Some days ago, I posted a question about models of arithmetic and incompleteness. I then made a mixture of too many scattered ideas. Thinking again about such matters, I realize that what really annoyed me was the assertion by Ken Kunen that the circularity in the informal definition of natural number (what one gets starting from 0 by iterating the successor operation a finite number of times) is broken “by formalizing the properties of the order relation on ω” ( page 23 of his “The Foundations of Mathematics”). What does actually “breaking the circularity” mean? Is there a precise model theoretic statement that expresses this meaning? And what about proving that statement? Is that possible?
Looking at the draft that was linked above, it's more clear what Kunen means. He is just saying that the informal "definition" of the natural numbers that you might think of in school is circular when examined closely. And it is, in the sense that you have to start with some undefined concept, be it "natural number", "finite set", "proof", etc., to capture finiteness.
However, Kunen does not dwell on that sort of philosohical point. He is simply saying that there is a formal and non-circular definition of ω in set theory, as the smallest infinite ordinal. This does give a rigorous definition, but it doesn't ensure that "finite" in an aribitrary model corresponds to our actual notion of finite. That is something that cannot be ensured in first-order logic.
I don't have that book, but as far as I can understand, the “circularity” must mean this: in the phrase “iterating the successor operation a finite number of times”, we should mean a number of times corresponding to a natural number. But since the natural numbers are what we are defining, this is circular. So one has to define the natural numbers without reference to the concept of “finite”. Where the circularity is broken is if you rewrite your definition as follows:
(All this stated in more technical language, of course.)
There is no reference to the notion of “finite” here. Instead, number 3 above gives us, by definition, the principle of induction. For example, how do you show that some object X is not a natural number? Well, if for some property P, you can show P(0) and you can also show that ∀n: P(n)⇒P(n') where the prime denotes the successor function, but X fails property P, then you can know for sure that X is not a natural number.
Edit: I see I did not answer all your questions. I am not a logician, so take this with a grain of salt. But basically, in first order logic (in which ZFC is expressed) it is impossible to make circular definitions, and if you can't make one, you can't repair it. The circularity, as I see it, all exists on the meta-level, before you have even gotten around to formalizing the theory. So “breaking the circularity” must in essence happen in the transition between the informal and the formal.
Strictly speaking, first order theories don't even allow definitions at all! What you have to do is to notice that there is a complicated formula NN(x) that we interpret as “x is the set of natural numbers”, and a theorem ∃!x NN(x) in ZFC (where ∃! is short for “there exists a unique …”); then we create a new theory by adding the symbol ω and adding the axiom NN(ω). Now, any formula A(ω) in the new theory can be rewritten in the old theory as ∃x:NN(x)∧A(x), so nothing new has really happened, except for a great amount of simplification.
protected by François G. Dorais♦ Jul 15 at 15:12
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