| viXra.org > Set Theory and Logic > 2010 |
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| viXra.org > Set Theory and Logic > 2010 |
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[5] viXra:2010.0159 [pdf] replaced on 2020-12-01 01:54:54
Authors: Antonio Leon
Comments: 13 Pages.
The argument of Thomson lamp and Benacerraf’s critique are reexamined from the perspective of the w-order legitimated by the hypothesis of the actual infinity subsumed into the Axiom of Infinite. The conclusions point to the inconsistency of that hypothesis.
Category: Set Theory and Logic
[4] viXra:2010.0151 [pdf] submitted on 2020-10-19 10:54:51
Authors: Antonio Leon
Comments: 3 Pages.
The discussions on Thomson’s lamp analyzed in the precedent chapter can be formalized (at least up to a certain point) by introducing a simple symbolic notation that allows to define the lamp and its functioning in abstract terms. The symbolic definition can then be used to develop formulas that represent the functioning laws of the lamp. Being independent of the number of times the lamp is turned on/off, these laws represent the universal attributes and the universal behaviour of a Thomson’s lamp. As we will see, some of those laws are not compatible with the assumption that a Thomson’s lamp can be switched infinitely many times during a finite interval of time. This conclusion proves that, as its author defended, Thomson supertask is inconsistent.
Category: Set Theory and Logic
[3] viXra:2010.0131 [pdf] submitted on 2020-10-18 12:33:45
Authors: Antonio Leon
Comments: 34 Pages.
It seems reasonable to assume that mathematical infinity was not the objective of Zeno’s Dichotomy (in any of its variants), however, a sort of mathematical infinity was already present in these celebrated arguments. Aristotle proposed a first solution to Zeno’s Dichotomy by introducing what we now call one-to-one correspondences, the key instrument of modern infinitist mathematics. But Aristotle, more naturalist than platonic, finally rejected the method of pairing the elements of two infinite collections (in this case of points and instants) and introduced instead the distinction between actual and potential infinities. Aristotle’s distinction served to define, gross modo, two opposite positions on the nature of infinity for more than twenty centuries. The actual infinity was finally mathematized through set theory in the first years of the XX century and the discussions on its potential or actual nature almost vanished. But, as we will see here, things still remain to be said on this issue.
Category: Set Theory and Logic
[2] viXra:2010.0088 [pdf] submitted on 2020-10-13 08:06:33
Authors: Egger Mielberg
Comments: 12 Pages.
Simple and readable diagrams of a complex set of any kind would allow a million of connections between elements of that set to be formulated and understood clearly. Current diagrams provide the
visual solution for the first four-five sets mostly, but it is unreadable for the number of sets in more than five. We propose a solution, sense diagrams, for visualization of multimillion sets using sense-to-sense paradigm [1]. The nature of the set can be any. The nature of the elements of a set may differ from each other. The main criterion of the use of this diagrams is the
presence of qualitative or/and quantitative properties of an element of the set.
Category: Set Theory and Logic
[1] viXra:2010.0009 [pdf] replaced on 2021-07-20 22:18:56
Authors: Udo E. Steinemann
Comments: 18 Pages.
If a variable is replace by its square and subsequently enlarged by a constant during a number of iteration-steps in quaternion-space, a network of (3) sets will be built gradually. As long as for the iteration-constant certain conditions are fulfilled, the network will consist of: an unbounded set (escape-set) with trajectories escaping to infinity during course of the iteration, a bounded set (prisoner-set) with trajectories tending to a sink-point and a further bounded one (JULIA-set) with a fixed-point as repeller having a repulsive effect on all points of both the other sets. The iteration will continue until the attracting sink-point of prisoner-set and the repelling fixed-point on JULIA-set have been found. This situation is reached if predecessor- and successor-state of the iteration became equal. The fixed-point-condition provisionally formulated in general terms of quaternions, can be separated into (3) sub-conditions. When heeding the HAMILTONian-rules for interactions of the imaginary sub-spaces of the quaternion-space, each sub-condition will be appropriate for one imaginary sub-spaces and independently debatable. Knowledge of fixed-points from this fundamental network will one enable to study the structure of a connected JULIA-set.
The Iteration will start from (1) on real-axis, this is not a restriction on generality because an appropriate scaling on real-axis can always be archived this way. It will become obvious, that the fixed-points in prisoner- and JULIA-set will depend on the iteration-constant only. Thus (16) different constants chosen appropriately will enable to arrange (16) fixed-points of JULIA-sets in the square-points of a hyper-cube and thereby together with the JULIA-sets to built a related JULIA-network. The symmetry-properties of this related JULIA-network can be studied on base of a hyper-cube's symmetry-group extended by some additional considerations.
Category: Set Theory and Logic
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