# Set Theory, Notes 1: Ordinals and Cardinals

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

**Axiom of Infinity**:

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

We can then define and iterate as before, whence by applying the axiom of infinity again we can obtain 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 iff

**Definition 1**. An ordinal is a **successor ordinal** iff A **limit ordinal** is an ordinal which is not a successor ordinal.

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

- and
- if is a limit ordinal and for all then

then

*Proof.* Since is a linear ordering on the ordinals, let be the least ordinal such that But then which is a contradiction. Hence

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 of ordinals where iff

**Definition 3.** Let be an nondecreasing sequence of ordinals and be a limit ordinal and 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 is **continuous** if for every limit ordinal in the indexing subclass we have

An example of a sequence which is not continuous may be one of the form where are both limit ordinals. So in this case

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

**Definition 4. (Ordinal Arithmetic)** We define

**Addition:**

**Multiplication:**

**Exponentiation:**

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

and

**Definition 5.** For a set we define its **cardinality**, denoted 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 then

*Proof.* For every define

Hence the mapping is a bijection between and

Hence in this context, Cantor’s theorem immediately follows:

**Proposition 7.** Let and Then

*Proof.* We have

Also if and are bijections, then we can define by and this is easily seen to be a bijection.

And if then we can define 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 and But

and

for any finite ordinal with Knuth notation

having raisings of Consider the following convention for defining countable infinite ordinals:

They are all called countable infinite ordinals since for any one of them To clarify when we are talking about cardinal numbers versus ordinal numbers, we will use aleph notation: From Cantor’s theorem above, we know that if then That is, the ordinal corresponding to must be greater than all of the countable ordinals above, otherwise its cardinality would be This necessitates the notion of **uncountable ordinals** and corresponding **uncountable cardinals**. We use subscripts to characterize these: etc.

**Definition 8.** An infinite cardinal is a **successor cardinal** iff is a successor ordinal, and it is a **limit cardinal** iff is a limit ordinal.

**Definition 9.** Let be a limit ordinal. An increasing –sequence with a limit ordinal is **cofinal** in if And if is an ordinal, then we define its **cofinality** as

It is easy to verify that iff is a successor ordinal. Also and for any finite ordinal

**Proposition 10.**

*Proof.* Let Then (in particular it is the smallest such ). Now if then certainly Now since is cofinal in a subsequence of indices is cofinal in (where cofinality can be chosen to be ). So

But then is cofinal in That is,

whence

**Definition 11.** An ordinal is **regular** if It is **singular** if it is not regular.

**Corollary 12.** If is a limit ordinal, then is a regular cardinal.

**Theorem 13.** If is an infinite cardinal, then

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

# Fractional Sobolev Spaces

Recall for a function for we define the **Fourier transform** of by

and the **inverse Fourier transform** by

If then a.e..

Now if and with we say is the **weak derivative** of provided

with and

The **Sobolev space** is then defined as the set of all functions such that for all These become Banach spaces under the norm

and Hilbert when whence we denote

For we can say that consists of such that

(using Fourier multipliers).

**Bessel Potential Spaces**:

We can generalize with and and define the **Bessel potential space** as all such that Note the fractional power of the Laplacian is defined by virtue of the fact that is dense in , so we say

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

**Sobolev-Slobodeckij Spaces:**

There is an alternative approach. Recall that the **Holder space** is defined as all functions such that

That is, it is the set of functions on which are and whose -th partial derivatives are bounded and Holder continuous of degree 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 with and by

The corresponding **Sobolev-Slobodeckij space** is defined as all functions such that

where This becomes a Banach space under the norm

[1] http://en.wikipedia.org/wiki/Sobolev_space (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 be a monoid. Then is a pointed set with point Let be a map satisfying

and

Such a map is called a **monoidal action**, and we correspondingly call an **-pointed set**. Note that since it is a pointed morphism. We will also write for short.

Suppose acts on and that is a differential pointed set (so ). Now define the pointed set

with point defined by for all Then this is a differential pointed set with differential defined by

since

for all so We then define the **cohomology** of with coefficients in as

where of course is the equivalence class of maps under the relation iff

Let be an -pointed set. Consider the two sets

where iff or for some nonzero (Note the “or” gives us symmetry). belongs to both of these (as a class of elements in the latter case such that (since is never true for nonzero and )). The point (or trivial class) in is called the **torsion subpointed set** of with respect to

**Proposition 1**. Let and the pointed natural numbers be -pointed monoids. Then

*Proof.* Let and and define be defined by (since ). The map is injective since if then for all but this just implies As is a monoid with one generator, any morphism is determined by the image of But such a map induces a preimage under And we have

So is a morphism.

We don’t however have that without some way of moving natural numbers between components of (i.e. a tensor product). We can define a **tensor product** of two -pointed sets and which are monoids, denoted as with the usual bilinear relations. We will denote simple elements by 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

*Proof.* Define by Suppose Then

Also if then since action for all we have a preimage And lastly so it is a morphism.

If the functors and are left (right) exact functors in the category of pointed sets and has an injective (projective) resolution, then we can define the **-pointed cohomology (homology)** via the right (left) derived functors ().

# 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 where denotes the trivial pointed set.

If and are pointed sets, then we define the intuitive **product pointed set** as the pointed set If and are also differential sets with differentials and then we define the **product differential** component-wise: This is clearly a differential on which we in turn call the **product differential set**.

If we then define the relation on where iff then we have Hence we have

We can similarly define a formal sum of pointed spaces whose corresponding homology is just defined as

So we have two forms of *additivity*. We also have *excision*, for if is a pair of sets with a sub-differential set of and 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 ). Let be a map between paired sets. We say this is a **paired set morphism** if If we furthermore have that these are **paired pointed sets** and with and sub-pointed sets, a paired set morphism between them is a **paired pointed morphism** if we also have that If both are differential sets as well, we can define a **paired differential morphism** if we further have Let be a paired differential morphism between such sets. Then we would like for its restriction 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 to be **homotopic mod** provided they agreed on

# Homology on Pointed Sets

Let be a pointed set (i.e. there is a nullary operation where ).

**Definition 1.** A pointed set is a **differential set** if there is an endomorphism (i.e. ) such that for all We say two elements are **homologous**, denoted if

**Proposition 2.** The homologous relation is an equivalence relation.

**Definition 3.** Elements of the equivalence class are called **cycles**. Elements of the set are called **boundaries**. The **homology** of is defined to be the set

Note that all boundaries are cycles, and so can have a pointed set structure with point Now let the quotient set be pointed as and note that can be pointed since

**Proposition 4.** as pointed sets.

*Proof.* Since the equivalence class it follows that the map is injective. This map is also clearly surjective since is defined on which is partitioned via And of course we also have

Now suppose we have a sequence of pointed sets and a sequence of pointed set homomorphisms such that for all We will denote such a collection by and call it a **pointed set complex**. We can also define the **th homology** of the complex as

where we have the intuitive equivalence relation on each set with We will also call a pointed set complex **exact** if for all

Let be a sub-differential set of a differential set That is, We will call a **normal** sub-differential set if for we have

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

**Theorem 6.** If is a normal sub-differential set of then there are homomorphisms such that

is an exact complex, where denotes the trivial pointed set .

*Proof.* Let then and is not the boundary of any element in (excluding ). Since let us just define This map is fine provided is not the boundary of any element in But this is not possible since is normal, so the map is well-defined.

Now let Then and is not the boundary of another element in We want to send it to a cycle in which is not the boundary of an element in We already have that can’t be the boundary of something in since it’s not the boundary of anything in So we will define by

The image points are of course boundaries since the final map sends everything to It follows that for 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 and Since the homology pointed sets will all have as their point, we will simply find the underlying sets. since is just the identity map, that is, For note that so we have that

Lastly we have and

So