Victella Website

This website is intended to hold articles and other information that may be useful to some members of the public. Currently, there are only two articles. However, the sub-website, ETAC, contains the details and download of a new programming language.

NOTE: This website is written in Australian English.

Articles

There are currently only three articles to download.

An Analysis of B-splines

This document presents an analysis of B-splines, especially uniform B-splines. It explains the nature of B-splines and how they are constructed, and also how B-splines can be merged, joined, and separated. The nature of B-splines is presented mathematically and geometrically. Methods for converting B-spline control points to Bezier control points are also presented, as well as emulating arbitrary curves by means of B-splines. A number of algorithms to implement B-splines is presented in an Appendix. The document also contains an appendix discussing Bezier curves and splines. Some familiarity with mathematics is a prerequisite.

To download the free article (An Analysis of B-splines), click the following link.

Download: An Analysis of B-splines revision 1 (DMY: 10/01/2023) (655 kB)

Transfinite Cardinals - RIP

The purpose of this article is to prove positively and rigorously that there are no transfinite cardinal numbers, other than À₀, by showing that the traditional arguments for their existence are logically invalid, and proving that there is only the one transfinite cardinal number, À₀, using the irrational number version of Dedekind cuts (the ultimate definition of an irrational number). A previous article, The Collapse of Transfinite Cardinals, proved only that the arguments for the higher transfinite cardinals are logically invalid. This article presents a devastating blow to those arguments using a different approach, AND also proves conclusively that all infinite sets are denumerable, in more than one way — guaranteeing the end of those factitious higher transfinite cardinals (RIP). The article uses mathematical logic, so the reader may need to be familiar with that subject. An appendix in the article shows the proper understanding and presentation of a proof by contradiction.

To download the free article (Transfinite Cardinals - RIP), click the following link.

Download: Transfinite Cardinals - RIP first published (DMY: 15/3/2025) (511 kB)

The Collapse of Transfinite Cardinals

This article is of interest mainly to pure mathematicians. The purpose of the article is to prove, rigorously, that the existence of transfinite cardinal numbers is untenable at this point in time because the traditional “proofs” of their existence are logically invalid. The only transfinite cardinal number that may be admitted is À₀. The definition of transfinite cardinal numbers depends upon Cantor's theorem and the non-denumerability of the real numbers. The “proofs” of those theorems depend upon Cantor's diagonal method. The article proves that the diagonal method is self-contradictory, and consequently, the existence of transfinite cardinal numbers is untenable (except for À₀). The article does not prove, with certainty, that the transfinite cardinal numbers do not exist, but that they have not yet been validly proven to exist. (Note that the word “exist” here has nothing to do with metaphysics; the word is a purely logical concept.)

To download the free article (The Collapse of Transfinite Cardinals), click the following link.

Download: The Collapse of Transfinite Cardinals revision 1 (DMY: 5/5/2007) (216 kB)

The downloadable document, The Collapse of Transfinite Cardinals, contains rigorous, strictly logical arguments that the traditional arguments for Cantor’s theorem and the non-denumerability of the real numbers are flawed. The following is the essence of the arguments contained in the document. Note that familiarity with Cantor's arguments is assumed.

Hypothesis: S = {r_i}_iÎN = [0, 1], where, for each r_i, r_j Î S, if r_i = r_j then i = j.

Note that S (= {r₀, r₁, r₂, …}), which is the set of real numbers from 0 to 1 inclusively, contains elements uniquely indexed by an integer. This is equivalent to having a one-to-one correspondence between the natural numbers, N, and the real numbers in the interval [0, 1].

The hypothesis implies that, for any real number r, r Î S.

Aim: “Construct” a number r such that r Î [0, 1] and r Ï S. Such an r would contradict the hypothesis, thereby the hypothesis would be rejected.

Case 1: The “construction” algorithm to construct the number r is self-contradictory.

The algorithm implies:

1a: r Î [0, 1]

1b: r Ï S. But r Ï S implies r Ï [0, 1], since S = [0, 1] by the hypothesis.

1a and 1b are contradictory, so the construction algorithm itself implies a contradiction and therefore fails. The stated aim has not been achieved.

Case 2: The “construction” algorithm to construct the number r is not self-contradictory.

The algorithm implies:

2a: r Î [0, 1]

2b: r Ï S^*, where S^* = {q_i}_iÎN = S\{r}, where, for each q_i, q_j Î S^*, if q_i = q_j then i = j (this is equivalent to having a one-to-one correspondence between the natural numbers, N, and the real numbers in S^*).

Note that S^* excludes the value r to avoid the contradiction at case 1. Therefore, S^*Ì S (S^* is a proper subset of S). Now, in order to establish the contradiction with the hypothesis, we need that r Ï S. However, in this case 2, the construction algorithm implies that r Ï S^*, but not that r Ï S (noting that S^* ¹ S). The contradiction with the hypothesis is not obtained; therefore, the stated aim has not been achieved.

Conclusion: Both cases 1 and 2 fail to achieve the stated aim. Therefore, Cantor’s diagonal algorithm for the non-denumerability of the real numbers is flawed.

Note that there was no need to specify the details of the actual “construction” algorithm itself; any similar algorithm will fail. Also, note that an analogous argument to the one above can be produced for Cantor’s power set theorem (|A| < |Ã(A)|), proving that Cantor’s argument for the power set theorem is also flawed (PROPOSITION 3 and the following text in the downloadable document also suggests that the theorem is actually false by providing an exception). In fact, any argument that tries to “construct” (or define) an entity belonging to a set of entities but different from all the members of that set will fail — either (1) the construction algorithm will be self-contradictory, or (2) the constructed entity cannot be a member of any proper subset of the set (ie: the entity is outside of all proper subsets of the set).

The lack of critical thinking by most mathematicians is astounding and also dangerous, especially when flawed mathematical arguments are considered valid by the mathematical community. Critical thinking requires pure logical thinking, not only intuitive thinking. Mathematicians probably failed to see the flawed Cantor’s arguments because they didn’t interpret those arguments in a purely logical way, but rather relied too much on intuitive thinking.

Websites

ETAC

This is the official sub-website for the ETAC programming language.