Question 1: Does there exist models of the Zermelo-Fraenkel set theory without the axiom of choice, but such that every indexed family of non-void sets whose index set has a well-orderable cardinal admits a choice function ?
Question 2: The same as question 1, with "a well-orderable cardinal" replaced by "a linearly ordered cardinal".
Gérard Lang



Existence of model of ZF without AC, but with many choice function
For the first question:


Yes. It is known that the axiom "Every well-orderable family of non-empty set has a choice function" implies $DC$ but not $DC\_{\aleph\_1}$.


You can find the proofs in Jech's "The Axiom of Choice" and in Felgner's "Models of ZF-Set Theory".


I am not sure about the second question, but I believe the answer should be "yes" as well.


In [Consequences of the Axiom of Choice](http://consequences.emich.edu/CONSEQ.HTM) you can find the following principles:


1. **Form 1** is the axiom of choice.
2. **Form 202** is the existence of a choice function for linearly ordered families of non-empty sets.
3. **Form 40** is the existence of a choice function for well-ordered families of non-empty sets.


Entering those three numbers in the table show that $1\implies 202\implies 40$ and neither implications is reversible. Although in the case of $202\implies 1$ this is in a weaker axioms system without regularity.




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**Edit:** I had a bit more time now, so I went to chase after the cited source for Form 202:



> 
> Truss, J. **The axiom of choice for linearly ordered families.** *Fund. Math.* \**99* (1978), no. 2, 133–139. ([MR480029](http://www.ams.org/mathscinet-getitem?mr=480029))
> 
> 
> 


Here is an excerpt from the review:



> 
> The axiom of choice for linearly ordered families asserts that any linearly ordered family of nonempty sets has a choice function. The author shows that in ZF this statement implies the full axiom of choice, while in FM (ZF without the axiom of foundation) it does not.
> 
> 
> 


So it seems that in ZF, requiring for linearly ordered families is enough to require the full axiom of choice.


