# 5.4 Nonstandard Models

Nonstandard models arise when a first order theory admits models that satisfy all its axioms but differ from the intended or standard interpretation, and their existence follows naturally from the compactness theorem and the Lowenheim Skolem theorems.

### Definition 5.22 (Standard Model)

A standard model is a structure that is intended to represent the usual interpretation of a theory.

For example, the standard model of arithmetic is:
$$
\mathbb{N} = \{0,1,2,\dots\}
$$
with the usual operations and relations.

A nonstandard model is any model of the same theory that is not isomorphic to the standard one.

### Construction via Compactness

We illustrate the construction of a nonstandard model of arithmetic.

Let $\mathcal{L}$ be the language of arithmetic, and extend it by adding a new constant symbol $c$.

Consider the set of sentences:
$$
\Gamma = \text{Th}(\mathbb{N}) \cup \{c > 0, c > 1, c > 2, \dots\}
$$

Here $\text{Th}(\mathbb{N})$ denotes all true sentences of arithmetic in the standard model.

Each finite subset of $\Gamma$ contains only finitely many inequalities:
$$
c > 0, c > 1, \dots, c > n
$$

Such a finite set can be satisfied in $\mathbb{N}$ by interpreting $c$ as a number greater than $n$.

Thus every finite subset of $\Gamma$ is satisfiable.

By compactness, $\Gamma$ has a model $\mathcal{M}$.

In $\mathcal{M}$, the element interpreting $c$ is greater than every standard natural number, and therefore $\mathcal{M}$ cannot be isomorphic to $\mathbb{N}$.

This gives a nonstandard model of arithmetic.

### Definition 5.23 (Nonstandard Element)

In a nonstandard model of arithmetic, an element is called nonstandard if it is greater than every element corresponding to a standard natural number.

Such elements behave like infinite numbers from the external point of view.

### Structure of Nonstandard Models

Nonstandard models of arithmetic have a rich internal structure.

They contain an initial segment that is isomorphic to the standard natural numbers:
$$
0,1,2,\dots
$$

Beyond this initial segment, there are nonstandard elements that extend the order indefinitely.

Between standard and nonstandard elements, there are no gaps, and the ordering behaves like a discrete linear order with additional infinite parts.

### Example 5.24 (Infinite Integers)

In a nonstandard model, one can consider an element $c$ such that:
$$
c > n
$$
for every standard natural number $n$.

This element behaves like an infinite integer.

One can also form expressions such as:
$$
c + 1, \quad c + 2, \quad 2c, \quad c^2
$$

All of these are also nonstandard elements, and they satisfy the same algebraic laws as ordinary natural numbers.

### Skolem Paradox Revisited

Nonstandard models help clarify the Skolem paradox.

A theory may prove the existence of uncountable sets, but still have a countable model.

Inside the model, there is no bijection between certain sets and the natural numbers, so those sets are considered uncountable within the model.

From the outside, however, the entire model is countable.

Thus the notion of countability depends on the perspective from which the structure is viewed.

### Transfer of First Order Properties

A key feature of nonstandard models is that they satisfy exactly the same first order sentences as the standard model, provided the theory is complete.

Thus any first order statement that is true in the standard model is also true in every nonstandard model, and vice versa.

This means that nonstandard models cannot be distinguished from the standard model by first order properties alone.

### Example 5.25 (Failure of Induction Beyond Standard Part)

Although induction holds in nonstandard models for all first order definable properties, there exist subsets of the domain that are not definable and for which induction may fail.

This shows that induction in first order logic applies only to definable properties, not to arbitrary subsets.

### Corollary 5.26 (Existence of Nonstandard Models)

If a first order theory has an infinite model, then it has a nonstandard model.

This follows from the Lowenheim Skolem theorem and compactness, since one can construct models of different sizes or extend the theory with additional elements that cannot be realized in the standard model.

### Conceptual Meaning

Nonstandard models demonstrate that first order logic cannot uniquely characterize many mathematical structures, because there are always alternative models that satisfy the same axioms but differ in important ways.

They also provide useful tools in mathematics, such as nonstandard analysis, where infinitesimal and infinite elements are introduced to study limits and continuity in a new way.

The existence of nonstandard models is therefore both a limitation and a source of new mathematical ideas, showing that logical systems often admit richer interpretations than initially expected.