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I'm interested in dream mathematics (https://ncatlab.org/nlab/show/dream+mathematics) as a foundation of "synthetic measure theory" in a similar vein as synthetic differential geometry, but I'm unclear on its relationship to ordinary mathematics, particularly as expressed through topos theory.

As far as I understand it, any model $V$ of ZFC with enough large cardinals allows for the construction of a model $W$ for dream mathematics by repeatedly applying the "definable subset" functor to the reals, as described here: https://en.wikipedia.org/wiki/L(R). This structure fulfills the axioms of ZF as well as determinacy, and thus has to be a topos.

I'd like to use internal reasoning within the topos of $W$ to derive results and export them to $V$ with some caveat like "these results hold for all definable subsets".

However, first I'd like to understand the relationship between $V$ and $W$ better. In particular, it seems that the embedding functor from $W$ to $V$ is neither a geometric nor a logical morphism of topoi.

Furthermore, I don't understand how to get from the topos structure of $V$ to that of $W$ since they have the same subobject classifier but different power object functors. For this to make any sense, the number of functions from the real numbers to the subobject classifier in $W$ has to be a subset of the functions in $V$, so the embedding of $W$ into $V$ cannot be full.

My questions are:

  1. Is there a clear or clearer description of which functions in $V$ make it over into $W$?

  2. Are there any transfer rules between $V$ and $W$ like those between models of synthetic differential geometry and normal differential geometry? Alternatively, are there any classes of preserved sentences, similar to the situation between nonstandard analysis and normal analysis?

  3. More generally, what is the categorical relationship between $V$ and $W$?

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    $\begingroup$ Of course the embedding is faithful. How could different functions in $W$ become the same function in $V$? The embedding is not full. $\endgroup$
    – Zhen Lin
    Commented Feb 2, 2023 at 12:41
  • $\begingroup$ @ZhenLin Yes, thanks, I inadvertently switched up the words. $\endgroup$ Commented Feb 2, 2023 at 12:44
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    $\begingroup$ One cannot produce a model of determinacy just from a model of ZFC, since the consistency strength of ZF+AD is far higher than ZFC. One needs infinitely many Woodin cardinals to build models of AD. $\endgroup$ Commented Feb 2, 2023 at 13:34
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    $\begingroup$ I see that you have edited, but your claims are still not exactly correct. The situation is more subtle than you describe. To get AD in L(ℝ) as an outright consequence of large cardinals, you need more than just infinitely many Woodin cardinals, although with infinitely many Woodin cardinals then one can construct a model of ZF+DC+AD in a forcing extension of the universe. Furthermore, to build Solovay's model of the dream theory, you need at least one inaccessible cardinal. $\endgroup$ Commented Feb 2, 2023 at 13:43
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    $\begingroup$ Solovay's construction uses an inaccessible to get a model of ZF+DC + every set is Lebesgue measurable, and Shelah proved that the inaccessible is necessary. If there are enough large cardinals (proper class of Woodin cardinals is enough), then L(ℝ) has all the properties you claim, and that is the model that set theorists usually work with. To get merely some model of ZF+DC+AD, then infinitely many Woodin cardinals suffices. $\endgroup$ Commented Feb 2, 2023 at 13:54

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