The Real Number System
Bare Bones Tentative Definition:
The real number system is a set of three elements.
The first element is a set, R (the numbers),
and the other two elements are functions from R x R to R,
which define addition and multiplication.
In symbols, this may be expressed by the rather daunting looking expression shown below. In ita-2-4-5-pg29, Rosenlict presents the real number system as described here - as a set of three elements. However, in ita-2-1-1-pg16, Rosenlicht puts the two functions into an ordered pair. For our purposes, the order of the functions does not seem to matter.
However, at this point, we don't know what is in the set R. For all we know R might be the null set. So let us assume that R is the null set. (In other words there are no real numbers, and - presumably - all numbers are imaginary. We can leave until later the question of whether or not imaginary numbers exist.) What I want to do here is look at the logical consequences of assuming that .
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If , then we need to figure out what is, since this cartesian product appears in the set
which now represents the real number system. But the cartesian product of the null set with itself is the set of ordered pairs both elements of which are in the null set. Since there are no such ordered pairs, we get , and the real number system set becomes
From the definition, we had the result that a "function from X to Y" is a subset of the cartesian product of X and Y. Then, in this case, the functions must reduce to the null set, because - in the last equation - both the "X" and the "Y" are the null set, so their cartesian product is the null set, and the only subset of the null set is the null set. Then our real number system set reduces to
It is customary, when listing sets, not to repeat identical elements. Therefore, it would seem to be better to put the functions into an ordered pair, as Rosenlicht did (ita-2-1-1-pg16) when he first introduced the real number system. We would then write
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This is my first reading of Rosenlicht's treatment of the real number system.
I recommend starting with my second reading, which begins here.
of the real number system:
We define the real number system to be a set R
together with an ordered pair of functions from RxR into R
that satisfy the seven properties
[linked below.] See ita-2-1-1-pg16.
(5) field properties :
(1) order property :
(1) least upper bound property :
The elements of R are called real numbers, or just numbers. The two functions are called addition and multiplication, and they make correspond to an element (a,b) in RxR specific elements of R that are denoted by a+b and axb respectively.
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The consequences of the seven properties that have been postulated are many and far reaching. Rosenlicht explores these consequences in detail, and he will show us that the set R is completely determined by the seven properties and two functions. But - at the moment - all that is obvious is that R contains at least two numbers, zero and one. Have another look at the seven properties, and - this time whenever you encounter a and b and c - think of the property as a conditional statement. The property is saying something that is true, if a,b etc are in R. For the logical exercise we are doing, keep in mind that - so far - we only have zero and one. And, concerning the two functions, we know only that they exist, and that they satisfy the requirements that are explicitly mentioned in the seven properties. The "usual" knowledge of arithmetic is yet to be derived.
The detailed discussion of the seven properties begins with the paragraph below and continues from the table of contents on the prerequisites page. It was desired, however, to state all the properties together without elaboration.
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Most of the rules of elementary algebra can be justified by these five field properties of the real number system. The main consequences of the field properties are given in paragraphs F1 through F10 [...] We shall employ the common notational conventions of elementary algebra when no confusion is possible
[but confusion is always possible if you're talking to me].
we often write ab for a·b.
One such convention is already implicit in the statement of the distributive property (Property III above), where the expression a · b + a · c is meaningless unless we know the order in which the various operations are to be performed, that is how parenthesis should be inserted;
by a · b + a · c we of course mean (a · b) + (a · c)
[, and the parentheses indicate that the multiplications are to be done before the addition, and - in general - parentheses are worked first, with inner parentheses before outer ones].