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Madeleine Birchfield (Sep 19 2024 at 21:04):In a Boolean topos and in a topos satisfying countable choice or , the Dedekind real numbers are the unique Cauchy complete Archimedean ordered field up to isomorphism and they coincide with the Cauchy real numbers. There exists other toposes where Dedekind real numbers do not coincide with the Cauchy real numbers. Are there any toposes where the Dedekind real numbers are the unique Cauchy complete Archimedean ordered field up to isomorphism, but do not coincide with the Cauchy real numbers?
Mike Shulman (Sep 19 2024 at 22:08):Is this property equivalent to the Dedekind reals coinciding with the Escardo-Simpson-HoTTBook reals?
Madeleine Birchfield (Sep 19 2024 at 22:19):Mike Shulman said:
Is this property equivalent to the Dedekind reals coinciding with the Escardo-Simpson-HoTTBook reals?
Yeah it would be.
The former is the terminal Cauchy complete Archimedean ordered field and the latter is the initial Cauchy complete Archimedean ordered field. Since every Archimedean ordered field is a subfield of the Dedekind reals whose canonical injection is given by the unique ring homomorphism into the Dedekind reals, the category of Cauchy complete Archimedean ordered fields and ring homomorphisms is a subcategory of the category of subsets of the Dedekind real numbers and so is thin. If a thin category has a zero object, then every object in the category is isomorphic to the zero object.
Mike Shulman (Sep 19 2024 at 22:44):I feel like the answer to your question is almost surely yes, but I don't know an example off the top of my head.
James E Hanson (Oct 04 2024 at 21:11):Is Lubarsky's model (the model of IZF he builds in section 6 of his paper with a symmetric submodel construction) the only known example of the Cauchy reals failing to be Cauchy-complete?