I have a question (or two) about recursive naming conventions. Consider the following recursive naming sequence:
Base step: Let S be any nonempty set. Let x be any arbitrary element of S. Let S* be {x}.
Recursive step: (i) Let 'x' now designate any element of S that is not an element of S* (ii) Let 'S**' designate the set theoretic union of S* and {x}. (recall that, from (i), 'x' no longer refers to the same thing as it did in the base step) (iii) Let 'S*' now designate S** (iv) repeat (i-iii) if and only if S* is not identical to S.
If S's cardinality is finite, then the recursion terminates and it makes perfect sense to talk about S* whenever you want to talk about S. That seems obvious.
But here's my question: suppose S's cardinality is uncountably infinite (it's the set of real numbers or something). Under this supposition, does it still make sense to talk about S* whenever you want to talk about S? Or is there a problem with having uncountably many iterations?
Another question: same question as above, except now we suppose that S's cardinality is countably infinite.
(i.e., the point of the recursive step is to "recover" S by changing what is meant by 'x' and 'S*' uncountably many times. Don't worry about WHY I want to do this; I'm just curious as to whether this works. I realize that I could just talk about S whenever I want to talk about S, but that's beside the point)
I suppose another way to frame the question is this... does the following loop make sense: (i) do something. (ii) do something else. (iii) repeat (i) and (ii) as many times as there are real numbers.
I hope these are easy questions. Thanks!
D
To talk about a construction that has "as many times as there are real numbers" you need to talk about ordinals.
You can carry out constructions on ordinals as long as you tell us what do to, given we have already taken care of smaller cases.
Usually this is split into 3 cases:
The $0$ stage, you tell us where to start.
Sucessor stages, a stage of the form $t+1$ for some stage $t$ (this is what your thing does, and, if you stopped here, you would have a normal inductive construction).
Limit stages, (non-zero) stages not of the form $t+1$. For instance you need to tell us what to do at the $\omega$ stage, after you have done your construction for the finite ordinals.
A very famous example is the construction of the cumulative hierarchy: $ V_0 = \varnothing$, $V_{\alpha+1} = \mathcal{P}(V_\alpha)$, $V_\beta = \bigcup_{\alpha< \beta}V_\alpha$ for $\beta$ a limit ordinal. Another example is the construction of $L$.
If you want something less set theoretic, Let $B_0$ be the collection of open subsets of $\mathbb{R}$. Let $B_{\alpha+1}$ be the collection of all countable unions, and complements of members of $B_\alpha$, and at limit stages $B_\beta$ take the union of all the previous things. You can see that every element of $B_\alpha$ is a Borel subset of $\mathbb{R}$, however you only get all the Borel sets when you have done $\omega_1$ iterations ($\omega_1$ is the first uncountable ordinal).