In [these slides of a talk](http://www.giovannicuri.com/Talks/Slides_Kanazawa2010.pdf) Giovanni Curi shows that the generalized uniformity principle follows from Troesltra’s uniformity principle and from the subcountability of all sets, which are both claimed to be consistent with CZF. Subcountability’s consistency with CZF is not surprising in light of counterintuitive results like that subsets of finite sets aren’t necessarily finite, but it seems to have a different flavor.


What are the intuitions or motivations for subcountability?


What references prove that subcountability is consistent with CZF?



Subcountability
An intuition for ESC (every set is subcountable, i.e., a subquotient of the natural numbers) in a predicative framework is that everything is built up from below starting with natural numbers, so we may assume that every set can be represented as a set of codes (natural numbers) quotiented out by an equivalence relation (denoting equality of whatever the codes represent).


For the consistency of CZF + ESC (indeed, CZF + REA + ESC), see Michael Rathjen's *Choice principles in constructive and classical set theories*, Thm. 8.3:
[http://www1.maths.leeds.ac.uk/~rathjen/acend.pdf](http://www1.maths.leeds.ac.uk/%7Erathjen/acend.pdf)


