Explicit constructions of models

 

Explicit constructions of models[edit]

We shall not prove that any models of the axioms are isomorphic. Such a proof can be found in any number of modern analysis or set theory textbooks. We will sketch the basic definitions and properties of a number of constructions, however, because each of these is important for both mathematical and historical reasons. The first three, due to Georg Cantor/Charles Méray, Richard Dedekind/Joseph Bertrand and Karl Weierstrass all occurred within a few years of each other. Each has advantages and disadvantages. A major motivation in all three cases was the instruction of mathematics students.

Construction from Cauchy sequences[edit]

A standard procedure to force all Cauchy sequences in a metric space to converge is adding new points to the metric space in a process called completion.

${\displaystyle \mathbb{�}}$ is defined as the completion of Q with respect to the metric |x-y|, as will be detailed below (for completions of Q with respect to other metrics, see p-adic numbers.)

Let R be the set of Cauchy sequences of rational numbers. That is, sequences

x₁, x₂, x₃,...

of rational numbers such that for every rational ε > 0, there exists an integer N such that for all natural numbers m,n > N, |x_m − x_n| < ε. Here the vertical bars denote the absolute value.

Cauchy sequences (x_n) and (y_n) can be added and multiplied as follows:

(x_n) + (y_n) = (x_n + y_n)

(x_n) × (y_n) = (x_n × y_n).

Two Cauchy sequences are called equivalent if and only if the difference between them tends to zero. This defines an equivalence relation that is compatible with the operations defined above, and the set R of all equivalence classes can be shown to satisfy all axioms of the real numbers. We can embed Q into R by identifying the rational number r with the equivalence class of the sequence (r,r,r, …).

Comparison between real numbers is obtained by defining the following comparison between Cauchy sequences: (x_n) ≥ (y_n) if and only if x is equivalent to y or there exists an integer N such that x_n ≥ y_n for all n > N.

By construction, every real number x is represented by a Cauchy sequence of rational numbers. This representation is far from unique; every rational sequence that converges to x is a representation of x. This reflects the observation that one can often use different sequences to approximate the same real number.^[5]

The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i.e. the least upper bound property. It can be proved as follows: Let S be a non-empty subset of R and U be an upper bound for S. Substituting a larger value if necessary, we may assume U is rational. Since S is non-empty, we can choose a rational number L such that L < s for some s in S. Now define sequences of rationals (u_n) and (l_n) as follows:

Set u₀ = U and l₀ = L.

For each n consider the number:

m_n = (u_n + l_n)/2

If m_n is an upper bound for S set:

u_n+1 = m_n and l_n+1 = l_n

Otherwise set:

l_n+1 = m_n and u_n+1 = u_n

This defines two Cauchy sequences of rationals, and so we have real numbers l = (l_n) and u = (u_n). It is easy to prove, by induction on n that:

u_n is an upper bound for S for all n

and:

l_n is never an upper bound for S for any n

Thus u is an upper bound for S. To see that it is a least upper bound, notice that the limit of (u_n − l_n) is 0, and so l = u. Now suppose b < u = l is a smaller upper bound for S. Since (l_n) is monotonic increasing it is easy to see that b < l_n for some n. But l_n is not an upper bound for S and so neither is b. Hence u is a least upper bound for S and ≤ is complete.

The usual decimal notation can be translated to Cauchy sequences in a natural way. For example, the notation π = 3.1415... means that π is the equivalence class of the Cauchy sequence (3, 3.1, 3.14, 3.141, 3.1415, ...). The equation 0.999... = 1 states that the sequences (0, 0.9, 0.99, 0.999,...) and (1, 1, 1, 1,...) are equivalent, i.e., their difference converges to 0.

An advantage of constructing R as the completion of Q is that this construction is not specific to one example; it is used for other metric spaces as well.

Construction by Dedekind cuts[edit]

Dedekind used his cut to construct the irrational, real numbers.

A Dedekind cut in an ordered field is a partition of it, (A, B), such that A is nonempty and closed downwards, B is nonempty and closed upwards, and A contains no greatest element. Real numbers can be constructed as Dedekind cuts of rational numbers.^[6][7]

For convenience we may take the lower set ${\displaystyle �}$ as the representative of any given Dedekind cut ${\displaystyle (�,�)}$, since ${\displaystyle �}$ completely determines ${\displaystyle �}$. By doing this we may think intuitively of a real number as being represented by the set of all smaller rational numbers. In more detail, a real number ${\displaystyle �}$ is any subset of the set ${\displaystyle \mathbf{\text{Q}}}$ of rational numbers that fulfills the following conditions:^[8]

1. ${\displaystyle �}$ is not empty
2. ${\displaystyle �\ne \mathbf{\text{Q}}}$
3. ${\displaystyle �}$ is closed downwards. In other words, for all ${\displaystyle �,�\in \mathbf{\text{Q}}}$ such that , if ${\displaystyle �\in �}$ then ${\displaystyle �\in �}$
4. ${\displaystyle �}$ contains no greatest element. In other words, there is no ${\displaystyle �\in �}$ such that for all ${\displaystyle �\in �}$, ${\displaystyle �\le �}$

• We form the set ${\displaystyle \mathbf{\text{R}}}$ of real numbers as the set of all Dedekind cuts ${\displaystyle �}$ of ${\displaystyle \mathbf{\text{Q}}}$, and define a total ordering on the real numbers as follows: ${\displaystyle �\le �\iff �\subseteq �}$
• We embed the rational numbers into the reals by identifying the rational number ${\displaystyle �}$ with the set of all smaller rational numbers .^[8] Since the rational numbers are dense, such a set can have no greatest element and thus fulfills the conditions for being a real number laid out above.
• Addition. ${\displaystyle �+�:=\{�+�:�\in �\wedge �\in �\}}$^[8]
• Subtraction. ${\displaystyle �-�:=\{�-�:�\in �\wedge �\in (\mathbf{\text{Q}}\setminus �)\}}$ where ${\displaystyle \mathbf{\text{Q}}\setminus �}$ denotes the relative complement of ${\displaystyle �}$ in ${\displaystyle \mathbf{\text{Q}}}$, ${\displaystyle \{�:�\in \mathbf{\text{Q}}\wedge �\notin �\}}$
• Negation is a special case of subtraction: 
• Defining multiplication is less straightforward.^[8]
  • if ${\displaystyle �,�\ge 0}$ then 
  • if either ${\displaystyle �}$ or ${\displaystyle �}$ is negative, we use the identities ${\displaystyle �\times �=-(�\times -�)=-(-�\times �)=(-�\times -�)}$ to convert ${\displaystyle �}$ and/or ${\displaystyle �}$ to positive numbers and then apply the definition above.
• We define division in a similar manner:
  • if  then ${\displaystyle �/�:=\{�/�:�\in �\wedge �\in (\mathbf{\text{Q}}\setminus �)\}}$
  • if either ${\displaystyle �}$ or ${\displaystyle �}$ is negative, we use the identities ${\displaystyle �/�=-(�/-�)=-(-�/�)=-�/-�}$ to convert ${\displaystyle �}$ to a non-negative number and/or ${\displaystyle �}$ to a positive number and then apply the definition above.
• Supremum. If a nonempty set ${\displaystyle �}$ of real numbers has any upper bound in ${\displaystyle \mathbf{\text{R}}}$, then it has a least upper bound in ${\displaystyle \mathbf{\text{R}}}$ that is equal to ${\displaystyle \bigcup �}$.^[8]

As an example of a Dedekind cut representing an irrational number, we may take the positive square root of 2. This can be defined by the set .^[9] It can be seen from the definitions above that ${\displaystyle �}$ is a real number, and that ${\displaystyle �\times �=2}$. However, neither claim is immediate. Showing that ${\displaystyle �}$ is real requires showing that ${\displaystyle �}$ has no greatest element, i.e. that for any positive rational ${\displaystyle �}$ with , there is a rational ${\displaystyle �}$ with  and  The choice ${\displaystyle �=\frac{2�+2}{�+2}}$ works. Then ${\displaystyle �\times �\le 2}$ but to show equality requires showing that if ${\displaystyle �}$ is any rational number with , then there is positive ${\displaystyle �}$ in ${\displaystyle �}$ with .

An advantage of this construction is that each real number corresponds to a unique cut. Furthermore, by relaxing the first two requirements of the definition of a cut, the extended real number system may be obtained by associating ${\displaystyle -\mathrm{\infty }}$ with the empty set and ${\displaystyle \mathrm{\infty }}$ with all of ${\displaystyle \mathbf{\text{Q}}}$.