AXIOMATIC FOUNDATIONS OF THE REAL NUMBER SYSTEM

axiomatic foundations of the real number system
the number system can be built up logically, starting from a basic set of axioms or ‘‘self-evident’’ truths, usually taken from experience, such as statements 1–9, page 2.
if we assume as given the natural numbers and the operations of addition and multiplication (although it is possible to start even further back with the concept of sets), we find that statements 1 through 6, page 2, with r as the set of natural numbers, hold, while 7 through 9 do not hold.
taking 7 and 8 as additional requirements, we introduce the numbers . . and 0. then
by taking 9 we introduce the rational numbers.
operations with these newly obtained numbers can be defined by adopting axioms 1 through 6,
where r is now the set of integers. these lead to proofs of statements and so on, which are usually taken for granted in elementary mathematics.
we can also introduce the concept of order or inequality for integers, and from these inequalities for rational numbers. for example, if a, b, c, d are positive integers, we define a=b > c=d if and only if ad > bc, with similar extensions to negative integers.
once we have the set of rational numbers and the rules of inequality concerning them, we can order them geometrically as points on the real axis, as already indicated. we can then show that there are points on the line which do not represent rational numbers (such as p2, , etc.). these irrational
numbers can be defined in various ways, one of which uses the idea of dedekind cuts (see problem 1.34). from this we can show that the usual rules of algebra apply to irrational numbers and that no further real numbers are possible.
point sets, intervals
a set of points (real numbers) located on the real axis is called a one-dimensional point set.
the set of points x such that a @ x @ b is called a closed interval and is denoted by. the seta < x < b is called an open interval, denoted by ?a b?. the sets a < x @ b and a @ x < b, denoted by and ½a b?, respectively, are called half open or half closed intervals.
the symbol x, which can represent any number or point of a set, is called a variable. the given numbers a or b are called constants.
letters were introduced to construct algebraic formulas around 1600. not long thereafter, thephilosopher-mathematician rene descartes suggested that the letters at the end of the alphabet be used
to represent variables and those at the beginning to represent constants. this was such a good idea thatit remains the custom.
example. the set of all x such that jxj < 4, i.e., 4 < x < 4, is represented by ? 4 4?, an open interval.
the set x > a can also be represented by a < x < 1. such a set is called an infinite or unbounded interval. similarly, epresents all real numbers x.
countability
a set is called countable or denumerable if its elements can be placed in 1-1 correspondence with thenatural numbers.
example. the even natural numbers 2 4 6 8 . . . is a countable set because of the 1-1 correspondence shown.
given set
natural numbers
a set is infinite if it can be placed in 1-1 correspondence with a subset of itself. an infinite set whichis countable is called countable infinite.
the set of rational numbers is countable infinite, while the set of irrational numbers or all real numbers is non-countably infinite (see problems 1.17 through 1.20).the number of elements in a set is called its cardinal number. aset which is countably infinite isassigned the cardinal number fo (the hebrew letter aleph-null). the set of real numbers (or any sets
which can be placed into 1-1 correspondence with this set) is given the cardinal number c, called the cardinality of the continuuum.
neighborhoods
the set of all points x such that jx aj < where > 0, is called a neighborhood of the point a.the set of all points x such that 0 < jx aj < in which x ¼ a is excluded, is called a deletingd neighborhood of a or an open ball of radius about a.
limit points
a limit point, point of accumulation, or cluster point of a set of numbers is a number l such thatevery deletingd neighborhood of l contains members of the set that is, no matter how small the radius of a ball about l there are points of the set within it. in other words for any > 0, however small, we canalways find a member x of the set which is not equal to l but which is such that jx lj < . byconsidering smaller and smaller values of we see that there must be infinitely many such values of x.
a finite set cannot have a limit point. an infinite set may or may not have a limit point. thus the natural numbers have no limit point while the set of rational numbers has infinitely many limit points.
a set containing all its limit points is called a closed set. the set of rational numbers is not a closed set since, for example, the limit point p2 is not a member of the set (problem 1.5). however, the set of all real numbers x such that 0 @ x @ 1 is a closed set.
bounds
if for all numbers x of a set there is a number m such that x @ m, the set is bounded above and m is called an upper bound. similarly if x a m, the set is bounded below and m is called a lower bound. iffor all x we have m @ x @ m, the set is called bounded.
if m is a number such that no member of the set is greater than m but there is at least one member which exceeds m for every > 0, then m is called the least upper bound (l.u.b.) of the set. similarly if no member of the set is smaller than m but at least one member is smaller than m ? for every then m is called the greatest lower bound (g.l.b.) of the set.