Re: Cantorian pseudomathematics

Han.deBruijn@xxxxxxxxxxxxxx wrote:
> cbrown@xxxxxxxxxxxxxxxxx didn't write:
>
> > those who lay the greatest stress on mathematical rigor are just the ones
> > who, lacking a sure sense of the real world, tie their arguments to unrealistic
> > premises and thus destroy their relevance.
>
> No. That's just another quote from Jaynes' book. And, Chas, that's
> precisely what you are doing.

Do you think that my premise that a random sequence not be generatable
by a non-random process is "unrealistic"?

Do you think my definition of "non-random" as "computable" is
unrealistic?

Could you identify which of my premises is "unrealistic"?

>
> > Jesse F. Hughes wrote:
> > > cbrown@xxxxxxxxxxxxxxxxx writes:
> > >
> > > > What is your definition of "x is selected from set X at random"?
> > >
> > > What is *your* definition of that?
> >
> > I've been thinking a bit about that; and I suppose the information
> > theory tinged ideas best suit my fancy.
>
> [ ... much snipped ... ]
>

Equals any and all content.

> > Now, I admit there's a few conceptual bumps here.
>
> A "few"? You're kidding.
>

Could you identify a specific problem you percieve?

> > I can see how to prove that a given sequence is /not/ fair and random;
> > but I don't see that that helps me prove that some sequence actually
> > does have the property.
> >
> > At this point, I don't even have a proof that such a sequence
> > neccessarily exists for finite X (although it seems inuitively obvious,
> > countability argument, waves hands in air).
>
> Here are three decimal sequences. Tell me if they are "fair and
> random":
>
> a) 1910484810053706146806749192781911979399520614196634287544
>
> b) 4121497099935831413222665927505592755799950501152782060571
>
> c) 4415557863469637694468527889283375102029398103480215402734
>

These are finite sequences. As in my example "is 3 a random number?", I
specified that a random sequence is an infinite sequence on some set X,
so no finite sequence can be random. I would be surprised if Jaynes
thought otherwise.

> d) Are the prime numbers "fair and random" ?
>

As a selection from what set? In any case, no: I can calculate the
prime numbers, in order. So the prime numbers are a computable
sequence. Theferefore, they are not random sequence. I would be
surprised if Jaynes thought otherwise.

> e) Does the Collatz sequence generate "fair and random" numbers?

What is "the" Collatz sequence? For any natural n, the Collatz sequence
starting with n is computable; therfore no it is not random sequence by
my definition. I would be surprised if Jaynes thought otherwise.

>
> Serious. You should read chapter 10 from Jaynes' book:
>
> http://omega.math.albany.edu:8008/JaynesBook.html
>

You have given me little incentive to.

Cheers - Chas

.

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