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Here the mathematician Caicedo recommended an article in the book Mathematical Logic where Shoenfield describes that the universe of sets is devided into stages. The first stage contains nothing. The second stage contains every collection of objects of the first stage. The third stage contains every collection of objects of the stages 1 and 2 and so on ...

He also writes:

Since we wish to allow a set to be as arbitrary a collection as possible, we agree that there shall be such stage whenever possible, i.e., whenever we can visualize a situation in which all the stages in the collection are completed.

What does he mean with "whenever we can visualize a situation in which all the stages in the collection are completed"? Is there a precise description of what he means? (In this form, the concept of "set" seams to be too ambigious to be a part of mathematics because I pretend that mathematics is not a vague subject.) I don't want to here an exact definition of what he meant but just an explanation what he means with "whenever we can visualize a situation in which all the stages in the collection are completed". What is he meaning with "all the stages in the collection are completed"? What does he mean with "completed"?

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    $\begingroup$ I think he is describing the von Neumann universe. In particular, I think with the "situation in which all the stages in the collection are completed" he refers to the stages corresponding to limit ordinals. $\endgroup$ Commented Dec 19, 2015 at 16:52
  • $\begingroup$ I know. But that does not help me. I want to get an intuition of the universe of sets (see the link). Therefore I need a description of that universe that does not use set-theoretical terms because that would be circular. So I don't want to hear: Every stage corresponds to an ordinal number. $\endgroup$ Commented Dec 19, 2015 at 16:56
  • $\begingroup$ @asefeQE, you do not need the Von Neumann style ranking system to understand sets. You only need the axioms of ZFC. Sets are just things that satisfy those axioms. $\endgroup$ Commented Dec 19, 2015 at 17:08
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    $\begingroup$ The precise description of what Schoenfield means relates to ordinals, and arguably can't be explained without redefining that concept. I'm sorry that is unsatisfying to you. $\endgroup$ Commented Dec 19, 2015 at 17:11

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I don't want to here an exact definition of what he meant but just an explanation what he means with "whenever we can visualize a situation in which all the stages in the collection are completed".

As you mention in your question, the first few stages are as follows:

  • Stage $1$: nothing
  • Stage $2$: collections of objects from stage $1$.
  • Stage $3$: collections of objects from stages $1$ and $2$.
  • ...

Now, I can visualize the collection of all of the objects in any Stage $n$ (for some positive integer $n$). Therefore, by "we agree that there shall be such stage whenever possible", there is a stage with that collection; let's call it "Stage $A_1$". Stage $A_1$ is the stage corresponding to "when all stages $n$ (for $n$ a positive integer) are completed".

"Completed" here just means something like "those objects have been thrown into the collection of all sets".

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Since "we agree that there shall be such stage whenever possible", there are:

  • Stage "$1$ after $A_1$": collections of objects from stage $A_1$.
  • Stage "$2$ after $A_1$": collections of objects from stage $1$ after $A$.
  • ...

And then a Stage with all of those, perhaps stage $A_2$. Stage $A_2$ would be "a situation in which all the stages in the collection of $n$ after $A_1$ have been 'completed'."

Analogously, we might imagine a situation in which all stages $A_n$ have been 'completed', defining a new stage (Stage $B_{1_1}$?) as the collection of all of the objects in stages of the form $A_n$.

I hope this clarifies what he meant by "completed".

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  • $\begingroup$ Your answer helps me very much. This is exactly what I wanted to know. Thank you so much!!! $\endgroup$ Commented Dec 19, 2015 at 18:39

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