We present a formalization in ACL2(r) of three proofs originally done by Cantor. The first two are different proofs of the nondenumerability of the reals. The first, which was described by Cantor in 1874, relies on the completeness of the real numbers, in the form that any infinite chain of closed, bounded intervals has a non-empty intersection. The second proof uses Cantor’s celebrated diagona...

(i) CAC iff every countable product of sequential metric spaces (sequentially closed subsets are closed) is a sequential metric space iff every complete metric space is Cantor complete. (ii) Every infinite subset X of R has a countably infinite subset iff every infinite sequentially closed subset of R includes an infinite closed subset. (iii) The statement “R is sequential” is equivalent to eac...

While there is, up to homeomorphism, only one Cantor space, i.e. one zero-dimensional, perfect, compact, nonempty metric space, there are many measures on Cantor space which are not topologically equivalent. The clopen values set for a full, nonatomic measure μ is the countable dense subset {μ(U) : U is clopen} of the unit interval. It is a topological invariant for the measure. For the class o...

In studying the early history of mathematical logic and set theory one typically reads that Georg Cantor discovered the so-called Burali-Forti (BF) paradox sometime in 1895, and that he offered his solution to it in his famous 1899 letter to Dedekind.! This account, however, leaves it something of a mystery why Cantor never discussed the paradox in his writings. Far from regarding the foundatio...

A chapter from his forthcoming book "Deciding for the Profoundly Mentally Disabled," Professor Norman Cantor argues persuasively for the right of incompetent persons to have a surrogate make critical medical decisions on their behalf, particularly in the context of refusing life-sustaining treatment. While abusive surrogate decision-making is always a concern, Professor Cantor recommends both s...

4. Ries LAG, Miller BA, Hankey BF, Kosary CL, Harras A, Edwards BK, eds. SEER Cancer Statistics Review, 1973-1991: Tables and Graphs. NIH Publ No. 94-2789. Bethesda, MD:National Cancer Institute, 1994. 5. Cantor KP. Drinking water and cancer. Cancer Causes Control 8(3):292-308 (1997). 6. Bates MN, Smith AH, Cantor KP. Case-control study of bladder cancer and arsenic in drinking water. Am J Epid...

In this paper, the question of which compact metric spaces can be attractors of hyperbolic iterated function systems on Euclidean space is studied. It is shown that given any finite-dimensional compact metric X , there is a Cantor set C such that the disjoint union of C and X is an attractor. In the process, it is proved that every such X is the Lipschitz image of a Cantor set.

This paper describes the theories and techniques for designing linear and planar fractal arrays and compare their radiation pattern with conventional arrays. Fractals are recursively generated object having fractional dimension. We have compared radiation pattern of cantor linear arrays with conventional linear arrays using MATLAB program. Another program was developed to characterize the radia...

4. Ries LAG, Miller BA, Hankey BF, Kosary CL, Harras A, Edwards BK, eds. SEER Cancer Statistics Review, 1973-1991: Tables and Graphs. NIH Publ No. 94-2789. Bethesda, MD:National Cancer Institute, 1994. 5. Cantor KP. Drinking water and cancer. Cancer Causes Control 8(3):292-308 (1997). 6. Bates MN, Smith AH, Cantor KP. Case-control study of bladder cancer and arsenic in drinking water. Am J Epid...

We analyze the structure and the regularity of a broad class of Cantor sets. We provide criteria, and non-trivial examples of regularity and of non-regularity, exhibiting new phenomena in dimension theory. We show that a Cantor set constructed with rectangles can be reduced to one constructed with aligned rectangles, having equal Hausdorr and box dimensions. This is proved showing the existence...