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universal relation (Definition)

If $ \Phi$ is a class of $ n$-ary relations with $ \vec{x}$ as the only free variables, an $ n+1$-ary formula $ \psi$ is universal for $ \Phi$ if for any $ \phi\in\Phi$ there is some $ e$ such that $ \psi(e,\vec{x})\leftrightarrow\phi(\vec{x})$. In other words, $ \psi$ can simulate any element of $ \Phi$.

Similarly, if $ \Phi$ is a class of function of $ \vec{x}$, a formula $ \psi$ is universal for $ \Phi$ if for any $ \phi\in\Phi$ there is some $ e$ such that $ \psi(e,\vec{x})=\phi(\vec{x})$.



"universal relation" is owned by Henry.
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Other names:  universal
Also defines:  universal function

Attachments:
universal relations exist for each level of the arithmetical hierarchy (Theorem) by Henry
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Cross-references: function, formula, free variables, relations, class
There are 8 references to this entry.

This is version 3 of universal relation, born on 2002-08-23, modified 2002-08-24.
Object id is 3342, canonical name is UniversalFormula.
Accessed 8333 times total.

Classification:
AMS MSC: 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)
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