Monthly Archives: April, 2012

Set Theory, Notes 1: Ordinals and Cardinals

Unless otherwise specified, we will assume ZF axioms.  Recall the following ZF axiom:

Axiom of Infinity:

(\exists\omega)(\varnothing\in\omega\wedge(\forall x\in\omega)(x\cup\{x\}\in\omega)).

We will call \omega the set of natural numbers.  That is,

\begin{array}{rcl}0&=&\varnothing\\1&=&0\cup\{0\}=\{0\}\\2&=&1\cup\{1\}=\{0,1\}\\&\vdots&\\n+1&=&n\cup\{n\}\\&\vdots&\end{array}

We can then define \omega+1=\omega\cup\{\omega\} and iterate as before, whence by applying the axiom of infinity again we can obtain \omega\cdot 2:=\omega+\omega, and so on.  All such sets generated by this process are called ordinals.  This in turn gives us the canonical linear ordering of the ordinals where n<m iff n\in m.

Definition 1.  An ordinal \alpha is a successor ordinal iff \alpha=\beta+1.  A limit ordinal is an ordinal which is not a successor ordinal.

Proposition 2.  (Transfinite Induction)  Let Ord denote the class of all ordinals (in accordance to VBG notion of class) and C be a class.  If

  1. 0\in C,
  2. \alpha\in C\Longrightarrow\alpha+1\in C, and
  3. if \gamma is a limit ordinal and \alpha\in C for all \alpha<\gamma, then \gamma\in C,

then C=Ord.

Proof.  Since < is a linear ordering on the ordinals, let \alpha be the least ordinal such that \alpha\notin C.  But then \alpha+1\in C which is a contradiction.  Hence C=Ord.

We can index ordinals with ordinals to generate the notion of a sequence of ordinals.  We define an nondecreasing (nonincreasing) sequence of ordinals an ordered set \{\gamma_\alpha\} of ordinals where \gamma_\alpha\leq\gamma_\beta iff \alpha\leq\beta.

Definition 3.  Let \{\gamma_\alpha\} be an nondecreasing sequence of ordinals and \xi be a limit ordinal and \alpha<\xi.  Then we define the limit of the sequence as

\displaystyle\lim_{\alpha\to\xi}\gamma_\alpha=\sup_{\alpha<\xi}\{\gamma_\alpha\}.

A dual definition can be defined for nonincreasing sequences, in which case the limits can be respectively distinguished as left and right limits.  A sequence \{\gamma_\alpha\} is continuous if for every limit ordinal \xi in the indexing subclass we have

\displaystyle\lim_{\alpha\to\xi}\gamma_\alpha=\gamma_\xi.

An example of a sequence which is not continuous may be one of the form S=(...,\gamma_{\beta},\gamma_{\beta+1},...) where \gamma_{\beta},\gamma_{\beta+1} are both limit ordinals.  So in this case

\displaystyle\lim_{\alpha\to\beta+1}\gamma_\alpha=\sup_{\alpha<\beta+1}\{\gamma_\alpha\}=\gamma_{\beta}\neq\gamma_{\beta+1}

(since the sup is actually a max in this case).

Definition 4.  (Ordinal Arithmetic)  We define

Addition:

    1. \alpha+0=\alpha,
    2. \alpha+(\beta+1)=(\alpha+\beta)+1,
    3. \alpha+\beta=\lim_{\gamma\to\beta}\alpha+\gamma.

Multiplication:

    1. \alpha\cdot 0=0,
    2. \alpha\cdot(\beta+1)=\alpha\cdot\beta+\alpha,
    3. \alpha\cdot\beta=\lim_{\gamma\to\beta}\alpha\cdot\gamma.

Exponentiation:

    1. \alpha^0=1,
    2. \alpha^{\beta+1}=\alpha^\beta\cdot\alpha,
    3. \alpha^\beta=\lim_{\gamma\to\beta}\alpha^\gamma.

It follows that addition and multiplication are both associative, but not commutative.  In particular one can see that

1+\omega=\omega\neq\omega+1

and

2\cdot\omega=\omega\neq\omega\cdot 2=\omega+\omega.

Definition 5.  For a set X, we define its cardinality, denoted |X|, as the unique ordinal with which the set has a bijection.  The corresponding subclass of ordinals is called the class of cardinals.

Proposition 6.  If |X|=\kappa, then |P(X)|=2^\kappa.

Proof.  For every A\subseteq X, define

\displaystyle\chi_A(x)=\left\{\begin{array}{ll}1&x\in A\\ 0&x\in X-A\end{array}\right..

Hence the mapping f:A\mapsto\chi_A(X) is a bijection between P(X) and \{0,1\}^X.

Hence in this context, Cantor’s theorem immediately follows: |X|<|P(X)|.

Proposition 7.  Let |A|=\kappa, |B|=\lambda, and A\cap B=\varnothing.  Then

  1. |A\cup B|=\kappa+\lambda,
  2. |A\times B|=\kappa\cdot\lambda,
  3. \left|A^B\right|=\kappa^\lambda.

Proof.  We have

|A\cup B|=|A\sqcup B|=|\kappa\sqcup\lambda|=\kappa+\lambda.

Also if f:A\to\kappa and B\to\lambda are bijections, then we can define h:A\times B\to\kappa\cdot\lambda by h:(a,b)\mapsto f(a)\cdot g(b)\in\kappa\cdot\lambda, and this is easily seen to be a bijection.

And if k\in A^B then we can define h:k\mapsto k(b)^b\in\kappa^\lambda, which is also seen to be a bijection.

It thus follows that addition and multiplication of cardinals are commutative (that is, ordinal operations are commutative on this subclass).

Above we said 1+\omega=\omega\neq\omega +1 and 2\cdot\omega=\omega\neq\omega\cdot 2.  But

\omega=|\omega|=|1+\omega|=|\omega+1|

and

\omega=|n\cdot\omega|=|\omega\cdot n|=\left|\omega\uparrow\uparrow n\right|

for any finite ordinal n with Knuth notation

\displaystyle\omega\uparrow\uparrow n=\omega^{\omega^{\cdot^{\cdot^{\cdot^\omega}}}}

having n raisings of \omega.  Consider the following convention for defining countable infinite ordinals:

\displaystyle \omega,\omega+1,...,\omega\cdot 2,...,\omega^2,...,\omega^\omega,...,\omega\uparrow\uparrow\omega=\varepsilon_0,...,\varepsilon_1=\varepsilon_0\uparrow\uparrow\varepsilon_0,...,\\\varepsilon_2=\varepsilon_1\uparrow\uparrow\varepsilon_1,...,\varepsilon_\omega,...,\varepsilon_{\varepsilon_0},...,\varepsilon_{\varepsilon_{\ddots}},...

They are all called countable infinite ordinals since for any one of them \alpha, |\alpha|=\omega.  To clarify when we are talking about cardinal numbers versus ordinal numbers, we will use aleph notation: \aleph_0=\omega.  From Cantor’s theorem above, we know that if |X|=\aleph_0, then |P(X)|=2^{\aleph_0}>\aleph_0.  That is, the ordinal corresponding to 2^{\aleph_0} must be greater than all of the countable ordinals above, otherwise its cardinality would be \aleph_0.  This necessitates the notion of uncountable ordinals and corresponding uncountable cardinals.  We use subscripts to characterize these: \aleph_1=\omega_1,\aleph_2=\omega_2, etc.

Definition 8.  An infinite cardinal \aleph_\alpha is a successor cardinal iff \alpha is a successor ordinal, and it is a limit cardinal iff \alpha is a limit ordinal.

Definition 9.  Let \alpha be a limit ordinal.  An increasing \deltasequence (\beta_\gamma)_{\gamma<\delta} with \delta a limit ordinal is cofinal in \alpha if \lim_{\gamma\to\delta}\beta_\gamma=\alpha.  And if \alpha is an ordinal, then we define its cofinality as

\displaystyle\text{cf}\,\alpha=\inf\left\{\delta:\lim_{\gamma\to\delta}\beta_\gamma=\alpha\right\}.

It is easy to verify that \text{cf}\,\alpha=1 iff \alpha is a successor ordinal.  Also \text{cf}\,0=0,\text{cf}\,\omega=\omega, and \text{cf}\,\omega_\alpha=\omega_\alpha for any finite ordinal \alpha.

Proposition 10.  \mbox{cf}\,\mbox{cf}\,\alpha=\mbox{cf}\,\alpha.

Proof.  Let \mbox{cf}\,\alpha=c.  Then \lim_{\gamma\to c}\beta_\gamma=\alpha (in particular it is the smallest such c).  Now if \mbox{cf}\,\mbox{cf}\,\alpha=\mbox{cf}\,c=d, then certainly d\leq c.  Now since (\beta_\gamma) is cofinal in \alpha, a subsequence of indices (\gamma_\delta) is cofinal in c (where cofinality can be chosen to be d).  So

\displaystyle\lim_{\delta\to d}\gamma_\delta=c.

But then \left(\beta_{\gamma_\delta}\right) is cofinal in \alpha.  That is,

\displaystyle\lim_{\delta\to d}\beta_{\gamma_\delta}=\alpha,

whence d\geq c.

Definition 11.  An ordinal \alpha is regular if \mbox{cf}\,\alpha=\alpha.  It is singular if it is not regular.

Corollary 12.  If \alpha is a limit ordinal, then \mbox{cf}\,\alpha is a regular cardinal.

Theorem 13.  If \kappa is an infinite cardinal, then \kappa<\kappa^{\mbox{cf}\,\kappa}.

[1]  Jech, Thomas.  Set Theory.  3rd Edition.  Springer Monographs in Mathematics.  Springer-Verlag.  2000.

Science in the US

As it is no surprise about the education in the US, science in particular has also been suffering.  Last week in the April meeting of the American Physics Society, a group of physicists made precisely this claim.  The primary data used in the basis of their argument was the decline in degrees in science.  This data is certainly consistent with the argument that the US is falling behind in science.  There is no debate that good performance in science predicates the choice of a degree in science.  That is to say, the decline in degrees in science is mostly a consequence of poor performance in science earlier–rather than the cause of it.  So the natural question is: why are kids less interested in science?

The everyday life of a kid is governed by three structures: school, parents, and other media they encounter (including television, internet, and social structure).  It follows that one should be able to attribute the decline in scientific interest to one or more of these structures.  Suppose we assume two things: (1) that there is a problem at some generation with respect to science advocacy in some of these components, and (2) that the extent to which kids assimilate to the scientific understanding of their mentors (i.e. parents and teachers) is strictly less than complete (that is, in an abstract isolated teacher-student situation, the student can only learn a proper subset of the teacher’s knowledge).  These two assumptions ensure a gradual decline in scientific understanding over the course of future generations.

These assumptions make it no surprise that deficiencies in science can only get worse over time.  This only leaves two more questions:  how did the deficiency begin, and how can it be fixed?  I might take a bold but not entirely unreasonable guess that much of it can be attributed to what I previously called curriculum that is not “readily applicable to everyday situations”.  Who is in charge of deciding this curriculum?  The answer to this is: precisely those who originally learned this curriculum.  So again, it is no surprise why nothing changes.  Many point to issues like parenting, television, and other media as the problem.  But each of these are easily traceable back to the fact that members of each of these components (parents and media affiliates) had a similar education anyway.  This further reinforces the claim that the curriculum itself might need to change.

Now if we change the curriculum, the question of teachers’ ability to implement it arises.  This can be addressed with changes in college curriculum that prepare emerging teachers for a new curriculum in the primary and secondary levels.  This seems like it might be the first promising step to address the issue in the long run.  In an era where governmental budget deficits are high, the appetite for further investment in education (itself a long-term yet critical investment) also seems to be on the decline.  It’s no surprise that a fair amount of politicians’ value placed on scientific investment puts it on the back burner in a budget crisis given that a vast majority of them have no background in science to begin with.  This is yet another reason that a change in curriculum seems more promising in my mind than further monetary investments in education.  And since the payoffs for this will not be immediate, it is all the more reason to begin a change in curriculum now rather than later.  If a scientific curriculum tied closer to reality and pragmatic obstacles implants itself in those who will become the future generations, the hope is that this will in turn resolve the future problems associated to the parents, media, and politicians.

[1]  http://news.yahoo.com/crisis-us-science-looming-physicists-warn-135405711.html

Fluid Mechanics in Seven Dimensions?

Cayley-Dickson Algebras:
The Cayley-Dickson algebras \mathbb{A}_n are special real algebras satisfying

\dim_\mathbb{R}\mathbb{A}_n=2^n

for n\geq 0.  If we let \{e_0,...,e_{2^n-1}\} be the basis of \mathbb{A}_n, then the algebra is generated by the relations

e_i^2=e_1\cdots e_{2^n-1}=-1

where i\geq 1.  Hence \mathbb{A}_0=\mathbb{R} and \mathbb{A}_1=\mathbb{C}.  \mathbb{A}_2 is called the quaternions and is often denoted \mathbb{H}.  \mathbb{A}_3 and \mathbb{A}_4 are called the octonions and sedenions respectively and are respectively also denoted \mathbb{O} and \mathbb{S}.

The Cayley-Dickson algebras can also be constructed inductively using a recursive unary operation (called conjugation).  It proceeds as follows, let \mathbb{A}_0=\mathbb{R}.  We define \mathbb{A}_1 constructively.  Let \mathbb{A}_1=\mathbb{R}\oplus\mathbb{R} and define the following operations:

\begin{array}{rcl}(a,b)+(c,d)&=&(a+b,c+d)\\(a,b)(c,d)&=&(ac-bd,ad+bc)\\(a,b)^*&=&(a,-b).\end{array}

One can verify this structure is isomorphic to \mathbb{C}.  Next we let \mathbb{A}_2=\mathbb{A}_1\oplus\mathbb{A}_1=\mathbb{C}\oplus\mathbb{C} and define its structure

\begin{array}{rcl}(a,b)(c,d)&=&(ac-d^*b,da+bc^*)\\(a,b)^*&=&(a^*,-b)\end{array}

and addition remaining the same and the inner conjugation being complex conjugation.  How does is this retain isomorphism to our original definition of \mathbb{A}_2?  Note the dimension is retained by the fact that

\mathbb{A}_2=\mathbb{C}\oplus\mathbb{C}=\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}\oplus\mathbb{R}.

Hence using this as our inclusion, we show the original requirements that

i^2=j^2=k^2=ijk=-1.

We have

\begin{array}{rcl}i^2&=&(0,1,0,0)^2\\&=&\left((0,1),(0,0)\right)^2\\&=&\left((0,1),(0,0)\right)\left((0,1),(0,0)\right)\\&=&\left((0,1)(0,1)-(0,0)^*(0,0),(0,0)(0,1)+(0,0)(0,1)^*\right)\\&=&\left((-1,0)-(0,0),(0,0)+(0,0)(0,-1)\right)\\&=&\left((-1,0),(0,0)\right)\\&=&\left(-1,0,0,0\right).\end{array}

Similar computations can be done for j and k and for showing that

(0,1,0,0)(0,0,1,0)(0,0,0,1)=(-1,0,0,0).

It turns out \mathbb{A}_3 and beyond are not associative, so in our initial requirement that

e_1\cdots e_{2^n-1}=-1,

we will clarify that we mean

\left(\cdots((e_1e_2)e_3)\cdots e_{2^n-2}\right)e_{2^n-1}=-1.

We continue inductively be setting \mathbb{A}_{n+1}=\mathbb{A}_n\oplus\mathbb{A}_n and defining product and conjugation as before (and componentwise addition).  One can establish the following table of properties

\begin{array}{|c|c|c|c|c|}\hline\mathbb{R}&\mathbb{C}&\mathbb{H}&\mathbb{O}&\mathbb{S}\\\hline\mbox{division algebra}&\mbox{division algebra}&\mbox{division algebra}&\mbox{division algebra}&\phantom{normed}\\\mbox{associative}&\mbox{associative}&\mbox{associative}&&\\\mbox{commutative}&\mbox{commutative}&&&\\\mbox{trivial conjugation}&&&&\\\end{array}

where associativity and commutativity refer to the multiplication (the addition is always both).  All of the above properties (save the division algebra structure) can at this point (albeit with some tediousness) be shown.  If a=(a_0,...,a_{2^n-1})\in\mathbb{A}_n, we can define its real part as

\mbox{Re}\,a=a_0.

Note for \mathbb{R} we have

a^*a=aa=a^2.

For a\in\mathbb{R} we can define \|a\|=\sqrt{\mbox{Re}\,a^*a}=\sqrt{a^2}=|a|.  For \mathbb{C} we have

(a,b)^*(a,b)=(a,-b)(a,b)=(a^2+b^2,ab-ba)=(a^2+b^2,0).

Similarly we can define \|(a,b)\|=\sqrt{\mbox{Re}\,(a,b)^*(a,b)}=\sqrt{a^2+b^2}.  For \mathbb{H} we have

(a,b)^*(a,b)=(a^*,-b)(a,b)=(a^*a+b^*b,a^*b-a^*b)=(\|a\|^2+\|b\|^2,0).

Hence \|(a,b,c,d)\|=\sqrt{a^2+b^2+c^2+d^2}.  And similarly for \mathbb{O} we have

(a,b)^*(a,b)=(a^*,-b)(a,b)=(\|a\|^2+\|b\|^2,0),

so \|(a_0,...,a_7)\|=\sqrt{a_0^2+\cdots+a_7^2}.  Inductively, one can see that the norm on \mathbb{A}_n coincides with the euclidean norm in \mathbb{R}^{2^n}.  We can also note that the multiplicative inverse of an element x\in\mathbb{A}_n is

x^{-1}=\frac{x^*}{\|x\|^2}.

Although it turns out that in the sedenions and above (\mathbb{A}_i for i\geq 4), there are zero divisors (for example, (e_3+e_{10})(e_6-e_{15})=0), so the algebra fails to be a division algebra.

Theorem 1.  (Hurwitz’ Theorem)  If A is a real normed division algebra with identity, then A\in\{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}.

Corollary 2.  (Frobenius’ Theorem)  If A is an associative real division algebra, then A\in\{\mathbb{R},\mathbb{C},\mathbb{H}\}.

Generalized Cross Products:
The cross product of two elements x,y\in\mathbb{R}^3 can be written

x\times y=\begin{vmatrix}e_1&e_2&e_3\\x_1&x_2&x_3\\y_1&y_2&y_3\end{vmatrix}=(x_2y_3-x_3y_2,x_3y_1-x_1y_3,x_1y_2-x_2y_1).

There is a natural way to define a cross product of n-1 elements in \mathbb{R}^n by

\mathsf{X}(v_1,...,v_{n-1})=\begin{vmatrix}e_1&\cdots&e_n\\v_{1_1}&\cdots&v_{1_n}\\\vdots&\ddots&\vdots\\v_{{n-1}_1}&\cdots&v_{{n-1}_n}\end{vmatrix}.

However, our ultimate objective is to define vorticity in other dimensions.  So we will want a binary cross product that we can apply to our differential operator and a vector field: \nabla\times f.  We want \times to satisfy some conditions

\begin{array}{rcl}(w+x)\times(y+z)&=&(w+x)\times y+(w+x)\times z=w\times y+x\times y+w\times z+x\times z\\ 0&=&x\cdot(x\times y)=(x\times y)\cdot y\\\|x\times y\|^2&=&\|x\|^2\|y\|^2-(x\cdot y)^2.\end{array}

These conditions are respectively called bilinearity, orthogonality, and magnitude.  Hence we want \times to be the product in a real normed algebra.  Moreover magnitude tells us that we want x\times x=0.  We also have e_i^2=-1 in our Cayley-Dickson algebras.  It turns out that the cross product in \mathbb{R}^3 can be modeled in \mathbb{H} as follows.  Define C:\mathbb{R}^3\to\mathbb{H} by

C(x_1,x_2,x_3)=x_1i+x_2j+x_3k

and C^{-1}:\mathbb{H}\to\mathbb{R}^3 by

C^{-1}(a+bi+cj+dk)=(b,c,d).

Then one can verify that for x,y\in\mathbb{R}^3

x\times y=C^{-1}\left(C(x)C(y)\right).

Note that in \mathbb{R}, the cross product is trivial r\times s=0 for all r,s\in\mathbb{R}.  This is obvious if we require it to satisfy orthogonality—where the dot product is just multiplication.  So the zero-product property implies the cross product must be 0.  It is compatible with our map as well (in this case C:\mathbb{R}\to\mathbb{C}):

r\times s=C^{-1}(C(r)C(s))=\mbox{Im}\,(ri)(si)=\mbox{Im}\,(-rs)=0.

Hence we are left with one remaining cross product: the cross product in \mathbb{R}^7 where, hereafter, C:\mathbb{R}^7\to\mathbb{O} is defined by

C(x_1,...,x_7)=x_1e_1+\cdots+x_7e_7

and C^{-1}:\mathbb{O}\to\mathbb{R}^7 is the imaginary map

C^{-1}(a_0e_0+\cdots+a_7e_7)=(a_1,...,a_7).

The cross product of x,y\in\mathbb{R}^7 is defined by

x\times y=C^{-1}(C(x)C(y)).

We will now define

C(\nabla)=e_1\frac{\partial}{\partial x_1}+\cdots+e_7\frac{\partial}{\partial x_7}.

If f:\mathbb{R}^7\to\mathbb{R}^7 is differentiable, then we define its curl by

\mbox{curl}\,f=\nabla\times f=C^{-1}(C(\nabla)C(f)).

Proposed Changes to Secondary Math and Language Arts Curriculum

It’s no controversy that education appears to be under par in comparison to what we might hope.  I claim this has not only to do with home/environmental conditions for individual students and the ever omnipresent media (and lately, social networking media as well), but also with the curriculum in the classes they take–particularly math and language arts courses.  What seems to be the case is that the math courses focus on topics that, while enhancing the mental discipline of students, fail to teach concepts that are readily applicable to everyday situations.  Language arts courses emphasize composition (and in particular, stress the ability to make effective arguments) under the premise that students are already well-versed in the grammatical aspects of language (an assumption so far from the truth that the very idea of expecting them to write cogent argumentative essays is ludicrous).

Math:

I’d recommend keeping the middle school curriculum essentially the same (introducing key concepts of elementary algebra and geometry), but I’d vehemently oppose the idea of continuing this sort of curriculum for another four years.  In 9th grade, students should be introduced to key concepts from set theory (set notations and operations, functions, equivalence relations, bijections and counting principles, and concepts from elementary number theory, including divisibility, prime numbers, and the fundamental theorem of arithmetic).  With this collection of concepts in their inventory, they can spend 10th grade covering concepts from logic and proof writing–and applying them to concepts they learned the previous year (as well as everyday phenomena).

11th grade could then be devoted to a more serious algebra course–covering concepts of operations more thoroughly than the 9th grade course, identities, concepts like associativity and commutativity, matrices, inverses, basic definitions and examples of groups, rings, and fields, polynomials, and comparisons between integers, rationals, reals, and complex numbers, and the fundamental theorem of algebra.  If students only took these courses and earned a C in them, I’d still argue they were better prepared for life than earning an A in the traditional courses.

An additional 12th grade class covering geometry more seriously could entail concepts of lengths, triangles, trigonometry, polygons, area of polygons, polytopes and hypervolume, axiomatic Euclidean, spherical, and hyperbolic geometry and basic properties of spherical and hyperbolic triangles, approximation of areas/volumes and method of exhaustion (leading up to a freshman course in real analysis (to replace calculus)).

Language Arts:

I’d recommend middle school curriculum to focus a little more on grammar and sentence structure.  A 9th grade course could solidify this with a rigorous treatment of the core concepts of language: parts of speech, morphology, syntax, and grammar.  Concepts like word families, clauses (independent, dependent, subject, and predicate), transitive and intransitive verbs, and sentence diagramming should be emphasized.

A 10th grade course emphasizing argumentative writing would appropriately accompany the 10th grade math curriculum in logic and proof writing.  I’d recommend abstaining from requiring students to read nontechnical works until the third year of language arts.  This way, rather than getting distracted by a fiction or nonfiction story as a platform for an argument, emphasis is instead placed on the concept of argumentative writing itself.  Smaller readings should be used in this course.

An 11th grade course could then more effectively do in one year what four years of language arts typically try to do.  This course could focus on work-based and research-based extended essay writing.

Curriculum comparable to the AP English Literature course could make an appropriate optional fourth year class.