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Concept
Disjunction and existence properties
Summary
In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories (Rathjen 2005).
The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem.
The existence property or witness property is satisfied by a theory if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other free variables, then there is some term t such that the theory proves A(t).
Rathjen (2005) lists five properties that a theory may possess. These include the disjunction property (DP), the existence property (EP), and three additional properties:
The numerical existence property (NEP) states that if the theory proves , where φ has no other free variables, then the theory proves for some Here is a term in representing the number n.
Church's rule (CR) states that if the theory proves then there is a natural number e such that, letting be the computable function with index e, the theory proves .
A variant of Church's rule, CR1, states that if the theory proves then there is a natural number e such that the theory proves is total and proves .
These properties can only be directly expressed for theories that have the ability to quantify over natural numbers and, for CR1, quantify over functions from to . In practice, one may say that a theory has one of these properties if a definitional extension of the theory has the property stated above (Rathjen 2005).
Almost by definition, a theory that accepts excluded middle while having independent statements does not have the disjunction property. So all classical theories expressing Robinson arithmetic do not have it. Most classical theories, such as Peano arithmetic and ZFC in turn do not validate the existence property either, e.g. because they validate the least number principle existence claim. But some classical theories, such as ZFC plus the axiom of constructibility, do have a weaker form of the existence property (Rathjen 2005).
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