Logic serves to organize the content of a subject and axiomatization serves to put
us at a vantage point to begin our odyssey of logical reasoning.
The axioms for probability theory [Boro, p.16, Definition 4] serves to find a common denominator for all types of probability problems.
We would like to reduce the amount of assumptions in an axiomatic system to the minimum (Compare [Boro, p.25, l.−11] with [Lin, p.161, l.10]).
The axiomatic interpretation [Lin, p.55, Theorem 5] of p.d.f. is less meaningful than its geometric interpretation [Lin, p.201, Fig. 7-2 & p.199, l.14].
[Lan8] shows that both the theory of real numbers and the theory of complex numbers can be founded
on the
five Peano axioms alone [Lan8, p.2]. Remark.
The scheme of axiomatization
If statement A' can be proved by a group of statements {A}, we say that statements {A} is more basic than statement A'.
As we search for more basic statements, we discover that there are certain
statements that can not be proved. We call them axioms. On the one hand, the
number of axioms should be as few as possible. On the other hand, the axioms
should be inclusive in the sense that every theorem in the theory of real
numbers can be proved by them. The goal of axiomatization of the real
number system serves to find those axioms that characterize the features of
the real number system and can be used to develop the entire theory of real
numbers. We first define
the operation of addition on natural numbers, then ordering can be defined
in terms of this operation [Lan8, p.9, Definition 2]. We use the the same method
to prove similar theorems. The theory development of fractions follows the
pattern of the theory development of natural numbers. For
addition, [Lan8, Theorems 21, 22, and 23] correspond to [Lam8, Theorems 54, 65,
and 66] respectively. For multiplication, [Lan8, Theorems 29, 30, 31, 32, 33,
34, 35, and 36] correspond to [Lam8, Theorems 69, 71, 70, 72, 73. 74, 75, and
76] respectively. In the theory of fractions, we do not prove
[Lan8, p.21, Theorem 41] by directly using the axiom of mathematical induction. Instead,
we transplant the essential content of [Lan8, p.9, Theorem 10] to the theory of
fractions. In fact, [Lan8, p.9, Theorem 10] is a copy (or direct consequence) of [Lan8, p.7, Theorem 9]
whose proof uses the axiom of mathematical induction. In other words, we restrict the
direct applications of the axiom of mathematical induction to the domain of
natural numbers to which the axiom belongs. Notice that even when we discuss the
ordering of the natural numbers [Lan8, chap. I, §3],
we do not directly use the mathematical induction to prove theorems except for
its equivalent: [Lan8, p.13, Theorem 27].
In order to avoid unnecessarily complicating a theory, we should use the
axiom of mathematical induction as infrequently as possible. The proof of [Lan8, p.13,
Theorem 27] and the proof of [Jaco, vol.1, p.9, Theorem O4] use the axiom of mathematical
induction only once. In contrast, Long unnecessarily uses the axiom of
mathematical induction too many times when he proves I1ÞI3
[Lon, p.20, l.-12-p.21, l.21].
Another
drawback of Long's proof is that the scope of the reduction to absurdity that he
uses in his proof is unnecessarily broad [Lon, p.21, l.14-l.21].¬