Monday, April 30, 2012

Quaternions

The history of mathematics began with the natural numbers. These numbers: 1,2,3... have a simple universal interpretation, i.e. counting a number of objects. This was followed by the use of fractions, or rational numbers, most notably by the Ancient Greeks (note that negative numbers, and even zero, were only accepted into use after fractions in most cultures) to accommodate the splitting of objects into portions. Gains and losses in finances contributed to the general acceptance of negative numbers (and zero) while the diagonals of squares forced square roots, and therefore irrational numbers, upon humanity.

The rational and irrational numbers form the reals, which collectively describe any point on the one-dimensional line and the manipulation of these quantities includes all of one-dimensional geometry. Similarly, taking the square root of a negative number brings about the introduction of i and the imaginary numbers (the basic description of which can be found here). Combining these with the real numbers, the complex numbers are formed, and operations on such numbers can be used to describe two-dimensional mappings of the plane. In fact, the general 2-manifold can be treated as having complex coordinates.

It follows naturally that one would next wonder whether a similar system for three dimensions exists. This same question was pondered by the mathematician William Hamilton in during the mid-19th century. He failed to invent a three-dimensional system that satisfied certain necessary constraints, including a way to multiply consistently, (a feat that is now known to be impossible) but created a similar four-dimensional system. Rather than "stumbling upon" a new number system through the application of simple operations to known numbers, (division of integers, square roots of negative, etc.) Hamilton simply defined two quantities, in addition to 1 and i, known as j and k. These new numbers were to be connected through the famous Brougham Bridge Equations,

i2=j2=k2=ijk=-1

The general quaternion is simply an arbitrary addition of multiples of the identity, 1, i, j, and k, namely a number of the form a+bi+cj+dk, where a, b, c, and d are real numbers (note that this is the same form as a complex number (a+bi).

Quaternions, like the complex numbers, follow certain rules that make them a consistent group. For any quaternions a, b, and c, the following axioms are satisfied, and can be verified through simple real number multiplications and additions. Among these axioms are the associativity of addition and multiplication,
a+(b+c) = (a+b)+c,
a(bc) = (ab)c,
the identity formulae of addition and multiplication,
a+0 = 0+a = a,
a1 = 1a = a,
and each quaternion a ≠ 0 has a unique inverse a*, such that
(a)(a*) = (a*)(a) = 1

The above equations qualify quaternions as a group† . In addition, quaternion addition is commutative, i.e. a+b = b+a. However, quaternion multiplication is not commutative, and in general ab ≠ ba. This fact, which is taken as obvious in arithmetic, may seem to make the quaternions unusable in application. Nevertheless, there is a redeeming factor, namely that the multiplication of quaternion basis elements is a special type of non-commutative: anti-commutative, this meaning that ab = -ba. Note that this does NOT generalize to all quaternions, ab is equal to -ba only when a and b are i, j, or k.

†The term group in this context, and in later appearances, is a slightly more casual and abstract use than the strict mathematical group, which requires a more rigorous proof. The above identities outline the framework of a group, together with the proviso that operations on elements of the quaternions (in this case addition and multiplication) produce only other quaternions. Assuming that the coefficients of 1, i, j, and k can vary over all real numbers, this is clearly satisfied.

The multiplication of quaternion basis elements is also cyclic, meaning that the product of any pair among i, j, and k will produce the third (or its negative). The six resulting identities are
ij = k
ji = -k
jk = i
kj = -i
ki = j
and
ik = -j

The reader may be curious as to how these mysterious abstract quantities define four-dimensional geometry. To those familiar with elementary vector algebra, the use of i, j, and k may recall that they seem eerily similar to the unit vectors i, j, and k, which point along the x-, y-, and z-axes, respectively.



An illustration of the unit vectors i, j, and k and their positions in relation to the Cartesian three-dimensional coordinate axes.

This is no coincidence. In fact, the quaternions are most commonly used in relation to rotations in three-dimensional space, rather than coordinates in four. There is a connection here to the complex numbers, in which multiplication by i represents a 90º counterclockwise rotation. In quaternion geometry, each of the basis elements stands for a rotation in three-dimensional space, keeping the axis on which the element is oriented still. For example, the quaternion j represents a rotation keeping the y-axis constant. However, for i, j, and k, the degree of rotation is 180º, rather than 90º as in complex numbers.

An easy way to visualize such rotations is to visually manipulate a three-dimensional object, specifically something in the shape of a rectangular prism. A commonly used example is a closed book. Start with the cover facing up, and the text oriented right side-up. The x-axis of the three-dimensional cartesian coordinate plane passes through the center of the book from increasing from left to right. Similarly, the y-axis passes through the center, increasing as it moves away from you. Finally, the z-axis pierces the back cover of the book, travels through the center, and then continues upward out of the center of the front cover. If you are having trouble visualizing this, simply try to superimpose the three-dimensional coordinate plane above on the book.

The rotation i is equivalent to rotating the book 180º about the x-axis and ending with the back cover facing upward, but the text, upside-down, in front of you. Similarly, j flips the book onto the back cover, though with text facing right side-up, and the k simply rotates the book through 180º without flipping it, resulting in the front cover, except with the text upside-down. Further experimentation confirms the identity ij=k and others. However, the difference between ji and ij (the first yielding -k, rather than k) is unclear in 3-space. To understand why this is, a type of quantities known as spinors must be considered, see here.

Sources: Road to Reality by Roger Penrose Chapter 11,
http://programmedlessons.org/VectorLessons/vch12/vch12_10.html.

Sunday, April 22, 2012

Mars Science Laboratory

Mars Science Laboratory (MSL) is a NASA mission to Mars, whose goal is to search for organic material and microbial life on the Martian surface. To do this, the mission landed the rover Curiosity on Mars, whence it will navigate the surface and conduct scientific experiments.



The spacecraft successfully launched from Cape Canaveral, Florida on November 26, 2011. After leaving the Earth's atmosphere, it began a cruise stage that lasted until MSL approaches Mars in the summer of 2012. It successfully landed on the red planet on August 6, 2012.

This mission makes use of innovative technologies essential for any future landings on Mars. At over 2000 pounds, Curiosity is by far the largest Mars rover ever to be constructed, five times the size of the rovers Spirit and Opportunity of the early 2000's. In order to land intact on the Martian surface, a precision landing was necessary.

Previous rovers used inflatable airbags to cushion their landings, and simply bounced until settling to their destination. This did not allow high precision in landing. However, Curiosity is too large for such techniques, and made use of a more complicated landing sequence.



The landing procedure that was used to lower the Curiosity rover to the surface (click to enlarge). In the atmosphere, parachutes and braking thrusts were used to decelerate the craft. Then, a device known as a  sky crane lowered the rover to the ground on cables to ensure proper orientation.  Once Curiosity was safely on the surface, the crane detached and propelled itself away, as to not interfere with the rover.

After the landing, the first images sent back from Curiosity confirmed its position.



One of the first images from Curiosity showing the Martian surface. The venue from which the rover explored its environment was Gale Crater. This crater was selected due to the exposed sediment along its banks, which hold millions of years of Martian geologic history.

Over the next few days, the rover, remaining stationary, tested its scientific equipment, checking all of its instruments and cameras in preparation for its first motorized movement on the surface of Mars, which occurred on August 29.

On August 19, Curiosity performed its first sample analysis, using its on-board laser and spectral analyzer to determine its composition. In the following months, the rover conducted numerous experiments involving samples, both of soil and of atmosphere. In early November 2012, the atmosphere was found to contain an unusual concentration of heavy isotopes of its constituent elements (mainly carbon and oxygen, forming CO2. This indicates that the lighter isotopes were lost to space in the distant past, and this could explain the thinness of the Martian atmosphere.

During the first few months of the mission, the rover also took weather data, identifying some meteorological events on Mars. Most changes were related to dust storms and whirlwinds, and in fact the dust was discovered to catalyze a convective process in the Martian atmosphere: the dust on the side of Mars facing the Sun is lifted by wind into the atmosphere where it warms it, causing a greater differential in temperature between one half of the Martian atmosphere and the other (where it is night). This causes a flow of air from the cool to the warm side, restarting the process.

In January 2013, Curiosity imaged rocks at night, illuminating them with lights on the spacecraft. Using ultraviolet lamps, the rover searched for fluorescent minerals during the Martian night, at which time they would be visible (see image below).



In early February 2013, the rover completed its first drilling, obtaining a sample from several inches below the surface of bedrock. After obtaining a good number of rock samples and performing numerous other tests at Gale Crater, the rover prepared in June to begin a journey to its next destination, Mount Sharp. By the time Curiosity had spent one year on Mars in August 2013, it had already transmitted over 70,000 images, and had traveled about a mile along Mars's surface.

In October 2013, Curiosity performed a detailed analysis of the argon present in the Martian atmosphere, identifying the abundance of different isotopes. From this result, scientists deduced that some meteorites on Earth do indeed have their origin on Mars. Also, the prevalence (relative to on Earth) of a heavier isotope of argon suggests that Mars did undergo massive atmospheric loss earlier in its history. Early that December, scientists, using data collected by Curiosity announced that the rover had discovered evidence of an ancient lake (which last existed about 3.7 billion years ago) in Gale crater. Remarkably, this lake would have had very low salinity, and so, unlike most other similar findings, would have been nearly freshwater. This is significant as freshwater lakes could have supported a wider variety of life.



In March 2014, Curiosity began moving toward an interesting geological site known as "the Kimberley" (shown above). This gave the rover its first opportunity to conduct geological investigations of the several sandstone varieties (rather than mudstone, which it had been previously examining) to be found at the new site. It arrived at the site at the beginning of April and began analysis, including a new drilling in May.

On June 24, 2014, Curiosity completed scheduled its primary mission of one Martian year (687 Earth days). During this time, Curiosity systematically investigated the soil composition, radiation exposure, and abundance of organic molecules. At the time of the primary mission's completion, the evidence gathered by the rover was enough to confirm that the environment of Mars had once been favorable for "supporting microbial life".

Later in June, the rover moved out of its landing area into new terrain. It ultimately arrived at the base of Aeolis Mons (also called Mount Sharp), the mountain at the center of Gale Crater, in August. Exploration of the mountain is the primary goal for MSL's first mission extension. The 5.5 km (18,000 ft) high mountain was captured in the image below, taken by the rover:



Data gathered on the lower slopes of Mount Sharp in late 2014 included a series of sediment deposits which indicated the presence of a large lake at Gale Crater early in Mars's history that lasted tens of millions of years or more. This was the first major evidence for such a long-lasting, stable body of water on the red planet.

By early 2015, Curiosity had moved out of the bottom 33 feet of altitude of Mount Sharp and had entered a region with prominent mineral veins (as shown in the image above taken on March 18, 2015, which includes a scale). Such mineral veins forms when fluids move through cracks in existing rock and leave deposits. The light and dark minerals indicate a variety of fluid compositions.

In April 2015, the rover made another exciting discovery. Data from Curiosity indicated that water vapor condenses into liquid water in the Martian soil every night and reevaporates in the morning. Even though the Martian night temperatures are well below the normal freezing point of water (they may drop to -100°F), perchlorate salts in the soil reduce the freezing point of water (just as salting roads prevents them from freezing) enough that liquid water can form. This unexpected discovery suggests that a great deal more water could exist on Mars than previously thought.

The rover spent several months investigating sand dunes as it continued its journey, learning a great deal concerning the wind patterns on the planet's surface through the inspection of sand dune "ripples" (see below). The appearance of the sand dunes was comparable in appearance to those found on Earth.



After the sand dune investigation, Curiosity crossed the Naukluft Plateau towards the upward slopes of the mountain. This journey encompassed much of the first half of 2016, during which time the rover analyzed a few additional rock samples. During its investigation of Mount Sharp, Curiosity aimed to determine what geological environments are most suitable for the preservation of organic compounds and to identify geological layers and transitions. As well as being informative in their own right, these new objectives will guide future Mars missions.

On October 1, 2016, a second extension of two years to the mission began, allowing the rover to travel further up Mount Sharp. Later that month, Curiosity made an interesting discovery: a golf-ball sized meteorite on the Martian surface.



Laser spectrometry of this darkly colored rock indicated that it was primarily composed of iron, along with some nickel and phosphorus. This type of meteorite is usually formed from the core of asteroids. Further, the study of Mars meteorites allows the comparison between its population of impacting bodies and Earth's, revealing a great deal about how the inner Solar System evolved over time.

Sources: Mars Science Laboratory, Wikipedia, http://marsprogram.jpl.nasa.gov/msl/, http://mars.jpl.nasa.gov/msl/, http://www.npr.org/blogs/thetwo-way/2013/12/09/249760330/curiosity-finds-evidence-of-ancient-fresh-water-lake-on-mars?utm_content=socialflow&utm_campaign=nprfacebook&utm_source=npr&utm_medium=facebook, http://upload.wikimedia.org/wikipedia/commons/6/65/673885main_PIA15986-full_full.jpg, http://mars.nasa.gov/files/msl/2014-MSL-extended-mission-plan.pdf, http://www.nasa.gov/press/2014/december/nasa-s-curiosity-rover-finds-clues-to-how-water-helped-shape-martian-landscape/#.VItLjov4tFL, http://www.vox.com/2015/4/13/8384337/mars-water-liquid-curiosity, http://mars.nasa.gov/msl/news/whatsnew/index.cfm?FuseAction=ShowNews&NewsID=1912, http://mars.nasa.gov/news/2016/mars-rock-ingredient-stew-seen-as-plus-for-habitability&s=2

Saturday, April 14, 2012

Infinity: The Aleph Sequence

This is the thirteenth and final post of the Infinity Series. For the first, see here. For all posts, see the Infinity Series Portal.

In the previous seven posts, the world of countable ordinals has been explored in depth, allowing one to taste its incomprehensible vastness. It is truly remarkable how merely taking 2 to the power of ℵ0 could produce a uncountable cardinal so "easily", but that no number of additions, multiplications, exponentiations, or functions on ω could yield an uncountable ordinal. It is through such an examination that one begins to appreciate how inconceivably numerous are the real numbers and the members of the continuum in comparison to the rational and even algebraic numbers. Yet it is possible to go still further.

The first uncountable ordinal is the set of all countable ordinals, which, not being a member of this set, confirms that it is not countable. It is denoted ω1 (note the similarity in notation to ω1CK, the Church-Kleene ordinal defined previously). The cardinality of ω1 is denoted ℵ1, the second ℵ number. Additionally, ω1 is what is called the initial ordinal of the cardinal number ℵ1, as it is the smallest ordinal with this cardinality (analogously, the initial ordinal of ℵ0 is ω).

Before going on, it is useful to describe another property that ordinals and cardinals may possess. Every ordinal (or cardinal) can be regular or singular. To determine whether any quantity is regular or singular, one must consider the set which has the given quantity as its supremum. For example, consider the ordinal ω2. It is equal to

sup{ω,ω*2,ω*3,...,ω*n,...},

and the set in question, namely {ω,ω*2,ω*3,...,ω*n,...}, has order type ω (recall that ordinals themselves each represent an order type, and each order type represents a special class of sets that can be reached from one another with a one-to-one order preserving correspondence). Since the order type of this set, ω, is not equal to the original ordinal, ω2, ω2 is singular. Note that, though, the set above has the same cardinality as ω2, it does not have the same order type. In contrast, ω is a regular ordinal. All the sets which have it as a supremum are infinite sets of natural numbers, and it is easily to see that they all have order type ω.

For cardinals, the definition is even simpler. One simply considers the sets of cardinals that have the cardinal as a supremum, and evaluate the cardinality of the sets. If these sets have the same cardinality of the number, it is regular. Otherwise, it is singular. For example, ℵ0 is the first infinite cardinal, and is the supremum of any infinite sequence of finite cardinal numbers, each of which have a cardinality of ℵ0. This cardinal is therefore regular. The order type or cardinality of the set of which the given quantity is a supremum is the quantity's cofinality. For a successor ordinal or cardinal, the cofinality is defined as 1, e.g. the ordinal 5 is the supremum of the set {4}, which has order type 1.

The first few regular ordinals are 0 (trivially), 1, ω, and ω1. ω1 can be seen as regular by the following argument. We know that countable sequences of countable ordinals have only countable ordinals as suprema. Therefore, the order type of a set {α,β,γ,...}, whose supremum is ω1, is an uncountable ordinal. However, if this ordinal were greater than ω1, some element of the set would also have to be greater than ω1. Therefore, the cofinality of ω1 is itself, and the first uncountable ordinal is regular. The cardinals corresponding to these four ordinals are also regular.

Recall the previous definition of the cardinal beth-one, which was that this cardinal was the cardinality of the real numbers, and that it was the cardinality of the power set of the set of all natural numbers, i.e. 20. It has been proven that the statement "beth-one is equal to aleph-one" is independent of ZF theory. The proposition that this statement is true is known as the continuum hypothesis. In other words, the continuum hypothesis states that

20 = ℵ1

This seems reasonable, given that both cardinals in the above equality were encountered in the context of being the "first" uncountable cardinal. For the purposes of this post, the two will be assumed to be equal simply in order to avoid dealing with the beth-sequence separately. Also, the generalized continuum hypothesis, that each subsequent member of the beth-sequence is equal to its corresponding aleph cardinal, will be assumed for clarity. However, the focus on the development of cardinals will be based on that of ordinals, and little is lost by assuming that the aleph- and beth-sequences are synonymous.

The next cardinal number after ℵ1 (assuming the continuum hypothesis), is ℵ2, which has an initial ordinal ω2, both of which are by definition regular for the same reasons as ω1. Similarly, all ordinals between ω1 and ω2, of which there are uncountably many, are singular.

Similarly, one can define a series of regular cardinals ℵ0, ℵ1, ℵ2, ℵ3, and in general ℵn for natural n, and a analogous sequence of ordinals ω, ω1, ω2, ω3,...,ωn.

The cardinal sup{ℵ0,ℵ1,ℵ2,...,ℵn,...} is denoted ℵω. The initial ordinal of this cardinal is ωω. Since the cofinality ωω is ω (there is one member in {ω,ω12,...,ωn,...} has one member for each natural n), this ordinal, and ℵω, are singular.

ω is also the first so-called inaccessible cardinal, meaning that its ordinal subscript is a limit ordinal. Similarly, larger cardinals, such as ℵω*2, and ℵω2, having limit ordinals as indices, are inaccessible. Next, we "number" these inaccessible cardinals with ordinals. For example, ℵω is identified with 1, ℵω*2 with 2, and so on. Therefore, ℵω2 is labeled as the ω-th inaccessible cardinal. Similarly, every cardinal ℵω*α is labeled α, as every limit ordinal is a multiple of ω.

A cardinal is considered 1-inaccessible if its initial ordinal is the same as its label on the list of inaccessible cardinals. This means that if the cardinal is of the form ℵω*α, its initial ordinal must be α. The first cardinal for which this holds is the supremum of the set

{ℵω,ℵωω,ℵωωω,...},
whose initial ordinal is ωωω....
Clearly, removing an "ω" from this infinite chain does not change the value, nor does multiplying by ω on the left. Therefore, this cardinal is 1-inaccessible. The first 1-inaccessible cardinal is the limit of the ℵ notation in a sense, as it requires an infinite number of "ω's" to write. Similarly, other 1-inaccessible cardinals exist, and can be numbered with ordinals in the same fashion. A cardinal whose initial ordinal is the same as its label on the list of 1-inaccessible cardinals is called 2-inaccessible. And so larger and larger cardinals are defined that are 3-inaccessible, 4-inaccessible. Clearly, for α-inaccessible cardinal κ, κ is also β-inaccessible for all β < α. Therefore, one can define a cardinal as ω-inaccessible if it is 1-, 2-, 3-,... inaccessible for all natural numbers.

Then, the notion of α-inaccessibility can be defined for any α, with the limit ordinal case being treated similarly to ω-for a limit ordinal α, a cardinal is α-inaccessible if it is also β-inaccessible for all β < α.

Next, a cardinal is hyper-inaccessible if it has an initial ordinal α, and is α-inaccessible. Such a cardinal is immensely large, as, for all cardinals yet considered, their initial ordinals were far higher than their degree of inaccessibility. For example, for the 1-accessible cardinal above, the degree of inaccessibility is only 1, while the initial ordinal α is so large that ωα = α. By substituting "hyper-inaccessible" for "inaccessible" in the definition of 1-inaccessible cardinals, i.e. by numbering the hyper-inaccessible cardinals, one obtains corresponding definitions for 1-hyper-inaccessible, 2-hyper-inaccessible, and, in general α-hyper-inaccessible cardinals. Finally, there are hyper-hyper-inaccessible cardinals, the process can repeat, generating hyper-hyper-hyper-inaccessible cardinals, and so on.

A Mahlo cardinal is then defined as a cardinal that is inaccessible, hyper-inaccessible, and so on for any finite number of "hyper-'s". This is an example of a cardinal that cannot be proved to exist without the use of a large cardinal axiom. Without such an axiom, it is impossible to prove that any cardinal satisfies the needed property to be considered Mahlo.

As one explores higher cardinals, and their corresponding ordinals, each type of cardinal tends to become associated with the axiom with which its existence can be proved. To confirm that larger cardinals exist, one generally needs stronger and stronger axioms. However, if an axiom becomes too strong, it can sometimes create an inconsistent system (paired with other axioms), where a statement and its negative can both be proved true. If such an axiom was confirmed to prove an inconsistency, it seems reasonable that it would be discarded, and with it, its corresponding cardinals.

Therefore, in a way, there does exist an "upper limit" for cardinals and ordinals based on the strength of their accompanying axioms. Note that this does not mean there is a highest ordinal or cardinal, because this is impossible (e.g., for any ordinal α, its successor α' is greater than α and is also an ordinal). The situation is analogous to that of countable ordinals: there is no highest countable ordinal, yet the set of countable ordinals does have an upper limit, ω1. In the case of the totality of cardinals and ordinals, there is probably even more "room to maneuver" below the upper limit, and plenty of cardinals not yet discovered.

The task of determining the hierarchy of large cardinals is ongoing and will not be discussed in detail here. However, the above description gives an idea of how such cardinals are defined. In conclusion, the realm of infinities, despite the insight provided by the use of cardinals and ordinals, is still not completely explored, and increasingly ingenious methods will be required to generate higher and higher quantities. The possibilities-as one would expect-are infinite!

Sources: Regular Cardinal, List of Cardinal Properties, etc., Wikipedia


Author's Note (March 29, 2012): Of all my endeavors concerning this blog to date, the Infinity Series was perhaps the most ambitious, and definitely the most difficult. To attempt to convey even a glimpse of infinite sets and set theory is a challenging task. The above is far from an authoritative reference, and much is omitted where explanation would require the addition of large amounts of new material. Moreover, the mathematical arguments are less than rigorous, and the so-called "proofs" are often but sketches of their full versions-which require whole sets of axioms to complete.

Nevertheless, I am hopeful that this nuanced approach to a very narrow area of mathematics allows at least limited accessibility to those not acquainted with the nuts and bolts of set theory. For all my visitors, thank you for reading.

Professor Quibb

Friday, April 6, 2012

Infinity: Beyond Recursive Ordinals

This is the twelfth and penultimate post of the Infinity Series. For all posts, see the Infinity Series Portal.

In the previous post, the idea of an ordinal collapsing function was expanded to produce a series of these functions, each corresponding to a natural number. One could go even further listing fixed points and identifying ordinals inaccessible from certain operations and functions, but this is tedious, and is of little interest. To identify further ordinals beyond those accessible with a method involving fixed points, one must use a totally new idea.

Though set theory was mentioned as the basis for all subject matter in this series, no rigorous definition was ever made of any part of this theory. Rather, it has been assumed that the sets mentioned can be defined and constructed. However, set theory is based on assuming a few intuitively obvious or underlying statements that are accepted as being true. These are known as axioms. Formulae derived from these axioms or combinations thereof are then true statements.

However, there is no one "correct" set of axioms from which to assemble a theory that simultaneously has two desirable qualities: consistency and completeness. A consistent theory contains no two statements contradicting one another, i.e. if, for some statement p, "p is true" is derivable from the axioms "p is false" is not. In contrast, a complete theory is such that every statement p can be proved either true or false from the axioms. It was proved by Gödel in his incompleteness theorems that a set theory incorporating arithmetic cannot be simultaneously consistent and complete.

Therefore, there is a multiplicity of possibilities for the selected axioms, with no one clearly the "best", and, correspondingly, there are many different varieties of set theory. Of interest here are the axioms involved in the construction of sets. One basic axiom present in many variations of set theory is the pairing axiom, which, in logical notation, is

ABCD(DC↔(D = AD = B)).

In this axiom, A, B, C, and D are set variables, meaning that they take as values sets only, ∀ is the universal quantifier, read "for all", ∃ is the existential qualifier, meaning "there exists", ∈ is the membership relation, ↔ is the biconditional meaning "if and only if", and ∨ is the logic gate "or". Therefore, in common english, the axiom of pairing states:

"For all sets A and B, there exists a set C such that for all sets D, D is a member of C if and only if D is equal to A or D is equal to B."

This can be further paraphrased to "For any two sets, there is a third containing only the two as members", which simply allows one to create a set with a given two members, hence the name "pairing". Together with the definition {S,S} = {S}, which simply means that redundant elements in a set are deleted, one can construct the ordinals 1 and 2 from 0 through the axiom of pairing. 1 = {0} is derived from the case A = B = 0 in the axiom, as C is then {0} = 1. Furthermore, if A = 0 and B = 1, then the axiom guarantees the existence of the set {0,1} = 2. However, the ordinal 3 cannot be constructed in the same fashion, as it has more than two members. Therefore, the axiom of union, another axiom useful in constructing sets, is introduced:

ABC(CB↔∃D(CDDA))

This axiom collects all sets C such that C is a member of a member of A into a set B, known as the union of A. Note the relation to the definition of union we have already seen: the operation of union, when applied to multiple sets, collects their members; similarly, the union of a single set is the union of all its members. For example, the union of the set {{Ø},{{Ø}},{Ø,{Ø}} = {{0},{1},{2}}, the members of which are all guaranteed to exist by the axiom of pairing is the same as the union of its members, namely {0}∪{1}∪{2} = {0,1,2} = 3. By use of these two axioms, any finite ordinal can be constructed.

There is a third important axiom, called the axiom of infinity, which grants the existence of an infinite set, specifically the ordinal ω. Without it, one can not prove that infinite ordinals exist, and ω, the first ordinal that cannot be proven to exist, is called the strength of the theory. Strictly speaking, even some theories without the axiom of infinity "know" that ω exists, i.e. know that there is a set of natural numbers, but cannot prove that it is well-ordered, and thus cannot confirm its status as an ordinal. Thus far, the axioms discussed have been part of the theory known as Zermelo-Frankel Set Theory, or ZF set theory. The strength of this theory is measured by an ordinal higher than any ordinal so far discussed, as ZF set theory includes the concepts of function, fixed point, and (to a limited extent) inaccessibility. Therefore, all the processes used to describe ordinals thus far are "embedded" in ZF, and all the ordinals stated thus far can be explicitly constructed.

Technically, this new ordinal, which we shall call the ZF-ordinal, is not an ordinal at all, at least in ZF theory, where we cannot even confirm its status as an ordinal. This is an example of incompleteness in ZF theory. In order to prove that the ZF-ordinal, which contains all countable ordinals constructed thus far, is an ordinal, one must add additional axioms to ZF to strengthen the theory, and augment its ability to construct ordinals. But what axioms should be added? The type of axioms that serve this purpose are called large cardinal axioms, and confirm the existence of cardinals with certain properties. It seems somewhat odd that axioms concerning cardinal numbers would have anything to do with constructing ordinals. However, of the most importance are the properties themselves of the cardinals, which outline new methods for constructing ordinals.

The precise forms of these axioms will not yet be explained, as they do pertain to large cardinals that have not yet been discussed, but we will outline the hierarchy which they form. The first large cardinal axiom will be called A1, and is independent of ZF set theory, i.e. cannot be proved true or false by the existing axioms. Therefore, a new theory consisting of the original axioms and A1 is formed, which will be denoted ZFA1. This new theory can construct the ZF-ordinal, and even higher countable ordinals at that. The set of ordinals that can be constructed in this theory is again an ordinal, and by definition, its status as an ordinal is indeterminate within ZFA1. It is called the ZFA1-ordinal. By assuming more axioms A2, A3,..., one can obtain higher ordinals, and correspondingly "higher" set theories ZFA2, ZFA3,..., each of which has a higher corresponding strength. The ordinals defined in this way are called (rather paradoxically) unrecursible recursive ordinals, referring to the fact that they cannot be defined by recursive functions within the given theory, but rather require the theory itself in their definition.

From here one can set up a hierarchy of axioms. For any two axioms of this type, X and Y, either the two are "equal" in that they can be proved to imply one another, or one is stronger than the other, and augments ZF to yet a stronger set theory. The limit of the strength of the successive theories in this way may be further increased by using so-called axiom schemata (singular schema), which introduce additional variables into the logical statement. An axiom schema asserts that the given statement is true for all values of the variable. Also, meta-axioms, namely procedures for finding more axioms can extend the hierarchy even further. Eventually, a point is reached where no strengthened version of ZF or any other type of function can generate an ordinal. This inaccessible limit is called the Church-Kleene ordinal, and is written ω1CK.

ω1CK is the set of all recursive ordinals, and is then an ordinal, although it cannot be proven that this is true in any of the theories yet described. One could say that this ordinal's definition is based on those smaller than it and is therefore recursive, but this is not true, as the system of recursion itself is within the definition. Additionally, ω1CK could have served as the Ω in the previously discussed ordinal collapsing functions (see here), though, technically speaking, the ZF-ordinal or any others of the "unrecursible recursive" variety would have served as well. ω1CK is often chosen simply because its definition is simpler and more elegant.

Remarkably, ω1CK is still countable. This can be seen by considering the fact that there are only countably many notations through which to define ordinals; for each function considered thus far, each can take only countably many arguments, and there are only countably many functions based off of countable ordinals. Since all recursive notations are exhausted on countable ordinals, ω1CK is countable. Nor is it the last countable ordinal. One can of course create the set including all recursive ordinals and ω1CK, which is the ordinal ω1CK + 1, and so on.

In fact, one can define the set of all ordinals accessible from functions on all ordinals including the Church-Kleene ordinal, which is some other limit ordinal that could have served as Ω2. Furthermore, by repeatedly define inaccessible ordinals, one can obtain a sequence of countable ordinals, each of which is greater than any recursive function on the previous one, and is parallel to the Ω sequence.

There is no end to this process, and one can generate as many countable ordinals as desired in a similar fashion. However, no matter how far one goes with defining countable ordinals, one can only reach an infinitesimal fraction of them, as functions defined within the countable ordinals always produce still more countable ordinals. The situation is analogous to that of whole numbers: one can always define a larger one, but there are always still infinitely more. How, then, can one work with uncountable ordinals and their cardinal counterparts? These problems will be addressed in the next post.

Sources; http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pl/1235418473&view=body&content-type=pdf_1,
Axiomatic Set Theory by Patrick Suppes, A New Kind of Science by Stephen Wolfram