The Iterative Conception of Set

1. What is a Set?
A set is a collection of elements, the members of the set. The elements can be anything, including other sets, except in a pure set theory, in which case they can only be sets. The nature of sets is determined by the principle of extensionality, which states that two sets are not distinct unless one has an element which is not in the other. (i.e. two sets are in fact the same set if and only if they have the same elements).

2. What sets are there?
The problem in set theory is not in knowing what a set is, but in knowing what sets there are (in the domain of discourse). This is a problem because the most natural assumption, that to every property of sets expressible in first order logic there corresponds a set whose elements are just those sets having that property, turns out to be inconsistent. (Russell's paradox demonstrates this).

There is no single answer to this question, there are many different interpretations of set theory. However, mathematics is normally conducted in the context of a set theory in which the sets are the pure well-founded collections and this domain is what is described by "the iterative conception of set".

3. A simple inductive description
A simple way one might hope to describe the iterative heirarchy is as follows:
  1. There is a unique set which has no members, the empty set..
  2. For any collection of sets there is a set containing just those sets.
  3. There are no sets which cannot be shown to be sets by application of rules 1 and 2.
The main reasons this doesn't really work are that:
  • there are too many possible sets to expect to be able to demonstrate their existence individually
  • and we want to allow sets with infinitely many members
  • the term "collection" which we used is arguably a surrogate for set, making the definition covertly circular
Another way of putting a similar idea is that the universe of sets is the smallest collection which contains the empty set and is closed under the operation of forming arbitrary new sets all of whose members are in the collection This gets over a similar idea without the suggestion that we can have a proof of the existence of each set.

4. The Best I Can Do
  1. We think of first order set theory, and we are attempting to define the intended or standard interpretation or interpretations of the language of set theory.
  2. It is not certain that there can be any single interpretation which quite does the trick, we might have to settle for a never-ending sequence of ever-closer approximations to the correct interpretation.
  3. An interpretation of set theory is formed in stages (which need not be construed as temporal, but must be well-ordered).
  4. At the first stage (V1) there is just the empty set.
  5. At each subsequent stage (Vn+1) there are all the sets which can be formed from sets available in the previous stages in the construction.
  6. Subsequent "stages" include limit points, where there is no immediately preceding stage but a new stage is formed to follow an infinite set of preceding stages. At the limit the universe consists of the union of all the previous stages.

5. Residual Problems
There are two main problems.
  1. What does "all" mean.
  2. How many stages are there.
The answer to problem 1 is pretty much that either you think this has a definite meaning and you are happy with the iterative conception of set, or you don't and you aren't.

The answer to problem 2 is: at least enough to make a model for the theory you want to work with (e.g. ZFC) but as many as you like after that.

6. The definition of truth
Instead of using the standard definition of truth, which I take to be the concept of validity, the following slight modification can be used.

A statement converges if after some point in the sequence of standard interpretations its truth value is always the same, and it converges to that value. A statement is true if it converges to true and false if it converges to false. Otherwise it has no truth value. This is only relevant at the judgement level, and doesn't affect the standard way in which the semantics of the propositional connectives is rendered. i.e. the logic is still two valued.

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