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I just today realized that the concept of ordinal definability is defined in a different way by vopenka-Balcar-Hajek ``The notion of effective sets and a new proof of the consistency of the axiom of choice'' (see the end of this question for the definition).

As they stated, their definition is equivalent to the one given by Myhill-Scott.

Unfortunately the definition is not enough clear to me, so my first question is the following:

Question 1. Can someone give in more details the definition of effective sets as given in the paper.

Question 2. In the note, they talk about a natural number p, whose existence seems interesting. How to prove the existence of such a p.

One can describe another model-class HEf and to describe uniquely a certain well-ordering of it. The class HEf is called the class of hereditarily effective sets. First, we shall define the class of effective sets. Roughly speaking, it is the closure of the class of transfinite power sets of the empty set with respect to the fundamental Gddelian operations. It seems to be interesting that natural number p can be found such that every effective set is obtained after at most p steps. Hereditarily effective sets are effective sets whose elements, the elements of whose elements, etc. are effective, too. The definition of the class HEf remarkable by the following fact: As soon as one describes uniquely some model-class and a certain well-ordering of it then one can prove that it is a subclass of HEf. Thus we can say that the class HEf is the maximal definable model-class with a definable well-ordering. The notion of effectivity is generalized to the notion of effectivity with respect to a class and some analogous theorems are proved. The notion of effectivity enables us to formulate some natural axioms of effectivity. (Received June 6, 1967.)

REMARK. In July, 1967, the third author was informed at the Los Angeles Summer Institute on the set theory about Myhill's and Scott's results concerning ordinal definable sets. The notions of ordinal definable and effective sets coincide and some results of Myhill and Scott and of ours are deeply analogous. Of course, their results had not been knov .i to the authors of the present communication.

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Regarding question 1, I interpret the phrase

the class of transfinite power sets of the empty set

to refer to the sets $V_\alpha$ appearing in the cumulative hierarchy. On this reading, the phrase

the closure of the class of transfinite power sets of the empty set with respect to the fundamental Gödelian operations

would seem to mean that $x$ is effective, if it can be obtained via the Gödel operations from parameters of the form $V_\alpha$.

Regarding question 2, the existence of the uniform bound $p$ on the number of steps required is a consequence of the fact that there is a definable Ord-enumeration of the effective sets.

That is, since the effective sets are all ordinal definable, it follows that every effective set $x$ is the $\alpha^{th}$ ordinal-definable set, and furthermore, by reflection, there is some $V_\theta$ such that $V_\theta$ thinks $x$ is the $\alpha^{th}$ set in OD. Since $\alpha$ is definable from $V_\alpha$, we can use parameter $V_\alpha$ in place of $\alpha$ in defining $x$ inside $V_\theta$.

Thus, starting with parameters $V_\theta$ and $V_\alpha$, we can run the definition inside $V_\theta$ and thereby construct $x$. Since the definition relativized to $V_\theta$ becomes expressible via the Gödel operations, the number of such operations is fixed, since we are always using the same definition, and only changing the parameters $V_\alpha$ and $V_\theta$.

So there is a uniform bound $p$ on the number of steps required.

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  • $\begingroup$ Thanks a lot for your nice and informative answer. $\endgroup$ – Mohammad Golshani Dec 18 '18 at 5:27

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