Kevin > Recent Status Updates

Ch11:Limits and Continuity of Functions;
This chapter is essentially the epsilon-delta definition of a limit introduced in Calculus I courses. The definition of continuity proceeds: A function takes an argument and produces a value. If, at every point within an interval, the function value has a limit which agrees when the argument is approached from "above or below", then the function is continuous on the interval.

Ch10:Limits and Continuity;
The definition of continuity given by Cantor and Dedekind is closer to the "absence of separateness" than the presence of some other properties. The naturals and rationals have separation between terms while the series of all reals does not. Also, the concept of a limit is not quantitative, but ordinal.(If there exists a finite number that cannot be succeeded, then an upper limit exists.)

Ch9:Infinite Series and Ordinals;
In addition to cardinal numbers that describe the size of a progression, there are ordinal numbers describing the order type of a well-ordered series("one in which every sub-class has a first term.") "An ordinal number means the relation-number of a well-ordered series." Ordinal and cardinal numbers obey their own set of arithmetic laws (commutative, associative,...)

Ch8:Infinite Cardinal Numbers;
It's not absolute that infinite collections exist. To assume so is the "axiom of infinity". But, there is "no logical reason against infinite collections" as mathematical induction shows that for any natural number n, one can choose n+1. Assigning a number to the size of the collection is called its cardinality. The natural numbers has cardinality Aleph_0 ;and all reals is 2^(Aleph_0)

Ch7:Rational, Real, and Complex Numbers;
With the definition of relations in hand, the theory of number types can be examined. The real rational numbers can be constructed with any ratio of the inductive numbers defined in chapters 2 and 3. Complex numbers arise in a similar, though not obvious, manner. This discussion seems to allude to the notion of algebraic numbers, but Russell doesn't call them algebraic.

Ch6:Similarity of Relations;
"...what matters in mathematics is not the intrinsic nature of our terms, but the logical nature of their interrelations."
When building definitions for our basis of mathematics, we must look for "likeness" (one-one relations and their correlates). Likeness, or "similarity", gives rise to structure. If I'm not mistaken, these terms are akin to isomorphs in modern parlance.

Ch4:Kinds of Relations;
All relations can be derived from "one-many" relations, i.e. "if x has relation in question to y, there is no other term x' which also has the relation to y."
One-one relations can be seen by conjunction of this with the converse: if y has relation to x, there is no y' with same relation to x.
Some definitions:"Referent" is the term from the relation; "relatum" is the term to which it goes.

Ch3: The Definition of Order;
"The order lies, not in the class of terms, but in a relation among the members of the class."

"mathematical induction is a definition, not a principle."

Ch3: Finitude and Mathematical Induction;
The three primitive axioms given by Peano are (1) "0" is a number (2) The successor of any number is a number (3) No two numbers have the same successor. This lays the groundwork for math induction provided that the series being generalized is of a finite quantity of terms. A process is inductive if it follows the hereditary property of axiom (2) and does not violate (3).

Ch2: Definition of Number;
Defining number is often seen as defining plurality, but this is not general enough to provide a basis. However, defining number as a class does. Classes group together items that share a set of characteristics that are peculiar to that set. The number 1 is the class containing all single objects, and the number 2 is the class of all couples. So, number is merely a definition of classes.

Ch1: The Series of Natural Numbers;
It can be quite difficult to explain simple concepts, often taken to be self-evident, such as arithmetic (and the basis for number sense). Yet, in order to have a deep understanding of higher mathematics, the capability of proving these simple concepts is necessary. We must start from a basis of definitions (axioms given by observation) and endeavor to minimize abstraction.

Giving this another shot. Hoping to work through this before school resumes.