Re: Infinite Binary Strings: A Question


"Herman Jurjus" <hjurjus@xxxxxxxxx> wrote in message news:48ce2837$0$27211$ba620dc5@xxxxxxxxxxxxxxxxxxxxxx
leon street wrote:

Part of what I have in mind is that an aperiodic string is in
general, if not always, chaotic or unpredictable. By that I suppose I
should mean something specific to the effect that there is no significantly
shorter way to determine the k-th digit (for some arbitrary, perhaps large
k) than to compute the string up to the k-th digit, or something like that.
At any rate, the last digit computed so far flips erratically between 0 and
1 as the string is explicated. Where is the precision in such a concept?

The answer to that last question is: there is none, and none is needed.
Set theory shows you how we can get away with -not- providing a precise concept.
It could be that we have two models of ZFC, one model assesses some string as valid, and the other doesn't. And whenever that's the case, one model will also have a real number set different from the one in the other model.


Now hold on there. The question is whether it is "constructible" in ZFC, not whether it is "valid".

The OP is correct when he says that if the last bit flips "erratically" (which I will take to mean "randomly") then there is no basis for constructing a number in this manner in ZF alone, as you need to make a simultaneous choice between a countably infinite sets of {0,1}. Choice does however allow you to postulate arbitrary binary strings, even if we can't construct them.


I also have in mind an argument which I couldn't lay out in a
couple of paragraphs but I can summarize here. (I might set it out in a
separate thread if I could get more confident about it.) I think there is a
difficulty with the notion of an arbitray infinite binary string,
understood as a point in a uniform combinatorial space of 2^infinity
possibilities. I believe an infinite binary string has to come from
somewhere, has to have something that produces it.

You can only construct countably many infinite binary strings in ZFC. They are produced by the operations of power set, union, etc. The others you can't produce, even with the axiom of choice; they have nowhere to come from.



In say, ZFC/FOL, they are 'produced' by some (fictitious) model (of ZFC). They're -assumed- to be there, just like the Euclidean axioms assumed points and lines to be there.

But perhaps your question is -why- it's reasonable to assume this, or why it's so necessary or handy for mathematics to have this available?

--
Cheers,
Herman Jurjus

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