Re: Infinite Binary Strings: A Question

Peter Webb wrote:

"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".

It can be present in one model, and absent from another. Hence, the axioms of ZFC in itself do not uniquely determine one (extension of the) concept 'infinite binary string'. (And, as said, fortunately you don't need one, anyway.)

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, [...]

Sure, but that's not what i was talking about.

Choice does however allow you to postulate arbitrary binary strings, even if we can't construct them.

And what does that mean, exactly? How could you 'postulate arbitrary binary strings'? Does ZFC manage to do that? And in a unique and unambiguous way?

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.

Sure; my hunch was that that would not satisfy the OP, but now that you mention it, i could very well be wrong. Mr. Street?

--
Cheers,
Herman Jurjus

.

Relevant Pages

  • Re: Well Ordering the Reals
    ... Ross A. Finlayson wrote: ... >> if there's an injection from the binary strings and from them to R, ... >> which is also consistent within ZFC. ... Unless you use a well-ordering that applies to an uncountable set, ...
    (sci.math)
  • Re: lwalke3 wrote
    ... WITHOUT ZFC OR CANTOR using only binary strings! ...
    (sci.math)
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