C'E ABC. Furthermore, every angle is congruent to itself. Postulate III.5. (SAS) If for two triangles ABC and A'B'C' the congruences ABHA'B', ACHA'C' and BAC H B'A'C' are valid, then the congruence ABC H A'B'C' is also satisfied. 4. Axiom of Parallels Postulate IV.1. Let a be any line and A a point not on it. Then there is at most one line in the plane that contains a and A that passes through A and does not intersect a. 5. Axioms of Continuity Postulate V.1. (Archimedes Axiom) If AB and CD are any segments, then there exists a number n such that n copies of CD constructed contiguously from A along the ray AB willl pass beyond the point B. Postulate V.2. (Line Completeness) An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence (Axioms I-III and V-1) is impossible. Bob Kolker
From: Chris Menzel on 20 Apr 2005 17:19 On 20 Apr 2005 12:43:47 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > If you want to quantify over sets in a set theory there's a universal > set. There can only be one proper class, or none. The powerset > operation on an n-set of ordinals introduces only one new ordinal. > There is nothing, greater than Ord....The universe is infinite, and > infinite sets are equivalent. Since your "null axiom" theory presumably has no (nonlogical) axiomss, they are obviously not theorems of the theory. So on what grounds are you asserting them? > Any theory with non-logical axioms is doomed by Goedel to > incompleteness. False. Chris Menzel
From: Ross A. Finlayson on 20 Apr 2005 17:54 Chris Menzel wrote: > On 20 Apr 2005 12:43:47 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > > If you want to quantify over sets in a set theory there's a universal > > set. There can only be one proper class, or none. The powerset > > operation on an n-set of ordinals introduces only one new ordinal. > > There is nothing, greater than Ord....The universe is infinite, and > > infinite sets are equivalent. > > Since your "null axiom" theory presumably has no (nonlogical) axiomss, > they are obviously not theorems of the theory. So on what grounds are > you asserting them? > > > Any theory with non-logical axioms is doomed by Goedel to > > incompleteness. > > False. > > Chris Menzel Hi, They're inferences. That is to say, for each of the non-logical axioms of ZF, for example, besides regularity, anything that could possibly be a set exists, that the set has the properties ascribed to them by those non-logical axioms is reasonable in that assuming them so, to be sets, does not lead to contradiction. The sets exists, in the set of all sets, because anything that is a set, basically via composition, is a set. It's a pure set theory, there are only sets, the minimal or ur-element is just a placeholder for any of the others, where any might mean each and any or any and every or only that satisfying all those, or none. I respectfully disagree about the falsehood of Goedel damning you to incompleteness in whatever you may believe. A reason to rationalize a way around the presence of non-logical axioms in a theory is that we're talking, in a sense, about the very most abstract, the limits of abstraction, or beyond so. To address these things which by their very nature, presumably, return to self-referential consequence, that very notion must itself be used to forestall its own dilemma (dilemna). Get with the Hilbert program. "Infinite sets are equivalent." - 1999 Ross
From: Chris Menzel on 20 Apr 2005 18:54 On 20 Apr 2005 14:54:20 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > Chris Menzel wrote: >> On 20 Apr 2005 12:43:47 -0700, Ross A. Finlayson > <raf(a)tiki-lounge.com> said: >> > If you want to quantify over sets in a set theory there's a > universal >> > set. There can only be one proper class, or none. The powerset >> > operation on an n-set of ordinals introduces only one new ordinal. >> > There is nothing, greater than Ord....The universe is infinite, and >> > infinite sets are equivalent. >> >> Since your "null axiom" theory presumably has no (nonlogical) axiomss, >> they are obviously not theorems of the theory. So on what grounds are >> you asserting them? >> >> > Any theory with non-logical axioms is doomed by Goedel to >> > incompleteness. >> >> False. >> >> Chris Menzel > They're inferences. From what? Look, all you need to do is demonstrate the inferences. Just one will do. Show us how to infer "Infinite sets are equivalent". From anything. > I respectfully disagree about the falsehood of Goedel damning you to > incompleteness in whatever you may believe. You are under the impression that this is a matter upon which reasonable people can disagree. In fact, to disagree on the matter in question is just ignorant foolishness. The existence of complete theories is a well-known, proven mathematical fact. Google "Presburger arithmetic". Chris Menzel
From: Ross A. Finlayson on 20 Apr 2005 21:10
Chris Menzel wrote: > On 20 Apr 2005 14:54:20 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > > Chris Menzel wrote: > >> On 20 Apr 2005 12:43:47 -0700, Ross A. Finlayson > > <raf(a)tiki-lounge.com> said: > >> > If you want to quantify over sets in a set theory there's a > > universal > >> > set. There can only be one proper class, or none. The powerset > >> > operation on an n-set of ordinals introduces only one new ordinal. > >> > There is nothing, greater than Ord....The universe is infinite, and > >> > infinite sets are equivalent. > >> > >> Since your "null axiom" theory presumably has no (nonlogical) axiomss, > >> they are obviously not theorems of the theory. So on what grounds are > >> you asserting them? > >> > >> > Any theory with non-logical axioms is doomed by Goedel to > >> > incompleteness. > >> > >> False. > >> > >> Chris Menzel > > > They're inferences. > > From what? > > Look, all you need to do is demonstrate the inferences. Just one will > do. Show us how to infer "Infinite sets are equivalent". From > anything. > > > I respectfully disagree about the falsehood of Goedel damning you to > > incompleteness in whatever you may believe. > > You are under the impression that this is a matter upon which reasonable > people can disagree. In fact, to disagree on the matter in question is > just ignorant foolishness. The existence of complete theories is a > well-known, proven mathematical fact. Google "Presburger arithmetic". > > Chris Menzel Hi, I read that the Presburger arithmetic is very similar to the Peano arithmetic except it has no multiplication. Addition it has, multiplication it does not have, although somebody said it could be defined as having multiplication but no addition. Now, I think that any statement that you can say about multiplication, that the same thing can be said using addition. 2 + 2 = 4 Now, 2*2 also equals 4, if you remove the word "multiplication" from the dictionary, in this arithmetic of integrs, I can still say "using addition, as many times is specified, reptition of addition as many times as is specified" wherever that word was excised from the vocabulary, and show the same things about the integers. That's to say, I don't see anything provable in Peano arithmetic, with addition and a special word for repeated addition called multiplication, that isn't provable in Presburger arithmetic. If that is so, and all true statements in Presburger are provable in Presburger, then all true statements provable in Peano are provable in Presburger, or Peano, where the Presburger arithmetic's "axioms" are a subset of Peano arithmetic's "axioms." The set of all sets is its own powerset, identity between them, tautology, is a bijection, from the existence of a universal set, an infinite set is equivalent to its powerset. To explain why any infinite set is equivalent to any other infinite set, besides via induction, that's kind of up in the air. I'll have to think about that, and what it means in terms of my little theory with no non-logical axioms. Basically the sets as ordinals lead to ubiquitous naturals themselves or ubiquitous ordinals as the cumulative hierarchy, and any infinite ordinal extracted in that way is irregular. Back to Goedel and non-logical axioms, in a way conceptually similar to talking about Cantor's first and second, you might want to consider Goedel's first and second incompleteness results, and why some people call them the second and first. I'm basically unimpressed with you calling these simple logical statements "ignorant foolishness." What can I say? I'm glad to see you remediating your position to consider a Quinean universal set, good luck on further developments. Ross |