The

The first element is a set,

and the other two elements are functions from

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

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 "

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

I recommend starting with my second reading, which begins here.

of the

together with an ordered pair of functions from

that satisfy the seven properties

(5) field properties

(1) order property

(1) least upper bound property

** R** are called

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

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.

**[...]** **[but confusion is always possible if you're talking to me]****.**
For example,
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]****.**

See ita-2-1-4-pg16.