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Any replacements are listed further down
[34] viXra:1112.0042 [pdf] submitted on 7 Dec
Authors: Dm. Vatolin
Comments: 19 Pages. In Russian
The unconditional verification of geometrical axiom which is enough to solve a continuum problem has been given at the present work. The work method is similar to forcing, but is differed by that ultrafilter is applied for direct proofs essentially resolved a continuum power problem.
Category: Set Theory and Logic
[33] viXra:1110.0055 [pdf] submitted on 18 Oct 2011
Authors: Thomas Evans
Comments: 19 pages
It is the underlying purpose of the author throughout this and subsequent related
papers to consider the examination of conjectures such as the Birch-Swinnerton-Dyer
conjecture, the Riemann Hypotheses, as well as a number of other misunderstood or
unacknowledged phenomena. It is the author's hope that through such considerations,
both autonomous and presented herein, that it may become evident that the introduction
of fundamental, new practices is a necessity to any advancement in the directions of the
aforementioned. This represents the first in a series of eight (8) papers regarding these
materials. Throughout the remaining 7 the author presents, to a much greater degree of
rigor, the basic theory of analytic gauge functions, associated phenomenology, and there
from a solution to the (two) above conjectures. This paper facilitates an introduction to
the theory of analytic gauges. In the first section the author presents a re-examination of
the concepts of geometries of connections. Very briefly introduced are the basic concepts
of analytic numbers, analytic fields, analytic gauge functions, etc.
Category: Set Theory and Logic
[32] viXra:1109.0017 [pdf] submitted on 8 Sep 2011
Authors: Andrew Banks
Comments: 10 pages.
It will be demonstrated that the property function defined in Cantor's theorem allows the possibility of the
name of the set being defined to appear free in the formula defining the set, which is not permitted under
the rules of set theory. Since the property function breaks the rules of set theory, then the conclusions of
Cantor's theorem are not proven. Additionally, Cantor's diagonal argument provides an indexing method
for an infinite list of numeric digits that claims to prove there exists a sequence of digits that is not in the
list. If the digits are allowed to range from 0 to 9 , the Cantor indexing method, if valid, also shows at any
stage of n decimal digits, there is a finite sequence of decimal digits that is not in the list. Hence, the
fractional component with this "number" cannot repeat since there is no finite sequence that makes up its
digits. Therefore, if valid, this would provide a justification for the existence of irrational numbers.
However, it will be demonstrated Cantor's indexing method contains a flaw that invalidates it conclusions.
By recursively defining the list of sequences, it will be proven the recursively defined list at the nth stage is
complete, meaning no sequence of length n is missing from the list making it impossible to construct a
Cantor number. Additionally, based on this recursive definition, it will be proven any arbitrary sequence of
length n is repeated an infinite number of times eliminating this methodology as a viable strategy for
proving the existence of an irrational number. Next, a second recursive definition is offered for the
construction of the power set of the natural numbers. Again, from this definition, it is proven for any initial
set of natural numbers from 1 to n, this definition provides a construction that is complete at the nth stage.
Also, this power set definition allows a proof that the power set of the natural numbers is a countable union
of finite sets and is therefore countable. Finally, in a similar manner, a recursive definition is offered for the
power set of ℵ0 X ℵ0 such that the definition is complete at any stage n and that it is also a countable union
of finite sets and thus is countable.
Category: Set Theory and Logic
[31] viXra:1108.0025 [pdf] submitted on 19 Aug 2011
Authors: Thomas Evans
Comments: 11 pages
I present extensions to logic theory whose utilitarian application contains itself
in the form of a developmental, logical framework determinant of all being, and then
derive several applications thereof to areas of general quantum theory and pure
mathematics, providing solutions to 2 longstanding relevant problems: P vs NP and the
Riemann Hypothesis.
Category: Set Theory and Logic
[30] viXra:1108.0011 [pdf] submitted on 4 Aug 2011
Authors: Andrew Schumann
Comments: 23 pages
We present a general way that allows to construct systematically analytic
calculi for a large family of non-Archimedean many-valued logics:
hyperrational-valued, hyperreal-valued, and p-adic valued logics characterized
by a special format of semantics with an appropriate rejection
of Archimedes' axiom. These logics are built as different extensions of
standard many-valued logics (namely, Lukasiewicz's, Gödel's, Product,
and Post's logics). The informal sense of Archimedes' axiom is that anything
can be measured by a ruler. Also logical multiple-validity without
Archimedes' axiom consists in that the set of truth values is infinite and
it is not well-founded and well-ordered. We consider two cases of
non-Archimedean multi-valued logics: the first with many-validity in the interval
[0; 1] of hypernumbers and the second with many-validity in the
ring Zp of p-adic integers. On the base of non-Archimedean valued logics,
we construct non-Archimedean valued interval neutrosophic logics by
which we can describe neutrality phenomena.
Category: Set Theory and Logic
[29] viXra:1107.0045 [pdf] submitted on 23 Jul 2011
Authors: Mauro Avon
Comments: 158 pages
This paper outlines an approach to mathematical logic which is different from the standard one. We
list the most relevant features of the system. In first-order logic there exist two different concepts of
term and formula, in place of these two concepts in our approach we have just one notion of
expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is
explicitly written in the expression). The meaning of a sentence will depend solely on the meaning
of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based
on a very simple definition of proof and provides a good model of human mathematical deductive
process. The soundness and consistency of the system are proved, as well as the fact that our system
is not affected by the most known types of paradox. The paper provides both the theoretical
material and two fully documented examples of deduction. The author has built the whole system
with the idea to provide a faithful model of human mathematical deductive process. He believes this
objective has been achieved but obviously the reader is free to examine the system and get his own
opinion about it.
Category: Set Theory and Logic
[28] viXra:1103.0099 [pdf] submitted on 24 Mar 2011
Authors: Giuseppe Iurato
Comments: 27 pages.
We discuss a central aspect of history of the concept
of differentiable manifold, as a proposal for using some quantitative
methods (drew from elementary Model Theory) in Mathematical Historiography.
Category: Set Theory and Logic
[27] viXra:1010.0052 [pdf] submitted on 20 Mar 2010
Authors: Florentin Smarandache
Comments: 3 pages
A paradox is called a statement <P> which is true and false in the same time.
Therefore, if we suppose that statement <P> is true, it results that <P> is false; and reciprocally,
if we suppose that <P> is false, it results that <P> is true.
Category: Set Theory and Logic
[26] viXra:1010.0002 [pdf] submitted on 1 Oct 2010
Authors: Dm. Vatolin
Comments: 11 pages, Russian.
In this paper the question is examined that a incompleteness
of enough advanced theories of arithmetics does not follow from the Gëdel statements.
Category: Set Theory and Logic
[25] viXra:1008.0091 [pdf] submitted on 31 Aug 2010
Authors: Florentin Smarandache
Comments: 157 pages
It was a surprise for me when in 1995 I received a manuscript from the mathematician,
experimental writer and innovative painter Florentin Smarandache, especially because the
treated subject was of philosophy - revealing paradoxes - and logics.
He had generalized the fuzzy logic, and introduced two new concepts:
a) "neutrosophy" - study of neutralities as an extension of dialectics;
b) and its derivative "neutrosophic", such as "neutrosophic logic", "neutrosophic set",
"neutrosophic probability", and "neutrosophic statistics" and thus opening new ways
of research in four fields: philosophy, logics, set theory, and probability/statistics.
Category: Set Theory and Logic
[24] viXra:1005.0059 [pdf] submitted on 14 May 2010
Authors: Dm. Vatolin
Comments: 15 pages, Russian.
This article formulates three geometrical axioms from which it follows
that the continuum power is greater then any well-ordered set power.
Category: Set Theory and Logic
[23] viXra:1005.0006 [pdf] submitted on 10 Mar 2010
Authors: Andrew Schumann, Florentin Smarandache
Comments: 121 pages
This book written by A. Schumann & F. Smarandache is devoted to advances
of non-Archimedean multiple-validity idea and its applications to logical reasoning.
Leibnitz was the first who proposed Archimedes' axiom to be rejected.
He postulated infinitesimals (infinitely small numbers) of the unit interval [0, 1]
which are larger than zero, but smaller than each positive real number. Robinson
applied this idea into modern mathematics in [117] and developed so-called
non-standard analysis. In the framework of non-standard analysis there were
obtained many interesting results examined in [37], [38], [74], [117].
Category: Set Theory and Logic
[22] viXra:1004.0112 [pdf] submitted on 21 Apr 2010
Authors: V. Veeramani, Roque Batulan
Comments: 7 pages
This paper contains the Basic Definitions of an Intuitionstic Fuzzy Set theory and
operations on it. Mainly we discussed the basic concepts of α - cut with examples
and Characterisations.
Category: Set Theory and Logic
[21] viXra:1004.0096 [pdf] submitted on 19 Apr 2010
Authors: Cheng-Gui Huang
Comments: 1 pages.
I claim that Neutrosophy, by Professor Florentin Smarandache, is a deep thought in human culture.
That gives advantage to break the mechanical understanding of human culture. For example,
according to the mechanical theory: existence and non-existence could not be simultaneously.
Actually existence and non-existence are simultaneously. Everyone knows that human life is
like a way in the empty space of a bird flying. Everyone can not see himself a second ago,
everyone can not see himself for the time being and everyone can not see himself a second
future. Everyone could not know what is the existence of self. Everyone is also difficult to
say the non-existence of self. So the existence and non-existence of self are simultaneously.
And the existence and nonexistence of everything are simultaneously, where, the law of excluded
middle does not apply. These basic facts express the depth of Smarandache's Neutrosophy. He
has a lot of friends in ancient and in nowadays, in the West and in the East.
Category: Set Theory and Logic
[20] viXra:1004.0092 [pdf] submitted on 19 Apr 2010
Authors: Feng Liu
Comments: 8 pages.
Logic should have been defined as the unity of contradiction between logic director and logic
implementation. Chinese Daoism asserts that everything is defined in the unity of opposites,
namely yin and yang, accordingly yang conducts change and yin brings it up (I-Ching, also known
as Book of Changes). In this way logic is redefined in an indeterminate style to facilitate
"both A and Anti-A" etc. in neutrosophics of logic. The unity of opposites is also described
as neutrality in neutrosophy. An intermediate multi-referential model of excitation and inhibition
is developed to derive a multiagent architecture of logic, based on Chinese yin-yang philosophy.
This methodology of excitation/inhibition suggests a rhymed way of logic, leading to a dynamic
methodology of weight strategy that links logic with neural network approach. It also confirms
the crucial role of indeterminacy in logic as a fatal criticism to classical mathematics and
current basis of science.
Category: Set Theory and Logic
[19] viXra:1004.0091 [pdf] submitted on 19 Apr 2010
Authors: Feng Liu
Comments: 12 pages.
As a philosophical analysis of some fatal paradoxes, the paper distinguishes the conceptual difference
between representation of truth and source of truth, and leads to the conclusion that in order to acquire
the genuine source of truth, independently of specific representations possibly belonging to different
worlds, one is necessary to ignore all the ideas, logics, conceptions, philosophies and representable
knowledge even himself belonging to those misleading worlds, returning to his infant nature, as a
preliminary step for his cultivation of unconstrained wisdom. It also carries out some coordinative
crucial issues as natural-doctrine, minded-unwitting, logic-infancy, conception-deconception,
determinacy-indeterminacy. The paper tries to verify the role of neutrosophy and neutrosophic logic in
religious issues and open a gateway toward the oriental classics, excavating the lost treasure.
Category: Set Theory and Logic
[18] viXra:1004.0065 [pdf] submitted on 10 Apr 2010
Authors: Florentin Smarandache
Comments: 8 pages
In this paper we introduce the operators of validation and invalidation of a proposition, and we
extend the operator of S-denying a proposition, or an axiomatic system, from the geometric space
to respectively any theory in any domain of knowledge, and show six examples in geometry, in
mathematical analysis, and in topology.
Category: Set Theory and Logic
[17] viXra:1004.0051 [pdf] submitted on 8 Mar 2010
Authors: Haibin Wang, Florentin Smarandache, Yan-Qing Zhang, Rajshekhar Sunderraman
Comments: 4 pages
Neutrosophic set is a part of neutrosophy which
studies the origin, nature, and scope of neutralities, as
well as their interactions with different ideational
spectra. Neutrosophic set is a powerful general formal
framework that has been recently proposed. However,
neutrosophic set needs to be specified from a technical
point of view. To this effect, we define the settheoretic
operators on an instance of neutrosophic set,
we call it single valued neutrosophic set (SVNS). We
provide various properties of SVNS, which are
connected to the operations and relations over SVNS.
Category: Set Theory and Logic
[16] viXra:1004.0026 [pdf] submitted on 3 Apr 2010
Authors: Florentin Smarandache
Comments: 14 pages
These paradoxes are called "neutrosophic" since they are based on indeterminacy (or neutrality,
i.e. neither true nor false), which is the third component in neutrosophic logic. We generalize the
Venn Diagram to a Neutrosophic Diagram, which deals with vague, inexact, ambiguous, illdefined
ideas, statements, notions, entities with unclear borders. We define the neutrosophic truth
table and introduce two neutrosophic operators (neuterization and antonymization operators)
give many classes of neutrosophic paradoxes.
Category: Set Theory and Logic
[15] viXra:1004.0016 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache
Comments: 11 pages
In this paper we present the N-norms/N-conorms in neutrosophic logic and set as extensions of
T-norms/T-conorms in fuzzy logic and set.
Also, as an extension of the Intuitionistic Fuzzy Topology we present the Neutrosophic
Topologies.
Category: Set Theory and Logic
[14] viXra:1004.0013 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache, V. Christianto
Comments: 15 pages
We extend Knuth's 16 Boolean binary logic operators to fuzzy logic and neutrosophic
logic binary operators. Then we generalize them to n-ary fuzzy logic and neutrosophic logic
operators using the smarandache codification of the Venn diagram and a defined vector
neutrosophic law. In such way, new operators in neutrosophic logic/set/probability are built.
Category: Set Theory and Logic
[13] viXra:1004.0010 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache
Comments: 6 pages
The paper presents an initial explorations on T, I, F operations based on genetic concept
hierarchy and genetic referential hierarchy, as a novel proposal to the indeterminacy issue in
neutrosophic logic, in contrast to the T, I, F values inherited from conventional logics in which
those values would fail to demonstrate the genetic aspect of a concept and accordingly loose the
connection between generality and practicality. Based on the novel definition of logic and on the
relativity of T, F concept, it illustrates that T, F are hierarchical operations which inter-consist and
inter-complement each other, that "I" relates to a learning behavior profiled by an inspiration from
I-ching, and that the neutralization operation, as the means to solve contradictions, will eventually
come to the unification of opposites, leading to the fundamental issues in Buddhism and such alike.
It also implies that Buddhism and Daoism are not religions.
Category: Set Theory and Logic
[12] viXra:1004.0006 [pdf] submitted on 8 Mar 2010
Authors: Florentin Smarandache
Comments: 10 pages
Based on the authors intensive investigation on the oriental dialectics, the paper presents a novel
theoretical frame of matter element in the world leading science, extenics dealing with inconsistency or
incompatibility, covering the widest range of application area from informatics, system engineering to
management and finance. The dialectic matter-element is defined as the integral of all existing and prospecting
ones based on all the infinite possible cognitive models. The novel model serves as the origin of constraint
matter elements, the unity of both state description and cognitive action (cognition force with respect to neural
science), a latent part of extenics, and possibly as essence of matter element. It explains, in a novel perspective,
the origin of a name, and uncovers the source of contradiction and even the impetus of cognition.
Category: Set Theory and Logic
[11] viXra:1003.0269 [pdf] submitted on 8 Mar 2010
Authors: C. Le
Comments: 4 pages, edited by C. Le, and translated into German by Bernd Hutschenreuther
The Smarandache's Class of Paradoxes are semantic paradoxes of the form "All is <A>, the
<nonA> too!", where <nonA> is what is not <A>. As a particular case, replacing <A> but an
attribute (or, in general, by an idea) it is well know the Smarandache semantic paradox:
"All is possible, the impossible too!" which is the motto of the Paradoxism movement in arts,
letters, and sciences.
Category: Set Theory and Logic
[10] viXra:1003.0224 [pdf] submitted on 7 Mar 2010
Authors: Charles Ashbacher
Comments: 145 pages
As someone who works heavily in both math and computers, I can truly appreciate the
role that logic plays in our modern world. One cannot understand the foundations of
mathematics while lacking knowledge of the basics of logic and how proofs are
constructed. Two of the first classes I took as a graduate student in mathematics were in
the foundations of mathematics, and hardly a day goes by where I do not use some topic
from those courses.
Category: Set Theory and Logic
[9] viXra:1003.0171 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 4 pages
In this paper we give two theorems from the Propositional Calculus of the
Boolean Logic with their consequences and applications and we prove them
axiomatically.
Category: Set Theory and Logic
[8] viXra:1003.0167 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 7 pages
In this article one builds a class of recursive sets, one establishes
properties of these sets and one proposes applications. This article widens
some results of [1].
Category: Set Theory and Logic
[7] viXra:1003.0119 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 6 pages
Thirty original and collected problems, puzzles, and paradoxes in mathematics and physics are
explained in this paper, taught by the author to the elementary and high school teachers at the
University of New Mexico - Gallup in 1997-8 and afterwards. They have a more educational
interest because make the students think different!
Category: Set Theory and Logic
[6] viXra:1003.0117 [pdf] submitted on 6 Mar 2010
Authors: Florentin Smarandache
Comments: 12 pages
Classes of linguistic paradoxes are introduced with examples and
explanations. They are part of the author's work on the Paradoxist
Philosophy based on mathematical logic.
The general cases exposed below are modeled on the English
language structure in a rigid way. In order to find nice
particular examples of such paradoxes one grammatically adjusts the
sentences.
Category: Set Theory and Logic
[5] viXra:1003.0065 [pdf] submitted on 6 Mar 2010
Authors: Feng Liu, Florentin Smarandache
Comments: 10 pages
The paper presents a fresh new comprehensive ideology on Neutrosophic Logic based
on contradiction study in a broad sense: general critics on conventional logic by examining the essence of
logic, fresh insights on logic definition based on Chinese philosophical survey, and a novel and genetic
logic model as the elementary cell against Von Neumann oriented ones based on this novel definition. As
for the logic definition, the paper illustrates that logic is rather a tradeoff between different factors than
truth and false abstraction. It is stressed that the kernel of any intelligent system is exactly a contradiction
model. The paper aims to solve the chaos of logic and exhibit the potential power of neutrosophy: a new
branch of scientific philosophy.
Category: Set Theory and Logic
[4] viXra:1003.0062 [pdf] submitted on 6 Mar 2010
Authors: Feng Liu, Florentin Smarandache
Comments: 7 pages
The paper presents a fresh new start on the neutrality of neutrosophy in
that "both A and Non-A" as an alternative to describe Neuter-A in that we conceptualize
things in both intentional and unintentional background. This unity of opposites
constitutes both objective world and subjective world. The whole induction of such
argument is based on the intensive study on Buddhism and Daoism including I-ching. In
addition, a framework of contradiction oriented learning philosophy inspired from the
Later Trigrams of King Wen in I-ching is meanwhile presented. It is shown that although
A and Non-A are logically inconsistent, but they are philosophically consistent in the
sense that Non-A can be the unintentionally instead of negation that leads to confusion. It
is also shown that Buddhism and Daoism play an important role in neutrosophy, and
should be extended in the way of neutrosophy to all sciences according to the original
intention of neutrosophy.
Category: Set Theory and Logic
[3] viXra:0912.0017 [pdf] submitted on 8 Dec 2009
Authors: Feng Xu
Comments: 6 pages, first published in 2006 in Hadronic Journal, volume 29, page 227
The set of all the subsets of a set is its power set, and the cardinality of the power set is always larger than the set and its subsets. Based on the definition and the inequality in cardinality, a set cannot include its power set as element, and a power set cannot include itself as element. "Russell's set" is a putative set of all the sets that don't include themselves as element. It can be shown, however, that "Russell's set" can never take in all such sets. This is because its own power set, which (like any power set) is a set that doesn't include itself (thus qualifies as an element for "Russell's set"), cannot (although should) be taken in due to the cardinality inequality. Thus "Russell's set" can never be formed. Without it, Russell's paradox, which forced the modification of Cantor's intuitive set theory into a more restricted axiomatic theory, can never be formulated. The reported approach to resolve Russell's paradox is fundamentally different from the conventional approaches. It may restore the self-consistency of Cantor's original set theory, make the Axiom of Regularity unnecessary, and expand the coverage of set to assemblies that include themselves as element.
Category: Set Theory and Logic
[2] viXra:0910.0041 [pdf] submitted on 21 Oct 2009
Authors: Amrit S. Sorli
Comments: 2 pages
In 1949, Gödel postulated a theorem that stated: "In any universe described by the theory of relativity,
time cannot exist". Gödel idea was that forth coordinate of space-time is not time. Fourth coordinate is
spatial too. In this article will be shown that on the base of elementary perception and experimental data
Gödel theorem is right. With eyes one observes universe is in a continuous change. A change n gets
transformed into a change n+1, the change n+1 into a change n+2 and so on. Clocks measure a
frequency, velocity and numerical order of change. Experimental date confirms that changes and clocks
do not run time; they run in space only. Time is not a part of space. Fourth coordinate of space-time is
spatial too. Space itself is timeless. Physical time that is clocks run is man created physical reality.
Fundamental arena of the universe is timeless space. In the timeless space into which massive bodies
and elementary particles move there is no past and no future. Past and future belong to the inner
neuronal space-time that is a result of neuronal activity of the brain.
Category: Set Theory and Logic
[1] viXra:0909.0039 [pdf] submitted on 16 Sep 2009
Authors: Victor Porton
Comments: 2 Pages.
In the framework of ZF formally considered generalizations, such as whole numbers generalizing
natural number, rational numbers generalizing whole numbers, real numbers generalizing
rational numbers, complex numbers generalizing real numbers, etc. The formal consideration
of this may be especially useful for computer proof assistants.
Category: Set Theory and Logic
[14] viXra:1109.0017 [pdf] replaced on 3 Nov 2011
Authors: Andrew Banks
Comments: 14 pages.
It will be demonstrated that the property function defined in Cantor's theorem allows the possibility of the
name of the set being defined to appear free in the formula defining the set, which is not permitted under
the rules of set theory. Since the property function breaks the rules of set theory, then the conclusions of
Cantor's theorem are not proven. Additionally, Cantor's diagonal argument provides an indexing method
for an infinite list of numeric digits that claims to prove there exists a sequence of digits that is not in the
list. If the digits are allowed to range from 0 to 9 , the Cantor indexing method, if valid, also shows at any
stage of n decimal digits, there is a finite sequence of decimal digits that is not in the list. Hence, the
fractional component with this "number" cannot repeat since there is no finite sequence that makes up its
digits. Therefore, if valid, this would provide a justification for the existence of irrational numbers.
However, it will be demonstrated Cantor's indexing method contains a flaw that invalidates it conclusions.
By recursively defining the list of sequences, it will be proven the recursively defined list at the nth stage is
complete, meaning no sequence of length n is missing from the list making it impossible to construct a
Cantor number. Additionally, based on this recursive definition, it will be proven any arbitrary sequence of
length n is repeated an infinite number of times eliminating this methodology as a viable strategy for
proving the existence of an irrational number. Next, a second recursive definition is offered for the
construction of the power set of the natural numbers. Again, from this definition, it is proven for any initial
set of natural numbers from 1 to n, this definition provides a construction that is complete at the nth stage.
Also, this power set definition allows a proof that the power set of the natural numbers is a countable union
of finite sets and is therefore countable. Finally, in a similar manner, a recursive definition is offered for the
power set of ℵ0 X ℵ0 such that the definition is complete at any stage n and that it is also a countable union
of finite sets and thus is countable.
Category: Set Theory and Logic
[13] viXra:1107.0045 [pdf] replaced on 2011-12-11 13:41:47
Authors: Mauro Avon
Comments: 159 Pages.
The paper is about an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. We list the most relevant features of the system. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one notion of expression. The set-builder notation is enclosed as an expression-building pattern. In our system we can easily express second-order and all-order conditions (the set to which a quantifier refers is explicitly written in the expression). The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved, as well as the fact that our system is not affected by the most known types of paradox. The paper provides both the theoretical material and two fully documented examples of deduction. The author believes his aims have been achieved but obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic
[12] viXra:1107.0045 [pdf] replaced on 8 Sep 2011
Authors: Mauro Avon
Comments: 159 pages
The paper is about an approach to logic that differs from the standard first-order logic and other
known approaches. It should be a new approach the author has created proposing to obtain a general
and unifying approach to logic and a faithful model of human mathematical deductive process. We
list the most relevant features of the system. In first-order logic there exist two different concepts of
term and formula, in place of these two concepts in our approach we have just one notion of
expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is
explicitly written in the expression). The meaning of a sentence will depend solely on the meaning
of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based
on a very simple definition of proof and provides a good model of human mathematical deductive
process. The soundness and consistency of the system are proved, as well as the fact that our system
is not affected by the most known types of paradox. The paper provides both the theoretical
material and two fully documented examples of deduction. The author believes his aims have been
achieved but obviously the reader is free to examine the system and get his own opinion about it.
Category: Set Theory and Logic
[11] viXra:1107.0045 [pdf] replaced on 12 Aug 2011
Authors: Mauro Avon
Comments: 159 pages
This paper outlines an approach to mathematical logic which is different from the standard one. We
list the most relevant features of the system. In first-order logic there exist two different concepts of
term and formula, in place of these two concepts in our approach we have just one notion of
expression. The set-builder notation is enclosed as an expression-building pattern. In our system we
can easily express second-order and all-order conditions (the set to which a quantifier refers is
explicitly written in the expression). The meaning of a sentence will depend solely on the meaning
of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based
on a very simple definition of proof and provides a good model of human mathematical deductive
process. The soundness and consistency of the system are proved, as well as the fact that our system
is not affected by the most known types of paradox. The paper provides both the theoretical
material and two fully documented examples of deduction. The author has built the whole system
with the idea to provide a faithful model of human mathematical deductive process. He believes this
objective has been achieved but obviously the reader is free to examine the system and get his own
opinion about it.
Category: Set Theory and Logic
[10] viXra:1103.0099 [pdf] replaced on 2 Jul 2011
Authors: Giuseppe Iurato
Comments: 44 pages.
We discuss a central aspect of history of the concept of a
differentiable manifold, as a proposal for to confirm the need of to use some
quantitative methods (drew from elementary Model Theory) in
Mathematical Historiography.
Category: Set Theory and Logic
[9] viXra:1103.0099 [pdf] replaced on 25 May 2011
Authors: Giuseppe Iurato
Comments: 34 pages.
We discuss a central aspect of history of the concept of a
differentiable manifold, as a proposal for to confirm the need of to use some
quantitative methods (drew from elementary Model Theory) in
Mathematical Historiography.
Category: Set Theory and Logic
[8] viXra:1103.0099 [pdf] replaced on 27 Apr 2011
Authors: Giuseppe Iurato
Comments: 33 pages.
We discuss a central aspect of history of the concept of
a differentiable manifold, as a proposal for to confirm the need of to
use some quantitative methods (drew from elementary Model Theory)
in Mathematical Historiography.
Category: Set Theory and Logic
[7] viXra:1103.0099 [pdf] replaced on 2 Apr 2011
Authors: Giuseppe Iurato
Comments: 31 pages.
We discuss a central aspect of history of the concept
of differentiable manifold, as a proposal for using some quantitative
methods (drew from elementary Model Theory) in Mathematical Historiography.
Category: Set Theory and Logic
[6] viXra:1010.0002 [pdf] replaced on 19 Apr 2011
Authors: Dm. Vatolin
Comments: 9 pages, in Russian.
In this paper the question is examined that an incompleteness
of advanced enough theories of arithmetic does not
follow from the Gëdel statements.
Category: Set Theory and Logic
[5] viXra:1010.0002 [pdf] replaced on 6 Apr 2011
Authors: Dm. Vatolin
Comments: 9 pages, Russian.
In this paper the question is examined that a incompleteness
of enough advanced theories of arithmetics does not follow from the Gëdel statements.
Category: Set Theory and Logic
[4] viXra:1010.0002 [pdf] replaced on 31 Oct 2010
Authors: Dm. Vatolin
Comments: 11 pages, Russian.
In this paper the question is examined that a incompleteness
of enough advanced theories of arithmetics does not follow from the Gëdel statements.
Category: Set Theory and Logic
[3] viXra:1005.0059 [pdf] replaced on 22 Nov 2010
Authors: Dm. Vatolin
Comments: 14 pages, Russian.
This article formulates three geometrical axioms from which it follows
that the continuum power is greater then any well-ordered set power.
Category: Set Theory and Logic
[2] viXra:1004.0026 [pdf] replaced on 22 Apr 2010
Authors: Florentin Smarandache
Comments: 14 pages
These paradoxes are called "neutrosophic" since they are based on indeterminacy (or neutrality,
i.e. neither true nor false), which is the third component in neutrosophic logic. We generalize the
Venn Diagram to a Neutrosophic Diagram, which deals with vague, inexact, ambiguous, illdefined
ideas, statements, notions, entities with unclear borders. We define the neutrosophic truth
table and introduce two neutrosophic operators (neuterization and antonymization operators)
give many classes of neutrosophic paradoxes.
Category: Set Theory and Logic
[1] viXra:1002.0003 [pdf] replaced on 15 Mar 2010
Authors: Willi Penker
Comments: 3 pages
To shift assignments between infinite sets is to create a
disturbance within the assignment itself that cannot be
removed. An assignment carrying such a disturbance cannot be
regarded as static.
Category: Set Theory and Logic