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This document discusses the Axiom of Choice (AC), a foundational principle in set theory that states that for any set of nonempty sets, there exists a function that chooses one element from each set. The document provides examples to illustrate AC with finite and infinite sets. It explains that while AC seems intuitive for finite sets, it leads to counterintuitive conclusions for infinite sets. The document also discusses the relationship between AC and Zermelo-Fraenkel set theory (ZF), noting that ZF does not prove or disprove AC, so its inclusion in ZF is a matter of mathematical preference.

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Axiom of choice presentation

The Axiom of Choice allows the selection of one element from each set in an infinite collection of non-empty sets. It is an axiom of set theory that is independent from the other axioms and its validity is controversial. The Axiom of Choice enables proofs of counterintuitive results like the Banach-Tarski Paradox, where a single ball can be decomposed and reassembled into two identical balls. While powerful, accepting or rejecting the Axiom of Choice can have implications in mathematics.

Neutral Geometry_part2.pptx

The document discusses Neutral Geometry, which is derived from Euclid's first four postulates without using the parallel postulate. It presents the Saccheri-Legendre theorem, which states that the sum of the interior angles of any triangle is less than or equal to 180 degrees. The proof of the theorem involves first establishing three lemmas and then proving it by contradiction. Neutral Geometry gives theorems that are common to both Euclidean and Hyperbolic geometries.

Modern geometry

Hyperbolic geometry was developed in the 19th century as a non-Euclidean geometry that discards one of Euclid's parallel postulate. It assumes that through a point not on a given line there are multiple parallel lines. This led to discoveries like triangles having interior angles summing to less than 180 degrees. Key figures who developed it include Gauss, Bolyai, Lobachevsky, and models include the Klein model, Poincaré disk model, and hyperboloid model.

Function or not function

The document discusses how to classify relations and functions based on their sets of ordered pairs and graphs. A relation is classified as a function if each element of its domain corresponds to exactly one element of its range. A graph is a function if it passes the vertical line test, where no vertical line intersects the graph in more than one point. The document also defines domain as the set of possible input values and range as the set of possible output values for a relation or function. It provides examples of different domains and ranges.

History Of Non Euclidean Geometry

This document provides a history of non-Euclidean geometry, beginning with Euclid's fifth postulate and early attempts to prove it from the other four postulates. In the early 19th century, Bolyai, Lobachevsky, and Gauss independently developed hyperbolic geometry by replacing the fifth postulate. However, their work was initially rejected by the mathematical community. Later, Riemann generalized the concept of geometry and Beltrami provided a model showing the consistency of non-Euclidean geometry. Klein classified the three types of geometry as hyperbolic, elliptic and Euclidean. Non-Euclidean geometry has since found applications in Einstein's theory of relativity and GPS systems.

1.3.1 Inductive and Deductive Reasoning

This document discusses inductive and deductive reasoning. It provides examples of using inductive reasoning to identify patterns, make conjectures, and find counterexamples. It also contrasts inductive and deductive reasoning, providing examples of each. Inductive reasoning involves drawing conclusions from specific observations, while deductive reasoning uses known facts or rules to draw conclusions. The document is intended to help students understand and apply different types of logical reasoning.

Sets of Axioms and Finite Geometries

The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.

5As Method of Lesson Plan on Ssolving systems of linear equations in two vari...

This lesson plan utilized 5As Method with a subject matter;
Solving Systems of Linear Equations in two Variables using Substitution Method.

Axiom of choice presentation

The Axiom of Choice allows the selection of one element from each set in an infinite collection of non-empty sets. It is an axiom of set theory that is independent from the other axioms and its validity is controversial. The Axiom of Choice enables proofs of counterintuitive results like the Banach-Tarski Paradox, where a single ball can be decomposed and reassembled into two identical balls. While powerful, accepting or rejecting the Axiom of Choice can have implications in mathematics.

Neutral Geometry_part2.pptx

The document discusses Neutral Geometry, which is derived from Euclid's first four postulates without using the parallel postulate. It presents the Saccheri-Legendre theorem, which states that the sum of the interior angles of any triangle is less than or equal to 180 degrees. The proof of the theorem involves first establishing three lemmas and then proving it by contradiction. Neutral Geometry gives theorems that are common to both Euclidean and Hyperbolic geometries.

Modern geometry

Hyperbolic geometry was developed in the 19th century as a non-Euclidean geometry that discards one of Euclid's parallel postulate. It assumes that through a point not on a given line there are multiple parallel lines. This led to discoveries like triangles having interior angles summing to less than 180 degrees. Key figures who developed it include Gauss, Bolyai, Lobachevsky, and models include the Klein model, Poincaré disk model, and hyperboloid model.

Function or not function

The document discusses how to classify relations and functions based on their sets of ordered pairs and graphs. A relation is classified as a function if each element of its domain corresponds to exactly one element of its range. A graph is a function if it passes the vertical line test, where no vertical line intersects the graph in more than one point. The document also defines domain as the set of possible input values and range as the set of possible output values for a relation or function. It provides examples of different domains and ranges.

History Of Non Euclidean Geometry

This document provides a history of non-Euclidean geometry, beginning with Euclid's fifth postulate and early attempts to prove it from the other four postulates. In the early 19th century, Bolyai, Lobachevsky, and Gauss independently developed hyperbolic geometry by replacing the fifth postulate. However, their work was initially rejected by the mathematical community. Later, Riemann generalized the concept of geometry and Beltrami provided a model showing the consistency of non-Euclidean geometry. Klein classified the three types of geometry as hyperbolic, elliptic and Euclidean. Non-Euclidean geometry has since found applications in Einstein's theory of relativity and GPS systems.

1.3.1 Inductive and Deductive Reasoning

This document discusses inductive and deductive reasoning. It provides examples of using inductive reasoning to identify patterns, make conjectures, and find counterexamples. It also contrasts inductive and deductive reasoning, providing examples of each. Inductive reasoning involves drawing conclusions from specific observations, while deductive reasoning uses known facts or rules to draw conclusions. The document is intended to help students understand and apply different types of logical reasoning.

Sets of Axioms and Finite Geometries

The document discusses sets of axioms and finite geometries. It begins by defining geometry as "earth measure" from its Greek roots and discusses some early examples of geometry like Eratosthenes' measurement of the circumference of the Earth. It then discusses the development of geometry through the Greeks and Euclid, including undefined terms, axioms, postulates, and theorems. It provides examples of Euclid's axioms and postulates and discusses modern refinements. Finally, it introduces finite geometries, which have a finite number of elements, and provides an example of a three-point geometry with its axioms and a proof of one of its theorems.

5As Method of Lesson Plan on Ssolving systems of linear equations in two vari...

This lesson plan utilized 5As Method with a subject matter;
Solving Systems of Linear Equations in two Variables using Substitution Method.

Introduction to Invariance Principle

Slides accompanying a 1-hr introduction to invariance principle. Uploaded for club members to access. Problem credits on last slide.
Freeman Cheng, Idris Tarwala.

Axiom of Choice

This document discusses the axiom of choice in set theory. It provides definitions of key terms like well-ordering, partial ordering, and Zorn's lemma. It also covers some equivalents and consequences of the axiom of choice, including the well-ordering principle and Banach-Tarski paradox. The axiom of choice allows choosing one element from each nonempty set in a collection of disjoint sets and guarantees the existence of a choice set.

Factoring Non-Perfect Square Trinomial Lesson Plan

This document contains a lesson plan for teaching factoring non-perfect trinomials in Math 8. The lesson plan outlines intended learning outcomes, learning content including subject matter and reference materials, learning experiences through various activities, an evaluation, and assignment. Students will learn to define trinomials, factor non-perfect square trinomials, and apply factoring trinomials to geometric figures through guided practice with algebra tiles and examples.

7-Experiment, Outcome and Sample Space.pptx

1) 1.56 and 1.0 cannot be probabilities because probabilities must be between 0 and 1.
2) 0.46, 0.09, 0.96, 0.25, 0.02 can be probabilities because they are between 0 and 1.
3) a) The probability of obtaining a number less than 4 is 3/6 = 1/2
b) The probability of obtaining a number between 3 and 6 is 4/6 = 2/3

Number Theory - Lesson 1 - Introduction to Number Theory

This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes

Math10 q2 mod3of8_theorems on chords, arcs, central angles and inscribed angl...

The document provides information on a mathematics module for 10th grade covering theorems related to chords, arcs, central angles, and inscribed angles. It includes the development team who created the module, learning objectives, and introduces key concepts and sample proofs involving chords, arcs, central angles, and inscribed angles using two-column proofs.

Perfect numbers and mersenne primes

The document discusses perfect numbers, abundant numbers, deficient numbers and their definitions. It then covers Mersenne primes and notes that the French monk Marin Mersenne stated some numbers of the form 2n - 1 are prime. The document shows there is a relationship between Mersenne primes and perfect numbers - if 2n - 1 is a Mersenne prime, then 2n-1 x 2n-1 will be a perfect number. This relationship is demonstrated for the first few Mersenne primes. The document encourages further research on Mersenne primes and perfect numbers.

Les5e ppt 03

The events are independent because the probability of tossing a head does not change based on the outcome of the roll, and vice versa. The probability of each event is unaffected by the other.
Solution:
Independent (the occurrence of A does not affect the probability of the occurrence of B)
© 2012 Pearson Education, Inc. All rights reserved. 42 of 88
The Multiplication Rule
Multiplication Rule
• For independent events A and B:
P(A and B) = P(A) × P(B)
• For dependent events A and B:
P(A and B) = P(A) × P(B|A)
© 2012 Pearson Education,

Polynomial Function and Synthetic Division

This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions.

Mathematics 9 Quadratic Functions (Module 1)

This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.

Axiom of Choice (2).pptx

This document discusses sets and axioms of set theory. It provides examples of sets such as a bag of potato chips or a university. It then discusses the axioms of set theory, including the axiom of extension, null set, pairing, union, power set, separation, and infinity. It provides definitions of well-ordered sets and discusses Zorn's lemma and its proof. The document also discusses concepts like maximal elements, linear orderings, and the axiom of choice.

Geometric Sequence

This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.

Math investigation (bounces)

This document summarizes an investigation into the path and number of bounces of a ball on a rectangular pool table represented by a dot grid. The investigation explored relationships between the number of dots in columns and rows and the resulting number of bounces and corner where the ball drops off. Several conjectures were proposed and tested, with some found to be true and others false. Key findings included formulas for determining the number of bounces based on the column and row sizes.

Geometric sequences and geometric means

This document discusses geometric sequences and geometric means. It defines a geometric sequence as a sequence where each term after the first is the product of the preceding term and a fixed number called the common ratio. It provides the formula for calculating the nth term and the sum of the first n terms of a geometric sequence. The terms between the first and last term of a geometric sequence are called the geometric means. It includes sample problems demonstrating how to find specific terms, the common ratio, the first term, geometric means, and the sum of terms for various geometric sequences.

Parabola

The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.

Probability (gr.11)

Grade 11 probability on Dependent and Independ events, Venn Diagrams (up to 3 events), and Tree diagrams

Geometric series

This document defines geometric series and provides formulas to calculate the sum of finite and infinite geometric series. It also provides examples of problems involving geometric series, such as calculating sums, determining convergence, and applying geometric series to real-world scenarios like compound interest, population growth, and bouncing balls.

Modern Geometry Topics

This powerpoint includes:
Triangles and Quadrangles
Definition, Types, Properties, Secondary part, Congruency and Area
Definitions of Triangles and Quadrangles
Desarguesian Plane
Mathematician Desargues and His Background
Harmonic Sequence of Points/Lines
Illustrations and Animated Lines.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, was developed by the French mathematician René Descartes. It uses two perpendicular number lines, the x-axis and y-axis, that intersect at the origin (0,0) to locate points in a plane. Each point is identified with an ordered pair of numbers known as Cartesian coordinates that represent the distance from the origin on the x-axis and y-axis. The system divides the plane into four quadrants and allows points to be easily plotted and located.

writing linear equation

This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
It begins with examples of rewriting equations from standard form to slope-intercept form and identifying the slope (m) and y-intercept (b). Then it provides examples of rewriting from slope-intercept form to standard form. Finally, it provides a series of practice problems for rewriting linear equations between the two forms.

The Mathematical Universe in a Nutshell

It gives me great comfort to visualize this universe as the surface of an ever expanding four-dimensional sphere originating from a distant, but finite, past and growing indefinitely for ever. In this idealized model it easy to calculate the age of the universe by observing the velocity of the receding stars and also to make several other interesting conclusions. For more details, continue reading the presentation.

mechanizing reasoning

Humans have been attempting to mechanize reasoning for thousands of years through formal axiomatic systems like those developed by Aristotle and Euclid. Euclid's system of geometry based on five axioms and five inference rules was able to prove hundreds of propositions, but was incomplete as it could not derive all true statements. Later systems by Frege, Russell, and Whitehead attempted to develop complete and consistent axiomatic systems but were hindered by paradoxes like Russell's paradox, which showed that certain self-referential statements could not be consistently defined.

Introduction to Invariance Principle

Slides accompanying a 1-hr introduction to invariance principle. Uploaded for club members to access. Problem credits on last slide.
Freeman Cheng, Idris Tarwala.

Axiom of Choice

This document discusses the axiom of choice in set theory. It provides definitions of key terms like well-ordering, partial ordering, and Zorn's lemma. It also covers some equivalents and consequences of the axiom of choice, including the well-ordering principle and Banach-Tarski paradox. The axiom of choice allows choosing one element from each nonempty set in a collection of disjoint sets and guarantees the existence of a choice set.

Factoring Non-Perfect Square Trinomial Lesson Plan

This document contains a lesson plan for teaching factoring non-perfect trinomials in Math 8. The lesson plan outlines intended learning outcomes, learning content including subject matter and reference materials, learning experiences through various activities, an evaluation, and assignment. Students will learn to define trinomials, factor non-perfect square trinomials, and apply factoring trinomials to geometric figures through guided practice with algebra tiles and examples.

7-Experiment, Outcome and Sample Space.pptx

1) 1.56 and 1.0 cannot be probabilities because probabilities must be between 0 and 1.
2) 0.46, 0.09, 0.96, 0.25, 0.02 can be probabilities because they are between 0 and 1.
3) a) The probability of obtaining a number less than 4 is 3/6 = 1/2
b) The probability of obtaining a number between 3 and 6 is 4/6 = 2/3

Number Theory - Lesson 1 - Introduction to Number Theory

This document provides an introduction to number theory, including:
- Number theory is the study of integers and their properties
- It discusses the origins and early developments of number theory in places like Mesopotamia, India, Greece, and Alexandria
- It defines different types of numbers like natural numbers, integers, rational numbers, irrational numbers, and describes properties like prime and composite numbers
- It discusses applications of number theory like public key cryptography and error-correcting codes

Math10 q2 mod3of8_theorems on chords, arcs, central angles and inscribed angl...

The document provides information on a mathematics module for 10th grade covering theorems related to chords, arcs, central angles, and inscribed angles. It includes the development team who created the module, learning objectives, and introduces key concepts and sample proofs involving chords, arcs, central angles, and inscribed angles using two-column proofs.

Perfect numbers and mersenne primes

The document discusses perfect numbers, abundant numbers, deficient numbers and their definitions. It then covers Mersenne primes and notes that the French monk Marin Mersenne stated some numbers of the form 2n - 1 are prime. The document shows there is a relationship between Mersenne primes and perfect numbers - if 2n - 1 is a Mersenne prime, then 2n-1 x 2n-1 will be a perfect number. This relationship is demonstrated for the first few Mersenne primes. The document encourages further research on Mersenne primes and perfect numbers.

Les5e ppt 03

The events are independent because the probability of tossing a head does not change based on the outcome of the roll, and vice versa. The probability of each event is unaffected by the other.
Solution:
Independent (the occurrence of A does not affect the probability of the occurrence of B)
© 2012 Pearson Education, Inc. All rights reserved. 42 of 88
The Multiplication Rule
Multiplication Rule
• For independent events A and B:
P(A and B) = P(A) × P(B)
• For dependent events A and B:
P(A and B) = P(A) × P(B|A)
© 2012 Pearson Education,

Polynomial Function and Synthetic Division

This presentation explains the basic information about Polynomial Function and Synthetic Division. Examples were given about easy ways to divide polynomial function using synthetic division. It also contains the steps on how to perform the division method of polynomial functions.

Mathematics 9 Quadratic Functions (Module 1)

This learner's module discusses or talks about the topic of Quadratic Functions. It also discusses what is Quadratic Functions. It also shows how to transform or rewrite the equation f(x)=ax2 + bx + c to f(x)= a(x-h)2 + k. It will also show the different characteristics of Quadratic Functions.

Axiom of Choice (2).pptx

This document discusses sets and axioms of set theory. It provides examples of sets such as a bag of potato chips or a university. It then discusses the axioms of set theory, including the axiom of extension, null set, pairing, union, power set, separation, and infinity. It provides definitions of well-ordered sets and discusses Zorn's lemma and its proof. The document also discusses concepts like maximal elements, linear orderings, and the axiom of choice.

Geometric Sequence

This document discusses geometric sequences and provides examples. It defines a geometric sequence as a sequence where each term is found by multiplying or dividing the same value from one term to the next. An example given is 2, 4, 8, 16, 32, 64, 128, etc. The document also provides the general formula for a geometric sequence as {a, ar, ar2, ar3, ...} where a is the first term and r is the common ratio between terms. It gives practice problems for finding missing terms and the common ratio of geometric sequences.

Math investigation (bounces)

This document summarizes an investigation into the path and number of bounces of a ball on a rectangular pool table represented by a dot grid. The investigation explored relationships between the number of dots in columns and rows and the resulting number of bounces and corner where the ball drops off. Several conjectures were proposed and tested, with some found to be true and others false. Key findings included formulas for determining the number of bounces based on the column and row sizes.

Geometric sequences and geometric means

This document discusses geometric sequences and geometric means. It defines a geometric sequence as a sequence where each term after the first is the product of the preceding term and a fixed number called the common ratio. It provides the formula for calculating the nth term and the sum of the first n terms of a geometric sequence. The terms between the first and last term of a geometric sequence are called the geometric means. It includes sample problems demonstrating how to find specific terms, the common ratio, the first term, geometric means, and the sum of terms for various geometric sequences.

Parabola

The document discusses parabolas and their key properties:
- A parabola is the set of all points equidistant from a fixed line called the directrix and a fixed point called the focus.
- The standard equation of a parabola depends on the orientation of its axis and vertex.
- Key properties include the axis of symmetry, direction of opening, and the length of the latus rectum.

Probability (gr.11)

Grade 11 probability on Dependent and Independ events, Venn Diagrams (up to 3 events), and Tree diagrams

Geometric series

This document defines geometric series and provides formulas to calculate the sum of finite and infinite geometric series. It also provides examples of problems involving geometric series, such as calculating sums, determining convergence, and applying geometric series to real-world scenarios like compound interest, population growth, and bouncing balls.

Modern Geometry Topics

This powerpoint includes:
Triangles and Quadrangles
Definition, Types, Properties, Secondary part, Congruency and Area
Definitions of Triangles and Quadrangles
Desarguesian Plane
Mathematician Desargues and His Background
Harmonic Sequence of Points/Lines
Illustrations and Animated Lines.

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian coordinate system, was developed by the French mathematician René Descartes. It uses two perpendicular number lines, the x-axis and y-axis, that intersect at the origin (0,0) to locate points in a plane. Each point is identified with an ordered pair of numbers known as Cartesian coordinates that represent the distance from the origin on the x-axis and y-axis. The system divides the plane into four quadrants and allows points to be easily plotted and located.

writing linear equation

This document provides examples for rewriting linear equations between the slope-intercept form (y=mx+b) and standard form (Ax + By = C).
It begins with examples of rewriting equations from standard form to slope-intercept form and identifying the slope (m) and y-intercept (b). Then it provides examples of rewriting from slope-intercept form to standard form. Finally, it provides a series of practice problems for rewriting linear equations between the two forms.

Introduction to Invariance Principle

Introduction to Invariance Principle

Axiom of Choice

Axiom of Choice

Factoring Non-Perfect Square Trinomial Lesson Plan

Factoring Non-Perfect Square Trinomial Lesson Plan

7-Experiment, Outcome and Sample Space.pptx

7-Experiment, Outcome and Sample Space.pptx

Number Theory - Lesson 1 - Introduction to Number Theory

Number Theory - Lesson 1 - Introduction to Number Theory

Math10 q2 mod3of8_theorems on chords, arcs, central angles and inscribed angl...

Math10 q2 mod3of8_theorems on chords, arcs, central angles and inscribed angl...

Perfect numbers and mersenne primes

Perfect numbers and mersenne primes

Les5e ppt 03

Les5e ppt 03

Polynomial Function and Synthetic Division

Polynomial Function and Synthetic Division

Mathematics 9 Quadratic Functions (Module 1)

Mathematics 9 Quadratic Functions (Module 1)

Axiom of Choice (2).pptx

Axiom of Choice (2).pptx

Geometric Sequence

Geometric Sequence

Math investigation (bounces)

Math investigation (bounces)

Geometric sequences and geometric means

Geometric sequences and geometric means

Parabola

Parabola

Probability (gr.11)

Probability (gr.11)

Geometric series

Geometric series

Modern Geometry Topics

Modern Geometry Topics

Rectangular Coordinate System

Rectangular Coordinate System

writing linear equation

writing linear equation

The Mathematical Universe in a Nutshell

It gives me great comfort to visualize this universe as the surface of an ever expanding four-dimensional sphere originating from a distant, but finite, past and growing indefinitely for ever. In this idealized model it easy to calculate the age of the universe by observing the velocity of the receding stars and also to make several other interesting conclusions. For more details, continue reading the presentation.

mechanizing reasoning

Humans have been attempting to mechanize reasoning for thousands of years through formal axiomatic systems like those developed by Aristotle and Euclid. Euclid's system of geometry based on five axioms and five inference rules was able to prove hundreds of propositions, but was incomplete as it could not derive all true statements. Later systems by Frege, Russell, and Whitehead attempted to develop complete and consistent axiomatic systems but were hindered by paradoxes like Russell's paradox, which showed that certain self-referential statements could not be consistently defined.

Presentation X-SHS - 27 oct 2015 - Topologie et perception

This document discusses the relationship between mathematics, perception, and cognition from multiple perspectives. It covers topics like:
- Pythagoras' and Fourier's views that mathematics compensates for the imperfection of the senses.
- Aristotle, Poincare, and others' ideas that experience and perception have a mathematical basis in principles like non-contradiction.
- Models of associative memory and complex energy landscapes from fields like neural networks and statistical mechanics.
- Geometry, from Euclid to Einstein's general relativity, and its relationship to perception through ideas like projective geometry and invariance under transformation groups.
- Information theory and its connections to measure theory and set theory through concepts like

What is an axiom?

This document discusses axioms and their role in formal mathematical systems. It uses Euclid's geometry as an example, outlining his five axioms including the first axiom that there is exactly one straight line between any two points. Definitions are distinguished from axioms. Contradictions to axioms may indicate either a logical error or that a different mathematical system is being described. The document concludes by promising to discuss Euclid's fifth postulate in more depth.

Zeno paradoxes

Zeno of Elea developed several paradoxes to argue against the concept of motion and plurality. The paradoxes include:
1) Achilles and the tortoise paradox which argues that Achilles can never overtake the tortoise due to the infinite divisibility of space and time.
2) The dichotomy paradox which argues that before reaching any distance, one must first reach the halfway point, and before that the quarter way point, and so on ad infinitum.
3) The arrow paradox which argues that at an instant of time, an arrow must be either moving or at rest, but cannot be both, so motion is impossible.
The plurality paradoxes argue that a group of objects cannot

paper publication

https://utilitasmathematica.com/index.php/Index
Our Journal has steadfast in its commitment to promoting justice, equity, diversity, and inclusion within the realm of statistics. Through collaborative efforts and a collective dedication to these principles, we believe in building a statistical community that not only advances the profession. Paper publication

Matricų kombinatorikos taikymas automatų rūšiavimui ir NP ir P automatų klasi...

Matricų kombinatorikos taikymas automatų rūšiavimui ir NP ir P automatų klasi...Lietuvos kompiuterininkų sąjunga

Dr. Andrius Kulikauskas proposes applying matrix combinatorics to classify and study automata based on their computational capabilities. He suggests a universal framework using matrix operations that can generate all types of automata, from finite state automata to Turing machines. This framework uses operations like matrix multiplication and cycle removal to model the different automata in a way that only increases complexity. Studying this may help determine if certain computationally difficult problems can be solved efficiently by nondeterministic machines through guessing, or if they require exponential resources deterministically.Category Theory made easy with (ugly) pictures

Introduction to very basic Category Theory blending rigorous definitions with pictorial intuition and Haskell code

Euclids postulates

Euclid was a Greek mathematician from Alexandria known as the "Father of Geometry". In his influential work Elements, he deduced the principles of Euclidean geometry from 5 postulates (axioms) for plane geometry related to drawing lines and circles. The postulates state that a line can be drawn between any two points, a line can be extended indefinitely, a circle can be drawn with any center and radius, all right angles are equal, and if two lines intersect another such that the interior angles on the same side sum to less than two right angles, the two lines will intersect on that side. Euclid's work was foundational and served as the main geometry textbook for over 2000 years.

Standrewstalk

This document discusses different views on doing mathematics, including studying things at a maximal level of generality (Grothendieck view) or focusing on examples (Thurston view). It then proposes combining these views by studying collections of events and phenomena (Pascal's view). The rest of the document explores this idea of studying "generic" or typical properties that emerge when considering random or generic examples of mathematical objects like matrices, groups, graphs, and manifolds. Several specific results are mentioned, but many open questions remain about computing properties of random higher-dimensional or more complex objects.

Basics of set theory

This document provides an introduction to set theory. It discusses that sets provide a useful vocabulary in mathematics and were originally studied by Georg Cantor in the late 19th century. Most mathematicians accept set theory as a foundation for mathematics, where all mathematical objects can be defined as sets. The document then discusses different ways to define sets, including listing elements, using properties to describe elements, and examples of common sets like real numbers and integers. It notes some key concepts like subsets, empty sets, and power sets. Finally, it discusses paradoxes that arise from naive set theory, such as Russell's paradox.

Fractals, Geometry of Nature and Logistic Model

Fractals are geometric shapes that exhibit self-similarity and complex patterns at every scale. Koch's snowflake is a famous fractal where the perimeter tends towards infinity as more iterations are done, even as the area approaches a limit. The Mandelbrot set is another well-known fractal that maps the behavior of values of c under a complex iterative function, resulting in diverse patterns when zooming in. Fractals are found throughout nature in shapes like clouds, coastlines, and Romanesco broccoli. They can also be generated through computer programs and used in applications like diagnosing skin cancer.

lect14-semantics.ppt

This document discusses logical representations of natural language. It introduces lambda calculus as a way to represent functions without names. Common logical connectives like AND, OR, and quantifiers like FOR ALL and THERE EXISTS are explained. The document also discusses using constants to represent objects and predicates in a logical language. Finally, it provides an example of how to logically represent the sentence "Gilly swallowed a goldfish" using a quantifier since "goldfish" refers to an unspecified entity.

True but Unprovable

Logicians sometimes talk about sentences being “true but unprovable." What does this mean? This presentation includes a fairly thorough introduction to mathematical logic.

Prolog 01

This document provides an introduction to Prolog programming through examples of syllogisms and modeling logic problems. It discusses SWI-Prolog, facts and rules, queries, variables, structures, predicates, programs, and modeling change. Key concepts are summarized such as backward chaining, cuts, dynamic facts, and modeling "real life" problems through games and adventure programs. The document uses a dragon adventure game to demonstrate Prolog programming concepts in action.

Fractals

Fractals are irregular patterns that are self-similar across different scales. They are a branch of mathematics concerned with shapes found in nature that have non-integer dimensions. Some early contributors to fractal concepts included Leibiz, Cantor, and Hausdorff, but it was Mandelbrot who coined the term "fractal" and brought greater attention to the field. One of the most basic and important fractals is the Mandelbrot set, which is a set of complex numbers that fulfill certain recursive conditions. Fractals can also be classified based on their level of self-similarity, from exact to statistical self-similarity. Fractals are found throughout nature and have applications in fields like astrophysics

Albert Einstein (2) Relativity Special And General Theory

This document provides instructions for classifying ebooks based on their file format and subject matter. It specifies that:
1) Ebooks should be in Adobe PDF or Tomeraider format, with txt files not considered ebooks.
2) The file name should include the classification in parenthesis - (ebook - File Format - Subject Matter).
3) The subject matter classification should be one of: Biography, Children, Fiction, Food, Games, Government, Health, Internet, Martial-Arts, Mathematics, Other, Programming, Reference, Religious, Science, Sci-Fi, Sex, or Software.
This standardization of ebook file names helps groups like Fink Crew

The quantum strategy of completeness

The Gödel incompleteness can be modeled on the alleged incompleteness of quantum mechanics
Then the proved and even experimentally confirmed completeness of quantum mechanics can be reversely interpreted as a strategy of completeness as to the foundation of mathematics
Infinity is equivalent to a second and independent finiteness
Two independent Peano arithmetics as well as one single Hilbert space as an unification of geometry and arithmetic are sufficient to the self-foundation of mathematics
Quantum mechanics is inseparable from the foundation of mathematics and thus from set theory particularly

Second_Update_to my 2013 Paper

This document discusses exploring different mathematical structures as potential candidates for the geometry of the physical universe. It begins by summarizing a previous paper exploring finite geometry approaches. The document then outlines a proposed method for attempting to locate the simplest mathematical structure that could describe the universe, and iteratively making small modifications to better match observations. As a starting point, it examines some "seven day creation stories", each beginning with a simple structure like sets and applying operations each day to generate more complex structures, with the goal of reaching a structure complex enough to shed light on theories like finite M-theory. It aims to provide visual feedback on candidate geometries using computer graphics to help guide the search.

Dialectica and Kolmogorov Problems

The document discusses finding the right abstractions for reasoning problems. It describes Andreas Blass' insight about a category called PV that models problems and reductions between them. PV objects are binary relations representing problems, with morphisms describing reductions. The talk discusses using this framework and Dialectica categories to model Kolmogorov's theory of problems from 1932 and Veloso's theory. It provides examples of modeling geometry and tangent plane problems as Kolmogorov problems and reductions between them.

The Mathematical Universe in a Nutshell

The Mathematical Universe in a Nutshell

mechanizing reasoning

mechanizing reasoning

Presentation X-SHS - 27 oct 2015 - Topologie et perception

Presentation X-SHS - 27 oct 2015 - Topologie et perception

What is an axiom?

What is an axiom?

Zeno paradoxes

Zeno paradoxes

paper publication

paper publication

Matricų kombinatorikos taikymas automatų rūšiavimui ir NP ir P automatų klasi...

Matricų kombinatorikos taikymas automatų rūšiavimui ir NP ir P automatų klasi...

Category Theory made easy with (ugly) pictures

Category Theory made easy with (ugly) pictures

Euclids postulates

Euclids postulates

Standrewstalk

Standrewstalk

Basics of set theory

Basics of set theory

Fractals, Geometry of Nature and Logistic Model

Fractals, Geometry of Nature and Logistic Model

lect14-semantics.ppt

lect14-semantics.ppt

True but Unprovable

True but Unprovable

Prolog 01

Prolog 01

Fractals

Fractals

Albert Einstein (2) Relativity Special And General Theory

Albert Einstein (2) Relativity Special And General Theory

The quantum strategy of completeness

The quantum strategy of completeness

Second_Update_to my 2013 Paper

Second_Update_to my 2013 Paper

Dialectica and Kolmogorov Problems

Dialectica and Kolmogorov Problems

The importance of being human 3

What makes us humans different from animals? Culture? The ability to make tools? The language? Morality? Art? This presentation will show us that these criteria alone are not enough to explain what makes us different from animals.

Can christian schools continue to teach only about traditional marriage

An approach to the big general issue whether faith schools should be allowed to teach based on their fundamental beliefs and discriminate amongst teachers and students because of their faith.

The unlimited desire IIS2019

1. The document discusses three potential traits that define humanity: understanding and language, freedom and self-responsibility, and unlimited desire.
2. It argues that unlimited desire, which drives humans to constantly search for meaning, happiness, and fulfillment through questions, choices, love and communion, may be the most fundamental human trait.
3. This trait of unlimited desire can justify fundamental personal rights like the right to life, freedom, and due process, which in turn other rights are derived from, in order to ensure humans can fulfill their desires. However, recognizing another's humanity ultimately requires choice and acknowledgement of their desires too.

Causality from outside Time

Causality from outside Time
Alfred Driessen
Talk presented at the 21th International Interdisciplinary Seminar, Science and Society: Defining what is human
Netherhall House, London, 5-1-2019
Content
Introduction
Time in Relativity
Time in Quantum Mechanics
Conclusions
Conclusions from this study:
There are causes beyond the realm of science,
- they are not observable by physical or scientific means
- the effects of these causes, however, are observable by physical and scientific means.
Physics is not complete.

Can computers replace teachers?

The presentation has two parts. In the first one, we review a series of studies that compare the efficacy of learning with "digital teachers" as opposed to learning with normal teachers. In the second part, we make several considerations from different points of view that may be helpful to answer the question.

Future contingents and the Multiverse

1) The document discusses future contingencies and the multiverse perspective from Antoine Suarez combining two philosophical views.
2) It describes Ernst Specker's work on quantum contextuality and includes photos of Specker with his son and colleague Simon Kochen.
3) Suarez identifies Hugh Everett's many worlds theory with thoughts in the "mind of God", finding a way to incorporate a deity into the picture according to a quote in Nature.

The CRISPR/CAS9 genome editing system and humans

This is a brief introduction to the CRISPR/Cas9 genome editing technique and a quick review of two articles that have to do with potential applications in humans. There is a draft for an ethical reflexion.

Achilles, the Tortoise and Quantum Mechanics

Achilles, the Tortoise and Quantum Mechanics
Alfred Driessen
prof. emer. University of Twente
In several places of his Physica Aristotle analyzes the famous antimony of Zeno about the competition between Achilles and the Tortoise. He emphasizes that any movement, or more general any change, is actually a continuum, i.e. an unity. It depends on the specific movement or change whether this continuum is potentially divisible in parts. In fact, there could be certain minima of the division. In line with this approach, Quantum Mechanics states that there are minima or quanta of movement (or change), with other words, there are no gradual changes in the world of micro- and nano-structures. This behavior is completely unexpected when starting with the mechanistic approach of classical physics.
Taking another finding of Aristotle, the four aspects of causality including final cause, one gets another ingredient of Quantum Mechanics. Movements and changes are not only influenced by the initial state -describing the present situation- but also by the final state which takes account of the future situation. As an example one may mention Fermi’s golden rule, where the initial and final state symmetrically determine the transition probability.
Bringing these two philosophical concepts of Aristotle together namely quanta of movement and final cause, a new light is shed on fundamental issues in Quantum Mechanics. One may mention the experimental evidence for contextuality, which is considered one of the weird phenomena in Quantum Mechanics. As illustration, some of the examples of experiments with optical microresonators are given.
This talk has been presented at the 20th International Interdisciplinary Seminar "Can Science and Technology Shape a New Humanity", Netherhall House, London, 5-1-2018

Transpostgenderism 5 jean_davidponci

This document discusses transhumanism and postgenderism. It summarizes views from gender studies that gender is more a social construct than biological essentialism. It explores how technologies may enable a postgender society without the gender binary, including artificial wombs, drugs controlling sexual behaviors and bonding, genetic engineering to change sex, and robots for sexual needs. However, the document concludes that pursuing postgenderism through technologies may dehumanize humanity and lose the richness of gender complementarity.

Transhumanism and brain

This presentation lists some brain-computer interface technologies that exist today and that could be attainable in future. At the end, philosophical comments about this kind of technology and transhumanism are purposed, in order to reveal the key difference between a humain brain and artificial intelligence.

Transhumanism and brain

The document discusses transhumanism and compares human and artificial intelligence. It covers current technologies like mini antennas and potential future technologies like telepathy and neuroreality. It then compares how humans and artificial systems learn, with humans able to reach an understanding of essence while artificial intelligence relies on computational power. The document concludes that hard transhumanism, with robots becoming the dominant lifeform, clashes with philosophical views, while soft transhumanism involves more integration between humans and robots but maintains humans as the primary agents.

Netherhall 2018 m fox

The document discusses experimental attempts to realize quantum computers. It begins with an introduction to quantum technologies and quantum bits or qubits. It then describes the current state of the art in quantum computing technologies, including ion traps, superconducting circuits, and linear optic quantum computing. The document provides examples of single qubit gates and two-qubit gates needed to build a universal quantum computer. It also summarizes different physical systems used to implement qubits and discusses challenges in scaling up quantum computers.

Sequential processing in nature

1. The document discusses the differences between sequential processing in nature compared to scientific computation. Protein folding occurs sequentially and efficiently in nature, while computation relies on approximation and linearization.
2. Artificial neural networks provide some similarities to natural processes by using weighted connections between nodes, but training requires vast resources compared to natural systems.
3. Quantum computing may provide solutions in a way analogous to how nature appears to "know" the right protein folding solution, but this ability is not well understood.

Infinite Chess: winning and draw

A presentation about Infinite Chess and the difference between man and machines. From works by C.D.A. Evans and J.D. Hamkins. Presented during the International Interdisciplinary Seminar of London, January 2018.

The importance of being human 3

The importance of being human 3

Can christian schools continue to teach only about traditional marriage

Can christian schools continue to teach only about traditional marriage

The unlimited desire IIS2019

The unlimited desire IIS2019

Causality from outside Time

Causality from outside Time

Can computers replace teachers?

Can computers replace teachers?

Future contingents and the Multiverse

Future contingents and the Multiverse

The CRISPR/CAS9 genome editing system and humans

The CRISPR/CAS9 genome editing system and humans

Achilles, the Tortoise and Quantum Mechanics

Achilles, the Tortoise and Quantum Mechanics

Transpostgenderism 5 jean_davidponci

Transpostgenderism 5 jean_davidponci

Transhumanism and brain

Transhumanism and brain

Transhumanism and brain

Transhumanism and brain

Netherhall 2018 m fox

Netherhall 2018 m fox

Sequential processing in nature

Sequential processing in nature

Infinite Chess: winning and draw

Infinite Chess: winning and draw

Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...

A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!

AJAY KUMAR NIET GreNo Guava Project File.pdf

AJAY KUMAR NIET GreNo Guava Project PDF File

HUMAN EYE By-R.M Class 10 phy best digital notes.pdf

Class 10 human eye notes physics
Handwritten best quality

11.1 Role of physical biological in deterioration of grains.pdf

Storagedeteriorationisanyformoflossinquantityandqualityofbio-materials.
Themajorcausesofdeteriorationinstorage
•Physical
•Biological
•Mechanical
•Chemical
Storageonlypreservesquality.Itneverimprovesquality.
Itisadvisabletostartstoragewithqualityfoodproduct.Productwithinitialpoorqualityquicklydepreciates

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

Centrifugation is a powerful technique used in laboratories to separate components of a heterogeneous mixture based on their density. This process utilizes centrifugal force to rapidly spin samples, causing denser particles to migrate outward more quickly than lighter ones. As a result, distinct layers form within the sample tube, allowing for easy isolation and purification of target substances.

23PH301 - Optics - Unit 1 - Optical Lenses

Under graduate Physics - Optics

快速办理(UAM毕业证书)马德里自治大学毕业证学位证一模一样

学校原件一模一样【微信：741003700 】《(UAM毕业证书)马德里自治大学毕业证学位证》【微信：741003700 】学位证，留信认证（真实可查，永久存档）原件一模一样纸张工艺/offer、雅思、外壳等材料/诚信可靠,可直接看成品样本，帮您解决无法毕业带来的各种难题！外壳，原版制作，诚信可靠，可直接看成品样本。行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备。十五年致力于帮助留学生解决难题，包您满意。
本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
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四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
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◇回国马上就要找工作，办给用人单位看；
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◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
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8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才

Lattice Defects in ionic solid compound.pptx

lattice of ionic solid

SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆

Context. The early-type galaxy SDSS J133519.91+072807.4 (hereafter SDSS1335+0728), which had exhibited no prior optical variations during the preceding two decades, began showing significant nuclear variability in the Zwicky Transient Facility (ZTF) alert stream from December 2019 (as ZTF19acnskyy). This variability behaviour, coupled with the host-galaxy properties, suggests that SDSS1335+0728 hosts a ∼ 106M⊙ black hole (BH) that is currently in the process of ‘turning on’. Aims. We present a multi-wavelength photometric analysis and spectroscopic follow-up performed with the aim of better understanding the origin of the nuclear variations detected in SDSS1335+0728. Methods. We used archival photometry (from WISE, 2MASS, SDSS, GALEX, eROSITA) and spectroscopic data (from SDSS and LAMOST) to study the state of SDSS1335+0728 prior to December 2019, and new observations from Swift, SOAR/Goodman, VLT/X-shooter, and Keck/LRIS taken after its turn-on to characterise its current state. We analysed the variability of SDSS1335+0728 in the X-ray/UV/optical/mid-infrared range, modelled its spectral energy distribution prior to and after December 2019, and studied the evolution of its UV/optical spectra. Results. From our multi-wavelength photometric analysis, we find that: (a) since 2021, the UV flux (from Swift/UVOT observations) is four times brighter than the flux reported by GALEX in 2004; (b) since June 2022, the mid-infrared flux has risen more than two times, and the W1−W2 WISE colour has become redder; and (c) since February 2024, the source has begun showing X-ray emission. From our spectroscopic follow-up, we see that (i) the narrow emission line ratios are now consistent with a more energetic ionising continuum; (ii) broad emission lines are not detected; and (iii) the [OIII] line increased its flux ∼ 3.6 years after the first ZTF alert, which implies a relatively compact narrow-line-emitting region. Conclusions. We conclude that the variations observed in SDSS1335+0728 could be either explained by a ∼ 106M⊙ AGN that is just turning on or by an exotic tidal disruption event (TDE). If the former is true, SDSS1335+0728 is one of the strongest cases of an AGNobserved in the process of activating. If the latter were found to be the case, it would correspond to the longest and faintest TDE ever observed (or another class of still unknown nuclear transient). Future observations of SDSS1335+0728 are crucial to further understand its behaviour. Key words. galaxies: active– accretion, accretion discs– galaxies: individual: SDSS J133519.91+072807.4

Gadgets for management of stored product pests_Dr.UPR.pdf

Insectsplayamajorroleinthedeteriorationoffoodgrainscausingbothquantitativeandqualitativelosses
Wellprovedthatnogranariescanbefilledwithgrainswithoutinsectsastheharvestedproducecontainegg(or)larvae(or)pupae(or)adultinsectinthembecauseoffieldcarryoverinfestationwhichcannotbeavoidedindevelopingcountrieslikeIndia
Simpletechnologiesfortimelydetectionofinsectsinthestoredproduceandtherebyplantimelycontrolmeasures

Post translation modification by Suyash Garg

overview of PTM helps to the students who wants to clear their basics about it.

Injection: Risks and challenges - Injection of CO2 into geological rock forma...

Presentation by Dr. Florian Krob, Workshop "Risks of CO2 storage - How can they be addressed and reduced?", 22.4.2024

Holsinger, Bruce W. - Music, body and desire in medieval culture [2001].pdf

Music and Medieval History

Methods of grain storage Structures in India.pdf

•Post-harvestlossesaccountforabout10%oftotalfoodgrainsduetounscientificstorage,insects,rodents,micro-organismsetc.,
•Totalfoodgrainproductioninindiais311milliontonnesandstorageis145mt.InIndia,annualstoragelosseshavebeenestimated14mtworthofRs.7,000croreinwhichinsectsaloneaccountfornearlyRs.1,300crores.
•InIndiaoutofthetotalproduction,about30%ismarketablesurplus
•Remaining70%isretainedandstoredbyfarmersforconsumption,seed,feed.Hence,growerneedstoragefacilitytoholdaportionofproducetosellwhenthemarketingpriceisfavourable
•TradersandCo-operativesatmarketcentresneedstoragestructurestoholdgrainswhenthetransportfacilityisinadequate

Signatures of wave erosion in Titan’s coasts

The shorelines of Titan’s hydrocarbon seas trace flooded erosional landforms such as river valleys; however, it isunclear whether coastal erosion has subsequently altered these shorelines. Spacecraft observations and theo-retical models suggest that wind may cause waves to form on Titan’s seas, potentially driving coastal erosion,but the observational evidence of waves is indirect, and the processes affecting shoreline evolution on Titanremain unknown. No widely accepted framework exists for using shoreline morphology to quantitatively dis-cern coastal erosion mechanisms, even on Earth, where the dominant mechanisms are known. We combinelandscape evolution models with measurements of shoreline shape on Earth to characterize how differentcoastal erosion mechanisms affect shoreline morphology. Applying this framework to Titan, we find that theshorelines of Titan’s seas are most consistent with flooded landscapes that subsequently have been eroded bywaves, rather than a uniform erosional process or no coastal erosion, particularly if wave growth saturates atfetch lengths of tens of kilometers.

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

Ahota Beel, nestled in Sootea Biswanath Assam , is celebrated for its extraordinary diversity of bird species. This wetland sanctuary supports a myriad of avian residents and migrants alike. Visitors can admire the elegant flights of migratory species such as the Northern Pintail and Eurasian Wigeon, alongside resident birds including the Asian Openbill and Pheasant-tailed Jacana. With its tranquil scenery and varied habitats, Ahota Beel offers a perfect haven for birdwatchers to appreciate and study the vibrant birdlife that thrives in this natural refuge.

(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...

(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...Scintica Instrumentation

Targeting Hsp90 and its pathogen Orthologs with Tethered Inhibitors as a Diagnostic and Therapeutic Strategy for cancer and infectious diseases with Dr. Timothy Haystead.Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

CSIR UGC NET CONTENT

Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...

Sexuality - Issues, Attitude and Behaviour - Applied Social Psychology - Psyc...

AJAY KUMAR NIET GreNo Guava Project File.pdf

AJAY KUMAR NIET GreNo Guava Project File.pdf

HUMAN EYE By-R.M Class 10 phy best digital notes.pdf

HUMAN EYE By-R.M Class 10 phy best digital notes.pdf

11.1 Role of physical biological in deterioration of grains.pdf

11.1 Role of physical biological in deterioration of grains.pdf

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

TOPIC OF DISCUSSION: CENTRIFUGATION SLIDESHARE.pptx

23PH301 - Optics - Unit 1 - Optical Lenses

23PH301 - Optics - Unit 1 - Optical Lenses

快速办理(UAM毕业证书)马德里自治大学毕业证学位证一模一样

快速办理(UAM毕业证书)马德里自治大学毕业证学位证一模一样

Lattice Defects in ionic solid compound.pptx

Lattice Defects in ionic solid compound.pptx

SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆

SDSS1335+0728: The awakening of a ∼ 106M⊙ black hole⋆

Juaristi, Jon. - El canon espanol. El legado de la cultura española a la civi...

Juaristi, Jon. - El canon espanol. El legado de la cultura española a la civi...

Gadgets for management of stored product pests_Dr.UPR.pdf

Gadgets for management of stored product pests_Dr.UPR.pdf

Post translation modification by Suyash Garg

Post translation modification by Suyash Garg

Injection: Risks and challenges - Injection of CO2 into geological rock forma...

Injection: Risks and challenges - Injection of CO2 into geological rock forma...

Immunotherapy presentation from clinical immunology

Immunotherapy presentation from clinical immunology

Holsinger, Bruce W. - Music, body and desire in medieval culture [2001].pdf

Holsinger, Bruce W. - Music, body and desire in medieval culture [2001].pdf

Methods of grain storage Structures in India.pdf

Methods of grain storage Structures in India.pdf

Signatures of wave erosion in Titan’s coasts

Signatures of wave erosion in Titan’s coasts

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

BIRDS DIVERSITY OF SOOTEA BISWANATH ASSAM.ppt.pptx

(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...

(June 12, 2024) Webinar: Development of PET theranostics targeting the molecu...

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

Flow chart.pdf LIFE SCIENCES CSIR UGC NET CONTENT

- 1. On the Axiom of Choice Flora Dellini Marco Natale Francesco Urso
- 2. Preliminary ›For every set S, a set U is a subset of S if, for every item in U, this item belongs to S too. ›For every set S, we define the set of the parts of S, P(S), as the set of all the possible subsets of S. Example: 𝑆 = 1,2,3 𝑃 𝑆 = {∅, 1 , 2 , 3 , 1,2 , 1,3 , 2,3 , {1,2,3}}
- 3. The Axiom of Choice (AC) › The Axiom of Choice is a statement about the existence of a certain kind of functions. › A choice function is a function which selects an item from a subset of a given set. › AC claims that, for every group of subsets of S, there exists a function of choice which selects a particular item from every given subset. ∀𝑆, ∀𝑈 ⊆ 𝑃 𝑆 ∅, ∃𝑓: 𝑈 → 𝑆 such that ∀𝑋 ∈ 𝑈, 𝑓(𝑋) ∈ 𝑋
- 4. AC: Examples with finite sets When a set is finite, everything is trivial. The existence of f is not disputable: we can actually show and build it! 𝑆 = 1,2,3,4,5 𝑈 = { 1 , 1,2,3 , 2,3 , 3,4,5 } ⊆ 𝑃(𝑆) 𝑓: 𝑈 ⊆ 𝑃(𝑆){∅} → 𝑆 𝑓 1 = 1 ∈ 1 𝑓 1,2,3 = 3 ∈ 1,2,3 𝑓 2,3 = 2 ∈ 2,3 𝑓 3,4,5 = 5 ∈ 3,4,5
- 5. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks › Let’s suppose there is a shop with infinite pair of shoes, we want to built an infinite set containing a shoe for each pair. › A Turing machine (i.e. a personal computer) can choose between right or left shoe because it can distinguish them. For example we can build the set of all right shoes. › Can we do the same with infinite pair of socks?
- 6. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks › A machine can not do it while a man could do. › The reason is: right and left shoes can be distinguished due to this feature. A man can do this, a machine can do it too. › It is not possible to choose right or left socks, because there are no such things as right or left socks! › As a consequence, in principle, a compuer can not build the wanted set.
- 7. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks ›Could a man build a set from a single sock for each pair? ›Actually, it is impossible, because we would die before we can name all the items in the set. ›But is it possible in principle? How can we distinguish between the socks of a pair? ›We do this by choosing “this one”, without any criterion but our free will.
- 8. AC: Examples of Infinite Sets Hilbert’s Shoes and Socks ›As human beings, it is in our everyday experience that we can distinguish between two socks by calling «this one» or «that one». ›Is it admissible that in maths it is also possible to act in such a way? ›The answer is far from trivial!
- 9. The Way of Formalism › The mathematical concept of Set raised up in XIX century by Georg Cantor (1845-1918). › Cantor’s idea of Set was a “collection” of “objects” which satisfy certain properties (e.g. “the set of all odd numbers”, “the set of all the right shoes”). › Paradoxes arise from a “too free” use of the concept of “property”.
- 10. The Way of Formalism › Russell Paradoxes: the set of all the sets which don’t contain themselves. › The problem is in the semantic. › D. Hilbert’s “Formalism” school proposed the reduction of mathematics to a pure “formal game”. In this way, Mathematics would have been stripped of all its “human components”.
- 11. The Way of Formalism › Zermelo and Fraenkel, following Hilbert’s intuition, began to develop the Formal Set Theory (also known as ZF). › In according to the formalistic concept of mathematics, the semantic in ZF is “eliminated” reducing the concept of proprieties to pure syntactic formulas, computable in principle by a machine. › The nature of Sets is so implicitly defined by syntactic formulas.
- 12. The Way of Formalism ›The validity of a formula must be determined through “propositional calculus”, an absolutely formal procedure which can be, theoretically, implemented on an ideal machine. ›E.g. instead of saying “there is the empty set”, we shall write the following formula: ∃𝑦 ∀𝑥 (𝑥 ∉ 𝑦)
- 13. The Axioms of Zermelo-Fraenkel 1. Axioms describing implicitly the concept of Set (Regularity and Extensionality Axioms). 2. Axiom of Existence. The unique axiom of existence in ZF is the Axiom of Infinity, which asserts that there exists an infinite set. 3. Axioms of Individuation, which allow us to “individuate” (i.e. build) new sets starting from ones already known (Axiom of Power Set, Axiom of Union and Axiom of Replacement).
- 14. Is AC compatible with ZF? › Being compatible means that, if we add AC to ZF, we cannot deduce a theorem and its negation. › ZF claims to describe all mathematical universe: “…with regard to ZF it’s hard to conceive of any other model”. P. Cohen. › Because we would like to proceed in maths as we do with socks, that is by choosing items as we want to, the compatibility of AC with ZF is highly desirable.
- 15. ZF does not disprove AC “Inside only ZF, it’s not possible to prove that AC is false” (Gödel, 1938). Main steps of proof: 1. Gödel added another axiom to ZF (“every set is constructible”), obtaining the stronger theory ZFL. 2. Gödel proved that ZFL is consistent. In this stronger theory he proved that every set can be well-ordered, that is a demonstration of AC. So, ZFL → AC.
- 16. ZF does not disprove AC 3. If, by contradiction, AC is false in ZF, it has to be false also in ZFL. But we have just seen that AC is true in ZFL! So AC could not be disproved in ZF. ZF does not disprove AC! This does NOT mean that AC is true in ZF !
- 17. ZF does not imply AC In 1960 Cohen has completed Gödel’s demonstration about independence of AC from ZF. So ZF does not imply AC.
- 18. Independence of AC from ZF As a consequence, we can add or remove AC from ZF as we like. So, its presence is actually a preference of the mathematician who can want a “richer” or “poorer” theory.
- 19. Theorems we lose without AC • Every non empty Vectorial Space has a base – i.e. imagine that, in the classic Euclidean 3D space, you don’t have the 𝑖, 𝑗, and 𝑘 vectors you use to build every other vector. • Every field has an algebraic closure – i.e. imagine that you could not define the complex numbers
- 20. What does it mean? “Simply” that there are sets whose volume is not invariant under translation and rotation. Strange! Isn’t it? This is why some mathematicians worry about the Axiom of Choice Decomposition of a ball into four pieces which, properly rotated and traslated, yield two balls Counterintuitive effects of AC: Banach-Tarski Paradox
- 21. Counterintuitive effects of AC: Zermelo’s Lemma This statement is equivalent to the Axiom of Choice: Every set S can be well ordered As a consequence, we could «well order» ℝ - i.e. defining an order in ℝ such that every subset of ℝ has a minimum. This order relation is strongly counterintuitive, as it implies that sets like (0,1), without a minimum – 0 ∉ (0,1) do not exist.