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# Mathematical Proof and the Principles of Mathematics

This wikibook aims to teach you the elements of pure mathematics in a self-contained, accessible style. The main objective is to introduce the reader to material usually found in an undergraduate course intended for mathematics majors at a university. In fact, many universities offer a course that serves as a transition from calculus to courses which involve more abstract concepts and writing proofs, and this book might serve as a text for such a course. It's also intended for people who are considering mathematics, especially pure mathematics, as a career or serious avocation, and wish to know what to expect as they advance to higher levels. We also intend to introduce the reader to style of proofs and rigor needed to read and write mathematical literature. The material is covered in greater detail and more rigorously than you may be used to. In fact much of the material will already be familiar, though the approach to it may not.
In the experience of most people, mathematics consists mostly of the mechanical application of rules of computation at various levels: arithmetic, solving equations, finding derivatives and integrals. But for a mathematician, mathematics is a process for discovering and establishing truths. It requires an analytical mind, but also a certain amount of creativity and intuition. It can also be, as we hope you'll discover through this book, very rewarding.

## Table of Contents

* Introduction
** The reason this book was written
** Objectives and prerequisites
* A history of mathematical rigor
** Euclid
** After Euclid, the next two thousand years
** The problem of parallels
** Infrastructure
** Four schools of thought
* Preliminaries
** What is mathematics?
** Mathematical Statements
** Mathematical proof
** Proof anatomy
* Logic
** Logical connectives
** Direct proofs for implication
** Indirect proofs
** Proofs with conjunction and disjunction
** Logical equivalence
** Non-classical logic (optional)
** Quantifiers and predicates
** The universal quantifier
** The existential quantifier
** Rules of inference summary
** Axioms and equality
* Sets
** History
** Elements and subsets
** Classes and collectivizing predicates
** Pairs
** Union and intersection
** Classes and foundation
** Power sets
** Natural numbers
** Replacement
** Axioms for sets
** Operations on sets
* Numbers
** Natural numbers
** Cardinality and counting
** Integers
** Rational numbers
** Real numbers
** Complex numbers
** Abstract number systems
* The scope of mathematics
** Algebra
** Analysis
** Geometry
** Number theory
** Combinatorics
Introduction →
# Mathematical Proof/Preface

Sometimes people read mathematical proofs and think they are reading a foreign language. This book describes the language used in a mathematical proof and also the different types of proofs used in math. This knowledge is essential to develop rigorous mathematics. As such, rigorous knowledge of math is not a prerequisite to reading this book. This book will use some set and logic notations for communication, and you should be familiar with those notations after learning more about set theory and logic in the first two chapters.
After introducing set theory informally (i.e. not emphasizing on axioms in set theory) and logic, we will be prepared to study methods of mathematical proof. After that, we will be discussing some fundamental concepts that are important for more advanced topics in mathematics.
# Mathematical Proof and the Principles of Mathematics/History

Missing from many books on mathematics is material on the history and development of subject covered. But not only does this do a disservice to the subject, leaving the impression that it's dry and academic, but it does a disservice to the reader in that it removes some of the potential avenues of understanding for a subject that may already be difficult.
Mathematics did not spring into being complete with proofs for every theorem and free of logical errors. The road to reach that stage was long, difficult, and full of wrong turns and surprises. Modern standards of mathematical proof mostly developed in the last 300 years, but the journey really started on the Greek peninsula some two and a half millennia ago.

## Chapter summary

A history of mathematics in general would take up many volumes, but what what we're mainly concerned with here is are the events that to the standards of mathematical rigor currently in common use. We start with Euclid who set a standard of mathematical proof that was not greatly improved upon for around two thousand years. During that time mathematics as a whole increased tremendously in scope, and eventually it became clear that a new standard would be needed if mathematics was to continue to grow while maintaining its prestige as the most logical of the sciences. The result was a new subject, the mathematics of mathematics itself, or metamathematics. This new subject was developed extensively in the 20th century, but yielded some results which seem to go against common sense, so the journey continues on.
# Mathematical Proof/TOC

* Mathematical Proof/Cover
* Mathematical Proof/Introduction to Set Theory
* Mathematical Proof/Logic
* Mathematical Proof/Methods of Proof
* Mathematical Proof/Relations
* Mathematical Proof/Functions
# Messier Index/M70

Messier 70 (also known as M70 or NGC 6681) is a w:globular cluster in the w:constellation Sagittarius. It was discovered by w:Charles Messier in w:1780.
M70 is at a distance of about 29,300 w:light years away from w:Earth and close to the w:Galactic Center. It is roughly the same size and w:luminosity as its neighbour in space, w:Globular Cluster M69. Only 2 w:variable stars are known within this cluster.

## External links

* Messier 70, SEDS Messier pages
* Messier 70, Galactic Globular Clusters Database page

## References

* 1 2 3 4 "SIMBAD Astronomical Database". Results for NGC 6681. Retrieved 2006-11-17.
* ↑ distance × sin( diameter_angle / 2 ) = 34 ly. radius
# Messier Index/M71

Messier 71 (also known as M71 or NGC 6838) is a w:globular cluster in the w:constellation w:Sagitta. It was discovered by w:Philippe Loys de Chéseaux in 1746 and included by w:Charles Messier in his catalog of w:comet-like objects in 1780. It was also noted by Koehler at w:Dresden around 1775.
M71 is at a distance of about 12,000 w:light years away from w:Earth and spans some 27 light years across. The irregular w:variable star Z Sagittae is a member of this cluster.
M71 was long thought (until the 1970s) to be a densely packed w:open cluster and was classified as such by leading w:astronomers in the field of star cluster research due to its lacking a dense central compression , its stars having more "metals" than is usual for an ancient globular cluster, and further its lacking the RR Lyrae "cluster" variable stars that are common in most globulars. However, modern photometric photometry has detected a short "w:horizontal branch" in the H-R diagram of M71, which is characteristic of a globular cluster. The shortness of the branch explains the lacking of the RR Lyrae variables and is due to the globular's relatively young age of 9-10 billion years. The relative youth of this globular also explains the abundance of "metals" in its stars. Hence today, M71 is designated as a very loosely concentrated globular cluster, much like M68 in Hydra. M71 has a luminosity of around 13,200 suns.

## External links

* Messier71 @ SEDS Messier pages
* Messier 71, Galactic Globular Clusters Database page
* Messier 71, LRGB CCD image based on two hours total exposure

## References

* 1 2 3 4 "SIMBAD Astronomical Database". Results for NGC 6838. Retrieved 2006-11-17.
* ↑ distance × sin( diameter_angle / 2 ) = 13 ly. radius
# Messier Index/M72

Messier 72 (also known as M72 or NGC 6981) is a w:globular cluster in the Aquarius constellation discovered by w:Pierre Méchain in w:August 29 w:1780. w:Charles Messier looked for it on the following w:October 4 and 5, and included it in his catalog. Both decided that it was a faint nebula not a cluster as is now believed. Using 10-inch (250 mm) telescopes, viewing the cluster is difficult resulting in only a view of a faint blurry picture, However using Kopernicks' 20-inch (510 mm) telescope resolution is highly increased. M72 is located at about 53,000 w:light-years away from w:Earth and lies in a considerable distance beyond the w:Galactic Center. Another source states that the cluster is 62,000 light-years away, with a diameter of 42 light-years. Generally considered a young cluster, the cluster has several blue giants, yet star clusters generally contain the oldest stars.

## External links

* Messier 72, SEDS Messier pages
* Messier 72, Galactic Globular Clusters Database page
* Messier 72, LRGB CCD image based on two hours total exposure
* M-72 Information

## References

* 1 2 3 "SIMBAD Astronomical Database". Results for NGC 6981. Retrieved 2006-11-16.
# Messier Index/M78

The nebula Messier 78 (also known as M 78 or NGC 2068) is a w:reflection nebula in the w:constellation Orion. It was discovered by w:Pierre Méchain in w:1780 and included by w:Charles Messier in his catalog of w:comet-like objects that same year.
M78 is the brightest diffuse reflection nebula of a group of nebulae that include NGC 2064, NGC 2067 and NGC 2071. This group belongs to the w:Orion Molecular Cloud Complex and is about 1,600 w:light years distant from w:Earth. M78 is easily found in small w:telescopes as a hazy patch and involves two w:stars of 10th magnitude. These two stars, HD 38563A and HD 38563B, are responsible for making the cloud of dust in M78 visible by reflecting their light.
About 45 w:variable stars of the T Tauri type, young stars still in the process of formation as well as some 17 w:Herbig-Haro objects are known in M78.

## External links

* Messier 78, SEDS Messier pages
* M78: Stardust and Starlight
* NightSkyInfo.com - M78
# Messier Index/M75

Messier 75 (also known as M75 or NGC 6864) is a w:globular cluster in the w:constellation Sagittarius. It was discovered by w:Pierre Méchain in w:1780 and included in w:Charles Messier's catalog of w:comet-like objects that same year.
M75 is at a distance of about 67,500 w:light years away from w:Earth and its apparent size on the sky translates to a true radius of some 67 w:light years. It is classified as class I, meaning it is one of the more densely concentrated globular clusters known. The w:absolute magnitude of M75 is about -8.5 or some 180,000 more luminous than the w:Sun.

## External links

* Messier 75 @ SEDS Messier pages
* Messier 75, Galactic Globular Clusters Database page

## References

* 1 2 3 4 "SIMBAD Astronomical Database". Results for NGC 6864. Retrieved 2006-11-16.
* ↑ distance × sin( diameter_angle / 2 ) = 67 ly. radius
# Messier Index/M76

The Little Dumbbell Nebula (also known as Messier 76, NGC 650/651, the Barbell Nebula, or the Cork Nebula) is a w:planetary nebula in the w:constellation Perseus. It was discovered by w:Pierre Méchain in 1780 and included in w:Charles Messier's catalog of w:comet-like objects as number 76. It was recognized as a planetary nebula in 1918 by the w:astronomer w:Heber Doust Curtis.
M76's distance is not well known, with estimates ranging from 1,700 to 15,000 light years, and consequently its dimensions are also not well known. The nebula shines at an w:apparent magnitude of +10.1 with a w:central star of magnitude +16.6. This star, whose expanding outer layers form the present nebula, has a surface temperature of 60,000 w:kelvins.
The Little Dumbbell Nebula got its name from its resemblance to the w:Dumbbell Nebula (M27) in w:Vulpecula. It was originally thought to consist of two separate nebulae and was thus given two catalog numbers in the NGC, 650 and 651. It is one of the faintest and hardest to see objects in Messier's list.

## External links

* The Little Dumbbell Nebula @ SEDS Messier pages
* NightSkyInfo.com - M76, the Little Dumbbell Nebula
* Little Dumbbell Nebula (M76, NGC 650 and 651)
* The little Dumbbell Nebula - Image of M 76 taken in an amateur-observatory with a semiprofessional optics]
# Messier Index/M77

Messier 77 (also known as NGC 1068) is a barred spiral galaxy about 47 million w:light-years away in the w:constellation w:Cetus. Messier 77 is an w:active galaxy with an w:Active Galactic Nucleus (AGN), which is obscured from view by astronomical dust at visible wavelengths. The diameter of the molecular disk and hot plasma associated with the obscuring material was first measured at radio wavelengths by the w:VLBA and w:VLA. The hot dust around the nucleus was subsequently measured in the mid-infrared by the MIDI instrument at the w:VLTI. It is the brightest w:Seyfert galaxy and is of type 2.
Messier 77's diameter is a 170,000 light-years.

## History

Messier 77 was discovered by w:Pierre Méchain in w:1780, who originally described it as a nebula. Méchain then communicated his discovery to w:Charles Messier, who subsequently listed the object in his catalog. Both Messier and w:William Herschel described this galaxy as a star cluster. Today, however, the object is known to be a galaxy.

## External links

* Spiral Galaxy M77 @ SEDS Messier pages
* VLBA image of the month: radio continuum and water masers of NGC 1068
* Press release about VLTI observations of NGC 1068

## References

* 1 2 3 4 5 6 7 "NASA/IPAC Extragalactic Database". Results for NGC 1068. Retrieved 2006-11-18.
* 1 2 "Pattern Speeds BIMA-SONG Galaxies with Molecule-Dominated ISMs Using the Tremaine-Weinberg Method". Bland-Hawthorn et al. (1997). Retrieved 2006-08-31.
* ↑ "Pattern Speeds BIMA-SONG Galaxies with Molecule-Dominated ISMs Using the Tremaine-Weinberg Method". (Schinnerer et al. 2000). Retrieved 2006-08-31.
* ↑ de Vaucouleurs, Gérard (w:April 1 w:1973), "Southern Galaxies.VI. Luminosity Distribution in the Seyfert Galaxy NGC 1566", Astrophysical Journal, 181: 31–50, doi:10.1086/152028 {{citation}}: Check date values in: |date= (help)CS1 maint: date and year (link)
* 1 2 K. G. Jones (1991). Messier's Nebulae and Star Clusters (2nd ed.). Cambridge: Cambridge University Press. ISBN 0-521-37079-5.
# Messier Index/M79

Messier 79 (also known as M79 or NGC 1904) is a w:globular cluster in the w:Lepus constellation. It was discovered by w:Pierre Méchain in w:1780. M79 is at a distance of about 41,000 w:light years away from w:Earth and 60,000 light years away from the w:Galactic Center.
Like w:Messier 54 (the other extragalactic globular on Messier's list), it is thought that M79 is not native to the w:Milky Way galaxy at all, but instead to the w:Canis Major Dwarf Galaxy which is currently experiencing a very close encounter with the Milky Way, one it is unlikely to survive intact. This is, however, a contentious subject as astronomers are still debating the nature of the Canis Major dwarf galaxy itself; care must therefore be taken when associating any object with the Canis Major dwarf.

## References

* 1 2 3 4 "SIMBAD Astronomical Database". Results for NGC 1904. Retrieved 2006-11-17.
* ↑ Re-affirming the connection between the Galactic stellar warp and the Canis Major over-density

## External links

* Messier 79, SEDS Messier pages
* Messier 79, Galactic Globular Clusters Database page
# Meteorology/Cyclone Formation

Cyclone formation is also known as cyclogenesis. There are several kinds of cyclogenesis. Usually cyclogenesis is used with reference to most cyclone patterns; however, if reference to tropical cyclones is intended, it is called tropical cyclogenesis. In the case of tornadoes, it is instead called tornadogenesis.
# Meteorology/Development

The Process of Tornado Development is called Tornadogenesis. Tornadoes are spawned from various thunderstorms, but more commonly the supercell. A supercell is a thunderstorm characterized by the strong updrafts, downdrafts, and windshear they contain. These are all parts of forming a tornado. Many of these things are caused by instability in the atmosphere from two airmasses colliding; one, a cold dry airmass, the other being a warm moist air mass. As the atmosphere begins to stabilize itself, this causes instability, thus resulting in the formation of wind shear and updrafts. Windshear is an upper level column of horizontally rotating air, that, when combined with an updraft, may form a tornado. Inflow, air that moves inyo or towards the base of the mesocyclone, often plays a role in the formation. This inflow feeds the storm, and brings it to intensify. A mesocyclone is a mass of rapidly rotating air, where many of natures most destructive tornadoes, such as the EF-5 are produced. This mass of rapidly rotating air can form a wall cloud, which also rotates with the rest of the storm. A wall cloud is a lowering in the clouds, and can often be identified by the rotation of this cloud, and how close to the ground it is. Sometimes, however, there are cases where the storm becomes rain wrapped, making us blind to the oncoming storm. The wall cloud is the updraft base of a storm, and where many tornadoes form.
# Meteorology/Definition

An air mass is a large volume of air in the atmosphere that is mostly uniform in temperature and moisture. They can extend thousands of miles across the surface and can go from the ground level to the stratosphere 16 kilometers (10 miles) into the atmosphere. Air masses can be classified as either as Arctic Air masses, Tropical Air masses, Polar Air masses, and Equatorial Air Masses.

## Sources

* ↑ "Air Mass". National Geographic Society. Retrieved 2023-05-09.
# Meteorology/F-Scale

The Fujita Scale is a measure of how powerful a tornado is, it is a scale between 0-5, 0 being barely any damage to 5 being very damaging. There are two Fujita scales the original and the Enhanced.

# Enhanced Fujita Scale

The Enhanced Fujita Scale also known as the EF scale had first become operational on February 1st, 2007, replacing the old Fujita Scale which was also known as the F scale. The Enhanced Fujita Scale was a revision of the Fujita Scale to better reflect the damage surveys to align the wind speeds more closely with the damage, the Enhanced Fujita Scale had also had to do with how the structures were designed. The Enhanced Fujita Scale rates a tornado from EF0 to EF5 based on the "estimated windspeeds" of the tornado and the damage it caused. When the damage caused by the Tornado, they use Damage Indicators (DI's) and Degrees of Damage (DoD's) that helps them estimate the wind speed of the tornado, to which then the tornado is given a rating.

## References

* ↑ The Enhanced Fujita Scale (EF Scale), retrieved 2023-05-09
# Meteorology/Dynamics

* Meteorology
* Introduction to Meteorology and the Atmosphere
* Dynamic Meteorology
* Pressure Measurement
* Kinematic Structures
* Dynamics and Forces
* Coriolis Force
* Ground Friction
* Pressure-Gradients
* Wind Energy
* Boundary Layers
* Turbulence
* Planetary Boundary Layer
* Heating and Temperature
* Moisture
* Atmospheric Stability
* Clouds and Precipitation
* Air Pressure
* Air Mass
* Circulation
* Patterns
* Thunderstorms
* Tornadoes
* Tropical Cyclones
* Pollution
* Optics

## Introduction

Before describing actual storms in all their complexity, this book aims to describe something much simpler: how the wind blows. Dynamic meteorology considers atmospheric motions as solutions of the fundamental equations of hydrodynamics: the equations of motion, the equation of continuity, the energy equation, the equation of state, and the equations of continuity for water substance. This book will develop these equations over the first several chapters. A rudimentary understanding of dynamic meteorology and atmospheric thermodynamics enables one to study storms: how they form, how they produce significant (and sometimes destructive) weather, and how they dissipate.
Even before considering the forces in the equation of state, a discussion of kinematics reveals mathematical conventions and vocabulary associated with dynamic meteorology.
Other systems of equations pertain to the statistical theory of turbulence and other special situations discussed in later chapters.
# Meteorology/El Niño and La Niña

El Niño and La Niña are part of the ENSO (El Niño Southern Oscillation) a weather system which alternates between 3 states of El Niño, Neutral, and La Niña.

# La Niña

During La Niña winds coming off the Andes push the warm surface waters to the West Pacific which strengthen the Trade wind. The sun warms up the surface waters off the coast of South America as the winds push it to the West Pacific they evaporate creating an overall low pressure in the West Pacific this rising air falls back down in the East Pacific which creates a high pressure system which pushes the trade winds even stronger across the Pacific until there is a stable locked in cycle of warm water being pushed westward. Usually this cycle is locked in once established for six months before the changing seasons can cause a reset. This causes a much stronger wet season for Tropical Regions in the West Pacific.

# Neutral

During a neutral phase of ENSO there is no obvious leaning towards La Niña or El Niño. The Trade winds are neither pushing warm waters off the South American coastline nor is it pushing it East from Australia and the West pacific.

# El Niño

During an El Niño phase the trade winds are reversed compared to the La Niña phase, this bring more wet weather to the East pacific and dryer weather to the West pacific as with the other phases they tend to get locked in once strongly established until a season change therefore during the summer in the southern hemisphere it will tend to be locked in.