scholarly journals The Future of Modern Set Theory

1994 ◽  
Vol 8 (4) ◽  
pp. 187-190
Author(s):  
James E. BAUMGARTNER
Keyword(s):  
Set Theory ◽  
The Future

scholarly journals Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism

2021 ◽  
Vol 10 (2) ◽  
pp. 351-372
Author(s):  
Jakub Gomułka
Keyword(s):  
Set Theory ◽  
Ludwig Wittgenstein ◽  
Number Line ◽  
Real Numbers ◽  
Mathematical Symbol ◽  
Continuous Sequence ◽  
Middle Period ◽  
The Future ◽  
George Lakoff ◽  
Static View

Cantor’s paradise from the perspective of non‐revisionist Wittgensteinianism: Ludwig Wittgenstein is known for his criticism of transfinite set theory. He forwards the claim that we tend to conceptualise infinity as an object due to the systematic confusion of extension with in‐ tension. There can be no mathematical symbol that directly refers to infinity: a rule is the only form by which the latter can appear in our symbolic operations. In consequence, Wittgenstein rejects such ideas as infinite cardinals, the Cantorian understanding of non‐denumerability, and the view of real numbers as a continuous sequence of points on a number line. Moreover, as he understands mathematics to be an anthropological phenomenon, he rejects set theory due to its lack of application. As I argue here, it is possible to defend Georg Cantor’s theory by taking a standpoint I call quietistic conventionalism. The standpoint broadly resembles Wittgenstein’s formalist middle period and allows us to view transfinite set theory as a result of a series of definitions established by arbitrary decisions that have no ontological consequences. I point to the fact that we are inclined to accept such definitions because of certain psycho‐ logical mechanisms such as the hypothetical Basic Metaphor of Infinity proposed by George Lakoff and Rafael E. Núñez. Regarding Wittgenstein’s criterion of applicability, I argue that it presupposes a static view of science. Therefore, we should not rely on it because we are unable to foresee what will turn out to be useful in the future.

scholarly journals The Prospects for Mathematical Logic in the Twenty-First Century

2001 ◽  
Vol 7 (2) ◽  
pp. 169-196 ◽  
Author(s):  
Samuel R. Buss ◽  
Alexander S. Kechris ◽  
Anand Pillay ◽  
Richard A. Shore
Keyword(s):  
Mathematical Logic ◽  
Computer Science ◽  
Set Theory ◽  
Model Theory ◽  
Proof Theory ◽  
Recursion Theory ◽  
First Century ◽  
Future Developments ◽  
The Future ◽  
Twenty First Century

AbstractThe four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.

Another Use of Set Theory

1996 ◽  
Vol 2 (4) ◽  
pp. 379-391 ◽  
Author(s):  
Patrick Dehornoy
Keyword(s):  
Set Theory ◽  
Theoretical Physics ◽  
Closed Domain ◽  
External Fields ◽  
The Future ◽  
Flexible Framework ◽  
The One

AbstractHere, we analyse some recent applications of set theory to topology and argue that set theory is not only the closed domain where mathematics is usually founded, but also a flexible framework where imperfect intuitions can be precisely formalized and technically elaborated before they possibly migrate toward other branches. This apparently new role is mostly reminiscent of the one played by other external fields like theoretical physics, and we think that it could contribute to revitalize the interest in set theory in the future.

scholarly journals Introduction of a Basic Theory of Objects

1970 ◽  
Vol 38 ◽  
pp. 185-225
Author(s):  
Katuzi Ono
Keyword(s):  
Set Theory ◽  
Intuitionistic Logic ◽  
Classical Logic ◽  
Mathematical Theory ◽  
Basic Theory ◽  
The Future ◽  
The Common ◽  
Formal Theories

In constructing various kind of mathematical theories on the basis of a common basic theory, it has been very usual to take up the set theory as the common basic theory. This approach has been already successful to a certain extent and looks like successfully developable in the future not only in constructing mathematical theories standing on the classical logic but also in constructing formal theories standing on weaker logics. In constructing mathematical theories standing on the classical logic, it has been successful in most cases only by interpreting mathematical notions in the set theory without defining any special interpretation of logical notions. In constructing any mathematical theory standing on weaker logics such as the intuitionistic logic, however, we have to give a special interpretation for logical notions, too.

International determination of the total motion of the pole

1961 ◽  
Vol 13 ◽  
pp. 29-41
Author(s):  
Wm. Markowitz
Keyword(s):  
Secular Motion ◽  
The Future

A symposium on the future of the International Latitude Service (I. L. S.) is to be held in Helsinki in July 1960. My report for the symposium consists of two parts. Part I, denoded (Mk I) was published [1] earlier in 1960 under the title “Latitude and Longitude, and the Secular Motion of the Pole”. Part II is the present paper, denoded (Mk II).

Status of the Lick Proper Motion Program

1978 ◽  
Vol 48 ◽  
pp. 387-388
Author(s):  
A. R. Klemola
Keyword(s):  
Proper Motion ◽  
The Future

Second-epoch photographs have now been obtained for nearly 850 of the 1246 fields of the proper motion program with centers at declination -20° and northwards. For the sky at 0° and northward only 130 fields remain to be taken in the next year or two. The 270 southern fields with centers at -5° to -20° remain for the future.

Some Structural Features of Metaphase Chromosomes of Mice and Men

1967 ◽  
Vol 25 ◽  
pp. 190-191
Author(s):  
Godfrey C. Hoskins ◽  
Betty B. Hoskins
Keyword(s):  
Electron Microscopy ◽  
Electron Micrographs ◽  
Structural Features ◽  
Metaphase Chromosomes ◽  
The Future ◽  
Of Mice And Men ◽  
Mouse Cells ◽  
Human And Mouse

Metaphase chromosomes from human and mouse cells in vitro are isolated by micrurgy, fixed, and placed on grids for electron microscopy. Interpretations of electron micrographs by current methods indicate the following structural features.Chromosomal spindle fibrils about 200Å thick form fascicles about 600Å thick, wrapped by dense spiraling fibrils (DSF) less than 100Å thick as they near the kinomere. Such a fascicle joins the future daughter kinomere of each metaphase chromatid with those of adjacent non-homologous chromatids to either side. Thus, four fascicles (SF, 1-4) attach to each metaphase kinomere (K). It is thought that fascicles extend from the kinomere poleward, fray out to let chromosomal fibrils act as traction fibrils against polar fibrils, then regroup to join the adjacent kinomere.

Freeze-fracture cytochemistry: A goal set by Russell Steere in 1957

1993 ◽  
Vol 51 ◽  
pp. 60-61
Author(s):  
Nicholas J Severs
Keyword(s):  
Chemical Nature ◽  
Biological Systems ◽  
Freeze Fracture ◽  
Structural Components ◽  
Major Goal ◽  
Membrane Biology ◽  
Routine Application ◽  
The Future ◽  
Freeze Etching

In his pioneering demonstration of the potential of freeze-etching in biological systems, Russell Steere assessed the future promise and limitations of the technique with remarkable foresight. Item 2 in his list of inherent difficulties as they then stood stated “The chemical nature of the objects seen in the replica cannot be determined”. This defined a major goal for practitioners of freeze-fracture which, for more than a decade, seemed unattainable. It was not until the introduction of the label-fracture-etch technique in the early 1970s that the mould was broken, and not until the following decade that the full scope of modern freeze-fracture cytochemistry took shape. The culmination of these developments in the 1990s now equips the researcher with a set of effective techniques for routine application in cell and membrane biology.Freeze-fracture cytochemical techniques are all designed to provide information on the chemical nature of structural components revealed by freeze-fracture, but differ in how this is achieved, in precisely what type of information is obtained, and in which types of specimen can be studied.

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