**Go backward to** 5.1 Proof of the Uncountability of the Reals

**Go up to** 5 More About the Specification BasicReal

**Go forward to** 5.3 The Specification of the Reals in Higher-Order
CASL

## 5.2 The Categoricity of the Reals

A specification is said to be categorical or monomorphic, if it has
exactly one model up to isomorphism. Neither the specification
BasicReal
nor the specifications of the reals in ZFC or HOL are categorical. Due
to the theorem of Löwenheim and Skolem, a categorical axiomatization
is impossible^{6}. However, it is possible to prove the *internal*
categoricity of the specification of the reals in ZFC (meaning that
within each model of ZFC plus two axiomatizations of the reals, the
two reals are isomorphic). A similar result should hold for the reals
in HOL (but we are not entirely sure).
BasicReal
is definitely too weak for such a result,
since there is not even an internalized notion
of functions on reals.

CoFI
Note: M-7 -- Version: 0.2 -- 13 April 1999.

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