We present a formal mechanical analysis using sweeping net methods to approximate surfacingsingularities of saddle maps. By constructing densified sweeping subnets for individual vertices and integrating them, we create a comprehensive approximation of singularities. This approach utilizes geometric concepts, analytical methods, and theorems that demonstrate the robustness and stabilityof the nets under perturbations. Through detailed proofs and visualizations, we provide a new perspective on singularities and their approximations in analytic geometry.

We continue with the development of the theory of problems and their solutions spaces cite{agama2023topology} and cite{agama2022theory}. We introduce and study the notion of verification and resolution time complexity of solutions and problem spaces.

This paper, building upon our previous work on Gödel category singularities (https://vixra.org/abs/2407.0164),presents a comprehensive geometric theory of Gödelian phenomena. By recasting logical structures as intricatemathematical landscapes, we offer a novel perspective on the nature of incompleteness and undecidability. Our approachsynthesizes concepts from category theory, algebraic topology, differential geometry, and dynamical systemsto create a rich, multidimensional view of logical spaces. We introduce the concept of Gödelian manifolds, wherestatements in formal systems are represented as points in a vast terrain. The elevations and contours of this landscapecorrespond to logical complexity and provability, with Gödelian singularities emerging as profound chasmsor peaks. This geometric framework allows us to apply tools from various mathematical disciplines to analyze thestructure of incompleteness. Our approach enables a nuanced analysis of different types of logical complexity. Wedevelop theoretical constructs to explore the nature of self-referential paradoxes, non-self-referential undecidability,and the characteristics of difficult but provable statements within our geometric model. This provides new mathematicalinsights into the structure of formal systems and the limits of provability. To illustrate the conceptualpower of this approach, we draw an analogy to the alleged "Gödel loophole" in the U.S. Constitution. While nota direct application, this metaphorical exploration demonstrates how our abstract framework can provide intuitiveunderstanding of complex logical structures, offering an accessible entry point for non-specialists to grasp theseintricate ideas.

An injection from the countable ordinals to the paths in the binary tree leads to a bijection between all paths in the binary tree and only those in the $aleph_1$-sized injection. Since there are $2^{aleph_0}$ many paths in the binary tree, this proves the Continuum Hypothesis, $2^{aleph_0} = aleph_1$.

This paper presents a novel geometric approach to Gödelian incompleteness phenomena using higher category theory and topos theory. We construct a hierarchy of (∞, 1)-categories that model formal systems as multidimensionalspaces, transforming logical structures into geometric objects. This framework allows us to represent Gödel’s incompleteness results as topological features—singularities or holes—in the fabric of mathematical space. Our use of (∞, 1)-categories is crucial for modeling the higher-order relationships between proofs and metaproofs, providing a natural setting for analyzing self-reference and reflection principles. These logical conceptsare transformed into geometric structures, offering new insights into the nature of incompleteness. We develop a topos-theoretic model that serves as a universal vantage point for surveying the landscape of formal systems. From this perspective, we prove a generalized incompleteness theorem that extends Gödel’sresults to a broader class of formal systems, now interpreted as geometric obstructions in the topos. Leveraging homotopy type theory, we establish a precise correspondence between proof-theoretic strength and homotopical complexity. This connection yields a novel complexity measure for formal systems based onthe geometric properties of their corresponding spaces. Our framework provides new insights into the nature of mathematical truth and the limits of formalization. It suggests a more nuanced view of the hierarchy of mathematical theories, where incompleteness manifests as an intrinsic topological feature of the space of theories.While primarily theoretical, our approach hints at potential applications in theoretical computer science, particularly in complexity theory. We also discuss speculative connections to fundamental questions in physics and cognitive science, presented as avenues for future research. By recasting Gödelian phenomena in geometric terms through higher category theory, we open new avenues for understanding the nature of mathematical reasoning and its inherent limitations. This geometric perspectiveoffers a powerful new language for exploring the foundations of mathematics and the boundaries of formal systems.

This document presents a comprehensive study of fractal partitioning and its application to subconvexity generalizations across various mathematical contexts. By utilizing a combination of advanced equations andinequalities, the paper develops robust models for partitioning sets into subsets of varying sizes, measuring the similarity and complexity within these partitions, and ensuring consistent interactions across boundaries. Special attention is given to computing the norm of differences betweensubsets and assessing their similarity, along with complexity measurements utilizing tensor equations and sums. These calculations provideinsights into the partitions’ fractal behavior and their probabilistic interactions.The document also delves into task scheduling algorithms based on SRPT, round-robin, and deadline-driven protocols, highlighting practical implications of fractal partitioning in optimizing resource management and minimizing distortions in dynamic systems. An emphasis is placedon ensuring the robustness and efficiency of fractal partitions through rigorousmathematical proofs and algorithmic implementations. By applyingthese models to data compression and analysis, the study demonstrates how fractal partitioning can efficiently represent complex data sets, expose hidden patterns, and identify anomalies in various domains such as finance and natural systems. Furthermore, the paper explores the concept of subconvexity in higher powers of the Riemann zeta function, establishing stronger forms of subconvexity conditions for different mathematical functions. This includesgeneralizations for cubic and higher powers of zeta functions, providing substantial evidence in support of hypotheses like the Riemann Hypothesis. The comprehensive approach combines theoretical constructs with practical algorithms, offering a powerful framework for analyzing and understanding complex mathematical and natural phenomena through fractalpartitioning and subconvexity measures.

In 1930 Gödel wrote a landmark paper showing that in any formal system there will always bestatements that cannot be proven. But the deficiency of formal systems goes much deeper. The same logically valid statement can be used in conjunction with two different sets simultaneously proving a true statement and a false statement. This result is profound. It explains why people can use the same sound argument to prove two contradictory statements. It is no wonder the most lucid arguments still sometimes result in hung juries and earnest people can disagree on the most fundamental issues. Truth is a much deeper concept than logical validity.

The Continuum Hypothesis has recently been proven in a form that might have been accepted had it appeared before ZFC but after Hilbert’s challenge in 1900. This work will develop the same technique to prove the Generalized Continuum Hypothesis by induction on aleph and beth subscripts.

This is a rendition of [2]. We study stringy motivic structures. This builds upon work dealing with $mathbb{F}_p$-modives for a suitable prime p. In our case, we let p be a long exact sequence spanning a path in a pre-geometric space. We superize a nerve from our previous study.

In this paper, we prove a joint generalization of Arslanov’s completenesscriterion and Visser’s ADN theorem for precomplete numberings. Then we considerthe properties of completeness and precompleteness of numberings in the context ofthe positivity property. We show that the completions of positive numberings are nottheir minimal covers and that the Turing completeness of any set A is equivalent to theexistence of a positive precomplete A-computable numbering of any infinite family withpositive A-computable numbering.

The paper studies Σ-0-n-computable families (n ⩾ 2) and their numberings. It is proved that any non-trivial Σ-0-n-computable family has a complete with respect to any of its elements Σ-0-n-computable non-principal numbering. It is established that if a Σ-0-n-computable family is not principal, then any of its Σ-0-n-computable numberings has a minimal cover.

We model an absolute reference frame using a pullback on a certain locally trivial line bundle. We demonstrate that this pullback is unrealizable in $mathbb{R}^4$. We devote section 3 to process-based thinking.

For a given infinite countable set A, we demonstrate that A is an infinite countableset if and only if A is equal to an infinite countable set indexed to the infinity. Saidotherwise we demonstrate that A is an infinite countable set iff there exists an infinite numberof non-empty, distinct elements a_i ǂ ∅, i ∈ N∗, ∀i, j ∈ N∗, i ǂ j, a_i ǂ a_j such thatA = U_{+∞}_{i=1} {a_i}. At this occasion, for infinite countable sets constituted by the union of two giveninfinite countable sets A and B, Au2032 = [A ∪ B]P(Au2032), we introduce the notion of undeterminedinfinite countable set in order to designate infinite countable sets for whichan explicit indexation is not determined meanwhile such indexation must necessarilyexist.

A dependent type theory is proposed as the foundation of mathematics. The formalism preserves the structure of mathematical thought, making it natural to use. The logical calculus of the type theory is proved to be syntactically complete. Therefore it does not suffer from the limitations imposed by Gödel’s incompleteness theorems. In particular, the concept of mathematical truth can be defined in terms of provability.

In this paper, we study the topology of problems and their solution spaces developed introduced in our first paper. We introduce and study the notion of separability and quotient problem and solution spaces. This notions will form a basic underpinning for further studies on this topic.

For a given infinite countable set constituted by the union of infinitely many non-empty, finite or infinite countable, disjoint sub-sets A = {i∈N*}[A_i]_P(A) , ∀i ∈ N* , A_i ≠ ∅, ∀i, j ∈ N* , i ≠ j, A_i ∩ A_j = ∅, we demonstrate that it is legitimate to partition A in a finite or infinite number of sub-sets that are themselves constituted by a finite or infinite number of sub-sets of A when the initial order of indexation of the sub-sets of A maintains a strictly increasing order in each sub-set.

We introduce and develop the logic of existence of solution to problems. We use this theory to answer the question of Florentin Smarandache in logic. We answer this question in the negative.

For a given infinite countable set A = U_{i∈N∗}{a_i }, ∀i ∈ N∗ , a_i≠∅, we demonstrate that it is legitimate to partition A in a finite or infinite number of infinite countable sub-sets when the initial order of indexation of the elements of A maintains a strictly increasing order in each sub-set. At this occasion we introduce two new formalism allowing to signify the fact that it is A that has been partitioned and the fact that A and the latter partition of A belong to the same class comprising infinitely many infinite countable sets constituted by the same infinite countable sub-sets that constitute the partition of A.

Arrew (Arrow Rewriter) is a mathematical system (theorem prover) that allows expressing and working with formal systems. It relies on a simple substitution rule and set equality to derive theorems.

If you study mathematics you are probably aware of the foundational crises that mathematics went through at the beginning of the 20th century. The three broad schools of thought namely constructivism, intuitionism and formalism collided and judging by the approach used today by most mathematicians, we can easily say that formalism emerged victorious in some sense. However while debates regarding the foundations of mathematics have subsided over the years, they aren’t dead. One such school of mathematics which still sees considerable traffic is finitism. In this article, we will be analysing the criticism of a finitist named Norman J Wildberger and trying to defend the current axiomatic mathematical systems against them.

Traditionally, the concept of an algorithm is introduced into the theory through a sequence of elementary steps leading to the solution of a problem, and parallel algorithms are considered as a technical solution external to the Theory of Algorithms, which allows speeding up the execution process. However, a number of physical processes currently used for computing, such as quantum computing, do not fit into the framework of the predictions of the Theory of Algorithms, in particular --- in terms of computational complexity, which suggests that our understanding of parallel computing processes, limited by the framework of the classical Theory of Algorithms, may not be complete. A qualitative leap in the Theory of Computability is possible if parallel algorithms are understood as a generalization of the classical ones within the framework of the hypothetical Theory of Parallel Algorithms. In this paper, pre-quantum physical processes are considered, which are already beyond the scope of the classical Theory of Algorithms. Conceptual primitives suitable for the analysis of parallel flows are proposed.

We reveal adjacent real points in the real set using a concise logical reference. This raises a paradox while the real set is believed as existing and complete. However, we prove each element in a totally ordered set has adjacent element(s); there is no densely ordered set. Furthermore, since the natural numbers can also be densely ordered under certain ordering, the set of natural numbers, which is involved with each infinite set in ZFC set theory, does not exist itself.

The set of all Real numbers, R, consists of all Rational numbers, Q, being any ratio of two Integer numbers that does not divide by 0. All other Real numbers that are not a Rational number are contained in the set of Irrational numbers, R/Q. These two subsets comprising all of the Real numbers are known to have distinct cardinalities of differing magnitudes of infinity[2]. When a consecutive ordering of all Rational numbers is established, whereby any unique Rational number can be shown to be disconnected from all other Rational numbers[3], a theorem regarding asymmetry on the Real number line is established. This theorem simplifies the necessary requirements to prove that the summation of two known Irrational numbers is Rational or Irrational.

For A an infinite countable set containing infinitely many distinct natural integersand B an infinite countable set containing infinitely many distinct natural integerssuch that ∀n ∈ A, n ∈ B and ∀m ∈ B, m ∈ A, we demonstrate that it is possiblethat A≠B by exposing infinitely many counter-examples in which, for each counter-example, A and B are respectively two sample spaces of two probability spaces havingdifferent probabilities for similar events. We thus prove that the axiom of extensionality is false for infinite countable sets.

We challenge Georg Cantor's theory about infinity. By attacking the concept of “countable/uncountable” and diagonal argument, we reveal the uncertainty, which is obscured by the lack of clarity. The problem arises from the basic understandings of infinity and continuum. We perform many thought experiments to refute current standard views. The results support the opinion that no potential infinity leads to an actual infinity, nor is there any continuum composed of indivisibles statically, nor is Cantor's theory consistent in itself.

This paper discusses Cantor’s paradox of the set all cardinals, and proves that in Cantor’s set theory every set of cardinal C originates at least 2C inconsistent infinite sets.

From different areas of mathematics, such as set theory, geometry, transfinite arithmetic or supertask theory, in this book more than forty arguments are developed about the inconsistency of the hypothesis of the actual infinity in contemporary mathematics. A hypothesis according to which the uncompletable lists, as the list of the natural numbers, exist as completed lists. The inconsistency of this hypothesis would have an enormous impact on physics, forcing us to change the continuum space-time for a discrete model, with indivisible units (atoms) of space and time. The discrete model would be a great simplification of physical theories, including relativity and quantum mechanics. It would also suppose the solution of the old problem of change, posed by the pre-Socratics philosophers twenty-seven centuries ago.

This paper proves a result on the decimal expansion of the rational numbers in the open rational interval (0, 1), which is subsequently used to discuss a reordering of the rows of a table T that is assumed to contain all rational numbers within (0, 1), in such a way that the diagonal of the reordered table T could be a rational number from which different rational antidiagonals (elements of (0, 1) that cannot be in T ) could be defined. If that were the case, and for the same reason as in Cantor’s diagonal argument, the open rational interval (0, 1) would be non-denumerable, and we would have a contradiction in set theory, because Cantor also proved the set of rational numbers is denumerable.

Inspired by the emblematic Hilbert Hotel, Hilbert machine is a conceptual super-machine whose functioning questions the consistency of the actual infinity hypothesis subsumed into the Axiom of Infinity.

This chapter introduces a formalized version of Zeno’s Dichotomy in its two variants (here referred to as Dichotomy I and II) based on the discreteness and separation of ω-order (Dichotomy I) and of ω∗-order (Dichotomy II) defined below in this section. Each of these formalized versions leads to a contradiction pointing to the inconsistency of the hypothesis of the actual infinity.

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.

I succeeded to give mathematical expressions to any correct Quranic Exegeses and define the Quranic correctness as the unique existence of Tahara I function. In a precise mathematical sense, the expressions and the definition are ill-defined however they might have meanings to prove the Quranic correctness.

The students in the surprise exam story reasoned that no surprise exam could take place on any day of the week. Actually, however, the students were surprised on Wednesday by the teacher's surprise exam. In this paper, we show where the students' reasoning went wrong and that students should be surprised on Wednesday or Tuesday.

In propositional logic, given a set of axioms, we can derive formulas. Here we present the derivations of some formulas without the use of the Deduction Theorem. The derivations are presented compactly with only few referrals to other theorems. Most textbooks in this subject avoid this kind of approach.

In standard construction of hyperrational numbers using ultrapower we assume that the ultrafilter is selective. It makes possible to assign real value to any finite hyperrational number. So, we can consider hyperrational numbers with selective ultrafilter as extension of traditional real numbers. Also proved the existence of strictly monotonic or stationary representing sequence for any hyperrational number.

What is mathematics? Why does it exist? Is it consistent? Is it complete? We answer these questions as well as resolve Russell’s Paradox and debunk Godel’s Incompleteness Theorem.

In the present study, for the medical decision making problem, the proposed technique related to the fuzzy soft set by Celik-Yamak through Sanchez’s method was used. The real dataset which is called Cleveland heart disease dataset applied in this problem.

A semantic analysis of formal systems is undertaken, wherein the duality of their symbolic definition based on the “State of Doing” and “State of Being” is brought out. We demonstrate that when these states are defined in a way that opposes each other, it leads to contradictions. This results in the incompleteness of formal systems as captured in the Gödel’s theorems. We then proceed to resolve the P-NP problem, which we show to be a manifestation of Gödel’s theorem itself. We then discuss the Zeno’s paradox and relate it to the same aforementioned duality, but as pertaining to discrete and continuous spaces. We prove an important theorem regarding representations of irrational numbers in continuous space. We extend the result to touch upon the Continuum Hypothesis and present a new symbolic conceptualization of space, which can address both discrete and continuous requirements. We term this new mathematical framework as “hybrid space”.

A proof of the Continuum Hypothesis as originally posed by Georg Cantor in 1878; that an uncountable set of real numbers has the same cardinality as the set of all real numbers. Any set of real numbers can be encoded by the infinite paths of a binary tree. If the binary tree has an uncountable node it must have a descendant with 2 uncountable successors. Each of those will have descendants with 2 uncountable successors, recursively. As a result the infinite paths of an uncountable binary tree will have the same cardinality as the set of all real numbers, as will the uncountable set of real numbers encoded by the tree.

Pythagorean fuzzy set (PFS) initially extended by Yager from intuitionistic fuzzy set (IFS), which can model uncertain information with more general conditions in the process of multi-criteria decision making (MCDM). The fuzzy decision analysis of this paper is mainly based on two expressions in Pythagorean fuzzy environment, namely, Pythagorean fuzzy number (PFN) and interval-valued Pythagorean fuzzy number (IVPFN). We initiate a novel axiomatic definition of Pythagorean fuzzy distance measure, including PFNs and IVPFNs, and put forward the corresponding theorems and prove them. Based on the defined distance measures, the closeness indexes are developed for PFNs and IVPFNs inspired by the idea of technique for order preference by similarity to ideal solution (TOPSIS) approach. After these basic definitions have been established, the hierarchical decision approach is presented to handle MCDM problems under Pythagorean fuzzy environment. To address hierarchical decision issues, the closeness index-based score function is defined to calculate the score of each permutation for the optimal alternative. To determine criterion weights, a new method based on the proposed similarity measure and aggregation operator of PFNs and IVPFNs is presented according to Pythagorean fuzzy information from decision matrix, rather than being provided in advance by decision makers, which can effectively reduce human subjectivity. An experimental case is conducted to demonstrate the applicability and flexibility of the proposed decision approach. Finally, the extension forms of Pythagorean fuzzy decision approach for heterogeneous information are briefly introduced as the further application in other uncertain information processing fields.

Finding whether a boolean formula is a tautology or not in a feasible time is an important problem of computer science. Many algorithms have been developed to solve this problem but none of them is a polynomial time algorithm. Our aim is to develop an algorithm that achieve this in polynomial time. In this article, we convert boolean functions to some graph forms in polynomial time. They are called two dimensional formulas and similar to AND-OR graphs except arches on them are bidirectional. Then these graphs are investigated to find properties which can be used to differentiate tautological formulas from non tautological ones.

Derived the Statistics of the un-solved problems (conjectures). The probability, what a conjecture will be solved is 50 %. The probability, that a conjecture is true is p=37 %. The probability, what we get to know the latter is psi=29 %....

We interpret 5 of the 10 axioms of ZFC (Zermelo-Fraenkel set theory with axiom of choice) as normal statements and proof them. So these 5 sentences don't need to be introduced as axioms, but can be used as proven statements.

This article is a mathematical experiment with the sets and the formulas. We consider new elements which are called the electors. The elector has the properties of the sets and the formulas.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤). The limit points are determined by the formula lim (X) = sup{a ∈ M | {x ∈ X| a ≤ x} ∈ D}, D where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter on X. We investigate lim (X) and topology atop(χ) properties. In particular, D we prove the compactness of the topology atop(χ).

Curry's paradox is generally considered to be one of the hardest paradoxes to solve. It is shown here that the paradox can be arrived in fewer steps and also for a different term of the original biconditional. Further, using different approaches, it is also shown that the conclusion of the paradox must always be false and this is not paradoxical but it is expected to be so. One of the approaches points out that the starting biconditional of the paradox amounts to a false definition or assertion which consequently leads to a false conclusion. Therefore, the solution is trivial and the paradox turns out to be no paradox at all. Despite that fact that verifying the truth value of the first biconditional of the paradox is trivial, mathematicians and logicians have failed to do so and merely assumed that it is true. Taking this into consideration that it is false, the paradox is however dismissed. This conclusion puts to rest an important paradox that preoccupies logicians and points out the importance of verifying one's assumptions.

A terminological framework is proposed for the mathematical examination and analysis of the Mandelbrot set's correlative ectocopial set. The Apeiropolis and anthropobrot multisets are defined and explained to be the mathematical entities underlying the well-known Buddhabrot visualization. The definitions are presented as tools conducive to finding novel approaches and generating discoveries that might otherwise be missed via a primarily programmatic approach. The anthropobrot multisets are introduced as a new, infinite repository of unique pareidolic figures as richly diverse as the Julia sets.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

In this paper we define an arithmetic theory PAM, which is an extension of Peano arithmetic PA, and prove that theory PAM has only one (up to isomorphism) model, which is the standard PA–model.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than Aleph Zero, the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but it is usually believed that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a degenerate case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental.

In this article some difficulties are deduced from the set of natural numbers. The demonstrated difficulties suggest that if the set of natural numbers exists it would conflict with the axiom of regularity. As a result, we have the conclusion that the class of natural numbers is not a set but a proper class.

In this paper we show the consistency of the essential part of Sergeyev's numerical methodology (\cite{Yarov 1}, \cite{Yarov 2}) by constructing a model of it within the framework of an ultrapower of the ordinary real number system.

The paper is about 'absolute logic': 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. 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. In our system the set-builder notation is an expression-building pattern. In our system we can easily express second-order, third order and any-order conditions. 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. We also discuss how our system relates to the most know types of paradoxes, from the discussion no specific vulnerability to paradoxes comes out. The paper provides both the theoretical material and a fully documented example of deduction.

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].