Pure Identity Theories
The signature of the pure identity theory is empty, with no functions, constants, or relations.
Pure identity theory has no (non-logical) axioms. It is decidable.
One of the few interesting properties that can be stated in the language of pure identity theory is that of being infinite. This is given by an infinite set of axioms stating there are at least 2 elements, there are at least 3 elements, and so on:
- ∃x1 ∃x2 ¬x1 = x2, ∃x1 ∃x2 ∃x3 ¬x1 = x2 ∧ ¬x1 = x3 ∧ ¬x2 = x3,...
These axioms define the theory of an infinite set.
The opposite property of being finite cannot be stated in first-order logic for any theory that has arbitrarily large finite models: in fact any such theory has infinite models by the compactness theorem. In general if a property can be stated by a finite number of sentences of first-order logic then the opposite property can also be stated in first-order logic, but if a property needs an infinite number of sentences then its opposite property cannot be stated in first-order logic.
Any statement of pure identity theory is equivalent to either σ(N) or to ¬σ(N) for some finite subset N of the non-negative integers, where σ(N) is the statement that the number of elements is in N. It is even possible to describe all possible theories in this language as follows. Any theory is either the theory of all sets of cardinality in N for some finite subset N of the non-negative integers, or the theory of all sets whose cardinality is not in N, for some finite or infinite subset N of the non-negative integers. (There are no theories whose models are exactly sets of cardinality N if N is an infinite subset of the integers.) The complete theories are the theories of sets of cardinality n for some finite n, and the theory of infinite sets.
One special case of this is the inconsistent theory defined by the axiom ∃x ¬x = x. It is a perfectly good theory with many good properties: it is complete, decidable, finitely axiomatizable, and so on. The only problem is that it has no models at all. By Gödel's completeness theorem, it is the only theory (for any given language) with no models.
Read more about this topic: List Of First-order Theories
Famous quotes containing the words pure, identity and/or theories:
“It seldom happens that any felicity comes so pure as not to be tempered and allayed by some mixture of sorrow.”
—Miguel De Cervantes (1547–1616)
“For the mother who has opted to stay home, the question remains: Having perfected her role as a caretaker, can she abdicate control to less practiced individuals? Having put all her identity eggs in one basket, can she hand over the basket freely? Having put aside her own ambitions, can she resist imposing them on her children? And having set one example, can she teach another?”
—Melinda M. Marshall (20th century)
“A work of art that contains theories is like an object on which the price tag has been left.”
—Marcel Proust (1871–1922)