Category Archives: Notes

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,


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


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


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


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

Definition 4.  (Ordinal Arithmetic)  We define


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


    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.


    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



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=|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


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.

Fractional Sobolev Spaces

Recall for a function f\in L^1(\Omega) for \Omega\subseteq\mathbb{R}^n we define the Fourier transform of f by

\displaystyle\hat{f}(x)=F(f)(x)=\int_\Omega e^{-2\pi ix\cdot\xi}f(\xi)\,d\xi

and the inverse Fourier transform by

\displaystyle \check{f}(x)=F^{-1}(f)(x)=\int_\Omega e^{2\pi i x\cdot\xi}f(\xi)\,d\xi.

If f,\hat{f}\in L^1(\Omega), then \check{\hat{f}}=\hat{\check{f}}=f a.e..

Now if f\in L^1 and D^\alpha (f)\in L^1 with \alpha\in\mathbb{Z}^n, we say D^\alpha f is the weak derivative of f provided

\displaystyle\int_\Omega f D^\alpha\phi\,dx=(-1)^{|\alpha|}\int_\Omega D^\alpha(f)\phi\,dx

with \phi\in C_C^\infty(\Omega), |\alpha|=\sum_i\alpha_i, and

\displaystyle D^\alpha\phi=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\cdots\partial x_n^{\alpha_n}}\phi.

The Sobolev space W^{k,p}(\Omega) is then defined as the set of all functions f\in L^p(\Omega) such that D^\alpha(f)\in L^p(\Omega) for all |\alpha|\leq k.  These become Banach spaces under the norm

\displaystyle\|f\|_{k,p}=\left(\sum_{|\alpha|\leq k}\int_\Omega \left|D^\alpha(f)\right|^p\,dx\right)^{1/p}

and Hilbert when p=2, whence we denote H^k(\Omega)=W^{k,2}(\Omega).

For 1<p<\infty we can say that W^{k,p}(\mathbb{R}^n) consists of f\in L^p(\mathbb{R}^n) such that

\displaystyle F^{-1}\circ(1+|2\pi x|^2)^{k/2}F\circ f=(1-\Delta)^{k/2}f(x)\in L^p(\mathbb{R}^n)

(using Fourier multipliers).

Bessel Potential Spaces:

We can generalize with r\in\mathbb{R} and 1<p<\infty and define the Bessel potential space W^{r,p}(\mathbb{R}^n) as all f\in L^p such that \|f\|_{r,p}:=\|(1-\Delta)^{r/2}f\|_p<\infty.  Note the fractional power of the Laplacian is defined by virtue of the fact that L^p\cap L^1 is dense in L^p, so we say

\displaystyle(1-\Delta)^{r/2}(f):=(F^{-1}\circ (1+|2\pi x|^2)\circ F)(f).

 These spaces satisfy our desire of being Banach (and Hilbert when p=2).

Sobolev-Slobodeckij Spaces:

There is an alternative approach.  Recall that the Holder space C^{k,\gamma}(\Omega) is defined as all functions f\in C^k(\Omega) such that

\displaystyle\|f\|_{C^{k,\gamma}}=\sum_{|\alpha|\leq k}\sup_{x\in\Omega}|D^\alpha|+\sum_{|\alpha|=k}\sup_{x,y\in\Omega}\frac{|D^\alpha f(x)-D^\alpha f(y)|}{|x-y|^\gamma}<\infty.

That is, it is the set of functions on \Omega which are C^k and whose k-th partial derivatives are bounded and Holder continuous of degree \gamma.  These spaces are Banach under the above norm.  We can generalize the Sobolev spaces to incorporate similar properties.  Let us define the Slobodeckij norm for f\in L^p(\Omega) with 1\leq p<\infty and \theta\in (0,1) by

\displaystyle [f]_{\theta,p}=\int_\Omega\int_\Omega\frac{|f(x)-f(y)|^p}{|x-y|^{\theta p+n}}\,dxdy.

The corresponding Sobolev-Slobodeckij space W^{s,p}(\Omega) is defined as all functions f\in W^{\lfloor s\rfloor,p}(\Omega) such that

\displaystyle\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}<\infty

where \theta=s-\lfloor s\rfloor\in(0,1).  This becomes a Banach space under the norm

\displaystyle\|f\|_{s,p}=\|f\|_{W^{\lfloor s\rfloor,p}}+\sup_{|\alpha|=\lfloor s\rfloor}[D^\alpha f]_{\theta,p}.

[1] (unclear text references)

[2]  Lieb, Elliot and Michael Loss.  Analysis.  2nd Edition.  Graduate Studies in Mathematics.  Vol. 14.  American Mathematical Society.  2001.

[3]  Evans, Lawrence.  Partial Differential Equations.  Graduate Studies in Mathematics.  Vol. 19.  American Mathematical Society.  1998.

Cohomology of Pointed Sets

Let M be a monoid.  Then M is a pointed set with point 1.  Let \varphi:(M,0)\times(X,x_0)\to (X,x_0) be a map satisfying

\varphi(1,x)=x, and


Such a map is called a monoidal action, and we correspondingly call X an M-pointed set.  Note that since \varphi(1,x_0)=x_0, it is a pointed morphism.  We will also write mx:=\varphi(m,x) for short.

Suppose M acts on X and that X is a differential pointed set (so d^2(x)=x_0).  Now define the pointed set

\displaystyle Hom_M(X,M)=\{f:X\to M:f(x_0)=1\}

with point 1:X\to M defined by 1(x)=1 for all x\in X.  Then this is a differential pointed set with differential defined by




for all x\in X, so d^2f=1.  We then define the cohomology of X with coefficients in M as


where of course [1] is the equivalence class of maps under the relation f\sim g iff df=dg=1.

Let X be an M-pointed set.  Consider the two sets

X^M=\{x\in X:mx=x\}


where x\sim y iff mx=y or x=my for some nonzero m\in M.  (Note the “or” gives us symmetry).  x_0 belongs to both of these (as a class of elements in the latter case such that mx=x_0 (since mx_0=x is never true for nonzero m and x\neq x_0)).  The point (or trivial class) in X_M is called the torsion subpointed set of X with respect to M.

Proposition 1.  Let X and the pointed natural numbers (\mathbb{N},0) be M-pointed monoids.  Then


Proof.  Let m\in M and n\in\mathbb{N} and define \phi:X^M\to Hom_M(\mathbb{N},X) be defined by m\phi(x)(n)=\sum_{i=1}^nx:=nx (since mx=x).  The map is injective since if \phi(x)=\phi(y), then nx=ny for all n, but this just implies x=y.  As \mathbb{N} is a monoid with one generator, any morphism \varphi:\mathbb{N}\to X is determined by the image of 1.  But such a map induces a preimage under \phi: \phi^{-1}(\varphi)=\varphi(1)\in X_M.  And we have


So \phi is a morphism.

We don’t however have that X_M=\mathbb{N}\times X without some way of moving natural numbers between components of \mathbb{N}\times X (i.e. a tensor product).  We can define a tensor product of two M-pointed sets X and Y which are monoids, denoted X\odot_M Y, as X\times Y/\sim with the usual bilinear relations.  We will denote simple elements by x\odot y.  It’s easy to verify that this is a monoid.  We then have the result:

Proposition 2.  Let the hypotheses of Proposition 1 hold, then

X_M=\mathbb{N}\odot_{\mathbb{N}[M]} X.

Proof.     Define \phi:\mathbb{N}\odot_M X\to X_M by \phi(n\odot x)=nx.  Suppose nx=n'y.  Then

\begin{array}{lcl}1\odot nx=1\odot n'y&\Rightarrow&1\odot n(1_Mx)=1\odot n'(1_My)\\&\Rightarrow&1\odot (n1_M)x=1\odot (n'1_M)y\\&\Rightarrow&n1_M\odot x=n'1_M\odot y\\&\Rightarrow&n\odot x=n'\odot y.\end{array}

Also if x\in X_M, then since action mx=x for all m\in M, we have a preimage \phi^{-1}(x)=1\odot x.  And lastly \phi(0,0_X)=00_X=0_X, so it is a morphism.

If the functors \cdot^M and \cdot_M are left (right) exact functors in the category of pointed sets and X has an injective (projective) resolution, then we can define the M-pointed cohomology (homology) via the right (left) derived functors D^n(F,M)=H^n(F(I)^M) (D_n(F,M)=H_n(F(P)_M)).

Homology on Pointed Sets 2

We wish to address four of the Eilenberg-Steenrod axioms for this category.  First note that for dimension we have H(x_0)=x_0 where x_0 denotes the trivial pointed set.

If (X,x_0) and (Y,y_0) are pointed sets, then we define the intuitive product pointed set as the pointed set (X\times Y,(x_0,y_0)).  If X and Y are also differential sets with differentials d_1 and d_2, then we define the product differential component-wise: d_1\times d_2 (x,y)=(d_1(x),d_2(y)).  This is clearly a differential on X\times Y, which we in turn call the product differential set.

If we then define the relation \sim on X\times Y where (x,y)\sim (x',y') iff (d_1(x),d_2(y))=(d_1(x'),d_2(y')), then we have [(x,y)]_{d_1\times d_2}=[x]_{d_1}\times [y]_{d_2}.  Hence we have

\begin{array}{lcl}H(X\times Y)&=&[(x,y)]-(d(X\times Y)-(x_0,y_0))\\&=&[x]\times[y]-\left((d_1(X),d_2(Y))-(x_0,y_0)\right)\\&=&\left([x]-(d_1(X)-\{x_0\}),[y]-(d_2(Y)-\{y_0\})\right)\\&=&H(X)\times H(Y).\end{array}

We can similarly define a formal sum of pointed spaces \sqcup_i(X_i,x_i) whose corresponding homology is just defined as


So we have two forms of additivity.  We also have excision, for if (X,A) is a pair of sets with A a sub-differential set of X and U\subseteq A, then


We lastly address homotopy and omit the long exact sequence as we have not developed a dimensional concept (although recall we constructed a short exact sequence under an assumption on A).  Let f:(X,A)\to (Y,B) be a map between paired sets.  We say this is a paired set morphism if f(A)\subseteq B.  If we furthermore have that these are paired pointed sets (X,A,x_0) and (Y,B,y_0) with A and B sub-pointed sets, a paired set morphism f between them is a paired pointed morphism if we also have that f(x_0)=y_0.  If both are differential sets as well, we can define a paired differential morphism if we further have f\circ d_1=d_2\circ f.  Let f be a paired differential morphism between such sets.  Then we would like for its restriction f:H(X,A)\to H(Y,B) to be well-defined.  Hopefully some combination of requirements like normality and being a paired differential morphism will work, but I have not yet convinced myself.  If it worked, we would in turn just define two maps f,g:(X,A)\to (Y,B) to be homotopic mod A provided they agreed on H(X,A).

Homology on Pointed Sets

Let (X,x_0) be a pointed set (i.e. there is a nullary operation x_0:\varnothing\to X where x_0(\,)=x_0).

Definition 1.  A pointed set is a differential set if there is an endomorphism d:X\to X (i.e. d(x_0)=x_0) such that d^2(x)=x_0 for all x\in X.  We say two elements x,y\in X are homologous, denoted x\sim y, if d(x)=d(y).

Proposition 2.  The homologous relation is an equivalence relation.

Definition 3.  Elements of the equivalence class [x_0] are called cycles.  Elements of the set d(X) are called boundaries.  The homology of X is defined to be the set


Note that all boundaries are cycles, and x_0\in H(X), so H(X) can have a pointed set structure with point x_0.  Now let the quotient set X/\sim be pointed as (X/\sim,[x_0]), and note that (d(X),x_0) can be pointed since d(x_0)=x_0.

Proposition 4.  (X/\sim,[x_0])\approx (d(X),x_0) as pointed sets.

Proof.     Since the equivalence class [x]=\{y:d(x)=d(y)\}, it follows that the map [x]\mapsto d(x) is injective.  This map is also clearly surjective since d is defined on X, which is partitioned via \sim.  And of course we also have [x_0]\mapsto x_0.

 Now suppose we have a sequence \{X_i,x_i\} of pointed sets and a sequence \{d_{i+1}:X_{i+1}\to X_i\} of pointed set homomorphisms such that d_i\circ d_{i+1}(x)=x_{i-1} for all x\in X_{i+1}.  We will denote such a collection by \{(X_i,x_i),d_i\} and call it a pointed set complex.  We can also define the ith homology of the complex as


where we have the intuitive equivalence relation on each set X_i with x\sim y\Leftrightarrow d_i(x)=d_i(y).  We will also call a pointed set complex exact if H_i(X)=(\{x_i\},x_i)=:0_i for all i.

Let (A,x_0) be a sub-differential set of a differential set (X,x_0).  That is, d(A)\subseteq A.  We will call A a normal sub-differential set if for x\in X we have d(x)\in A\Rightarrow x\in A.

Definition 5.  We define the relative homology of X with respect to A as the set


Theorem 6.  If A is a normal sub-differential set of X, then there are homomorphisms i,j such that

x_0\longrightarrow H(A)\stackrel{i}{\longrightarrow}H(X)\stackrel{j}{\longrightarrow}H(X,A)\to x_0

is an exact complex, where x_0 denotes the trivial pointed set (\{x_0\},x_0).

Proof.     Let x\in H(A), then x\in A, d(x)=x_0, and x is not the boundary of any element in A (excluding x_0).  Since x\in A, let us just define i(x)=x.  This map is fine provided x is not the boundary of any element in X-A.  But this is not possible since A is normal, so the map is well-defined.

Now let x\in H(X).  Then x\in X, d(x)=x_0, and x is not the boundary of another element in X.  We want to send it to a cycle in X-(A-\{x_0\}) which is not the boundary of an element in X-A.  We already have that x can’t be the boundary of something in X-A since it’s not the boundary of anything in X.  So we will define j by

j(x)=\left\{\begin{array}{lcl}x&\mbox{if}&x\in X-A\\x_0&\mbox{if}&x\in A\end{array}\right.

The image points are of course boundaries since the final map sends everything to x_0.  It follows that j\circ i(x)=x_0 for x\in H(A), so we have a differential complex of homology pointed sets.  To prevent confusion, we will denote the homology sets of this complex of homology sets by H_2,H_1 and H_0.  Since the homology pointed sets will all have x_0 as their point, we will simply find the underlying sets.  H_2=x_0 since i is just the identity map, that is, [x_0]_i=\{x_0\}.  For H_1 note that [x_0]_j=H(A), so we have that


Lastly we have [x_0]_j=H(X,A) and