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| − | Informally, the "real numbers" are the [[rational number]]s with all
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| − | the holes plugged.
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| − | Before we proceed with any formalism, let's exhibit an example of a
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| − | "hole" in the rational numbers. Take, for our "hole", the square root
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| − | of 2. Suppose <math>x^2 = 2</math>. If x is a rational number then
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| − | we can express it as a fraction, and what's more, we can express it as
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| − | a ''reduced'' fraction. Let '''I''' = the set of [[integer]]s. Then:
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| − | :(1) <math>x = {n \over m},\ \ n,m \in \mathbb{I}</math>
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| − | and
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| − | :(2) <math>\not\exists p,q,k \in \mathbb{I} \left (k > 1 \ \and \ n = p \cdot k \ \and \ m = q \cdot k \right )</math>
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| − | Then we must have
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| − | :(3) <math> {\left ({ n \over m} \right ) }^2 = 2</math>
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| − | :(4) <math> {{n^2} \over {m^2}} = 2</math>
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| − | :(5) <math> n^2 = 2 \cdot m^2</math>
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| − | And so <math>n^2</math> must be ''even''. But since the square of any ''odd'' integer is also ''odd'', we see that '''n''' must ''also'' be even.
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| − | If n is even, then clearly <math>n^2</math> must be divisible by
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| − | ''4''. In that case, n<sup>2</sup>/2 must also be even,
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| − | and since we have
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| − | :(6) <math> {{n^2} \over 2} = m^2</math>
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| − | we see that <math>m^2</math> is even also. By the same argument,
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| − | therefore, m must be even. But if both n and m are even, we must be
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| − | able to find p and q such that,
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| − | :(7) <math>p,q \in \mathbb{I}\ (n = 2 \cdot p \ \and \ m = 2 \cdot q)</math>
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| − | But this contradicts (2), and we conclude that <math>\sqrt 2</math>
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| − | must not be expressible as a rational number.
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| − | The real numbers add the ''continuum axiom'' to the axioms satisfied by
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| − | the rational numbers:
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| − | :(A.1) '''Any nonempty subset of the real numbers which is bounded above has a least upper bound.'''
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| − | There are actually several equivalent ways to state the continuum
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| − | axiom. The statement I've just given, though perhaps not maximally
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| − | intuitive, is convenient for use in the construction of the real
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| − | numbers.
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| − | We've just shown, above, that the rational numbers
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| − | do ''not'' satisfy the continuum axiom. For consider the set:
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| − | :<math>S \equiv \{x \ | \ x^2 < 2\}</math>
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| − | This is certainly nonempty: <math>1 \in S</math>. It's certainly bounded
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| − | above: 3 is larger than any element of S. But the least upper bound
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| − | of the set is <math>\sqrt 2</math> which is not a rational number.
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| − | Very well, so the rationals don't satisfy (A.1) -- but how do we
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| − | know ''any'' set will satisfy it? How do we know the real numbers
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| − | exist? We will construct them ... or, rather, we will sketch the
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| − | construction of the reals; the details actually fill a slim book (see
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| − | references).
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| − | We will start with the rational numbers, which we call Q. We first
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| − | define a ''cut'' (also called a "Dedekind cut"). If a nonempty subset
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| − | of the rationals, Γ, is a ''cut'', then it divides the rational
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| − | numbers in two:
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| − | :(C.1) <math>x \in \Gamma \ \and \ y < x \Rightarrow y \in \Gamma</math>
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| − | :(C.2) <math>x \not\in \Gamma \ \and \ y > x \Rightarrow y \not\in \Gamma</math>
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| − | :(C.3) <math>(y \not\in \Gamma \Rightarrow y \geq x) \Rightarrow x \in \Gamma</math>
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| − | Property (C.1) says a cut contains a continuous block of numbers, extending to the "left". Property (C.2) says that the numbers that are ''not'' in the cut are also a continuous block, extending to the ''right''.
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| − | Property (C.3) says that, if a cut has a rational least upper bound,
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| − | then that bound is a member of the cut set.
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| − | We observe immediately that each rational number, q, corresponds to a
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| − | cut:
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| − | :<math>\Gamma(q) \equiv \{ x \in Q \ | \ x \leq q \}</math>
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| − | From here on, we will use rational numbers interchangeably with the
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| − | cuts corresponding to them. In particular, we'll use "0", "1", and
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| − | "-1" to refer to Γ(0), Γ(1), and Γ(-1), as the need
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| − | arises.
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| − | At this point, we observe that the set '''S''', defined above, is also a
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| − | cut, and it corresponds to <math>\sqrt 2</math>. If we view the
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| − | rational numbers as being contained in the set of cuts, via the
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| − | correspondence <math>q \leftrightarrow \Gamma(q)</math>, then the
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| − | set of cuts must therefore be a ''proper'' superset of the rational
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| − | numbers.
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| − | The set of all cuts will be our model of the real numbers. But we're
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| − | not done yet: we still need to define comparisons, and we need to
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| − | define multiplication and addition.
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| − | Comparisons are easy. It's simpler to define <math>\leq</math> than
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| − | <math> < </math>, so that's what we'll do. For two cuts, G and H,
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| − | :<math>G \leq H \equiv x \in G \ \Rightarrow \ x \in H</math>
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| − | Addition is easy, too.
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| − | :<math> G + H \equiv \{x+y\ | x \in G\ \and\ y \in H\}</math>
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| − | Multiplication is a bit trickier. The problem is that all our cuts
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| − | have "tails" extending arbitrarily far to the left, and if we just
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| − | multiply all the members of two cuts we'll get something that has a
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| − | tail extending arbitrarily far to the ''right''. So we need to be
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| − | cleverer than that.
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| − | We can define the product of two ''non-negative'' cuts:
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| − | :<math>G,H \geq 0 \Rightarrow G \cdot H = \{x\ |\ \exists y \in G, z \in H (y \geq 0 \ \and \ z \geq 0 \ \and \ x \leq yz)\}</math>
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| − | Now, before we define the product of two ''general'' cuts, we need a few
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| − | "helper definitions". The first of these is ''negation'':
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| − | :<math>-G = \{x\ |\ \forall y \in G (x \leq -y) \}</math>
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| − | We can define absolute value in terms of negation:
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| − | :<math>|G| = \begin{cases} G, & \mbox{if } G \geq 0 \\ -G, & \mbox{if } G < 0 \end{cases}</math>
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| − | We'll define one "helper function":
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| − | :<math>Sgn(G) \equiv \begin{cases}1, & \mbox{if } G \ \geq 0 \\ -1, & \mbox{if } G < 0 \end{cases} </math>
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| − | and
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| − | :<math>Sgn(G,H) \equiv Sgn(G) \cdot Sgn(H)</math>
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| − | We also need to define multiplication of a cut by ''-1'':
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| − | :<math>-1 \cdot G \equiv -G</math>
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| − | And finally, we can define multiplication of two general cuts:
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| − | :<math>G \cdot H \equiv Sgn(G,H) \cdot (|G| \cdot |H|)</math>
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| − | This completes the creation of the model. We have defined comparisons,
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| − | addition, and multiplication. The rationals are embedded in the new
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| − | model, and the comparison operation is clearly an extension of the
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| − | comparison operation for the rationals. The definitions for addition
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| − | and multiplication are pretty clearly the appropriate extensions from
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| − | the rationals but rigorous proofs of those claims, as well as the claim that the model satisfies the axioms of the reals (the [[field axioms]]) would take more work than what I have displayed here.
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| − | The continuum axiom, as I stated it here, just describes a cut of the
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| − | rationals. In our model for the real numbers, the least upper bound
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| − | of such a set is the cut itself. So the model does indeed satisfy the
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| − | continuum axiom.
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| − | '''The Axiom of Choice'''
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| − | It's worth calling some attention to one of the steps I glossed over.
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| − | I rather casually said we would form the set of all cuts of the
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| − | rationals. That's actually an enormous set, and it's formed as a
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| − | subset of an equally enormous set, which is the [[power set]] of the
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| − | rational numbers. It actually requires an [[uncountably infinite]]
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| − | number of operations to form the set of all cuts. The assertion that
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| − | we can form that set depends on the [[axiom of choice]]. This leads
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| − | to the real numbers having some rather peculiar properties, and not
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| − | everyone feels it's entirely legitimate.
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| − | It is possible to develop a version of analysis that doesn't depend on
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| − | the axiom of choice, based on what are called the [[constructible real]]s.
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| − | -----------
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| − | References:
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| − | See Edmund Landau, Foundations of Analysis, for a thorough exposition
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| − | of the construction of the real numbers. I've glossed over a lot and
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| − | left out a great deal on this page. Landau does it "right". If you
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| − | can find it in the library, that's the way to go -- it's well worth
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| − | reading once, but it's not a book you'll refer to a lot.
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| − | --[[User:sal|SAL]] 03:39, 19 Nov 2004 (UTC)
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