Homotopy Type Theory Dedekind real numbers > history (Rev #7, changes)

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Definitions

Axiomatic Locally definiton𝒰\mathcal{U}-small Dedekind real numbers

The Dedekind real numbers is aDedekind complete Archimedean ordered fieldlocally 𝒰\mathcal{U}-small Dedekind real numbers for a universe 𝒰\mathcal{U} is defined as the Archimedean ordered integral domain 𝒰\mathbb{R}_\mathcal{U} with a strictly monotonic function? i:𝕀 𝒰 𝒰i:\mathbb{I}_\mathcal{U} \to \mathbb{R}_\mathcal{U} from the locally 𝒰\mathcal{U}-small Dedekind real unit interval to 𝒰\mathbb{R}_\mathcal{U} such that i(0)=0i(0) = 0 and i(1)=1i(1) = 1.

Large Dedekind real numbers

The 𝒰\mathcal{U}-large Dedekind real numbers for a universe 𝒰\mathcal{U} is defined as the type of 𝒰\mathcal{U}-Dedekind cuts on the rational numbers \mathbb{Q} in a universe: 𝒰DedekindCut 𝒰()\mathbb{R}_\mathcal{U} \coloneqq DedekindCut_\mathcal{U}(\mathbb{Q}).

Sigma-Dedekind real numbers

The Σ\Sigma-Dedekind real numbers for a $\sigma$-frame Σ\Sigma is defined as the type of Σ\Sigma-Dedekind cuts on the rational numbers \mathbb{Q}: ΣDedekindCut Σ()\mathbb{R}_\Sigma \coloneqq DedekindCut_\Sigma(\mathbb{Q}).

See also

References

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