Re: does sqrt(2) exist in CM?
From: David C. Ullrich (ullrich_at_math.okstate.edu)
Date: 02/18/05
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Date: Fri, 18 Feb 2005 06:19:43 -0600
On 17 Feb 2005 10:06:29 -0800, examachine@gmail.com wrote:
>David C. Ullrich wrote:
>> On 16 Feb 2005 03:05:53 -0800, examachine@gmail.com wrote:
>>
>> >Ullrich,
>> >
>> >You don't understand what you are doing
>> >
>> >You accuse me of things that I haven't done:
>> >> >> On 13 Feb 2005 18:43:41 -0800, examachine@gmail.com wrote:
>> >> >> You've said that all the AIT definitions are equivalent,
>> >> >> and you've given one definition of "random real" as
>> >
>> >I wasn't even speaking of the AIT definitions. I doubt that you know
>> >what Martin-Lof randomness is, otherwise you would not be
>insistently
>> >claiming that "random real" is an AIT term! Which is precisely what
>is
>> >so annoying about your rambling.
>>
>> Guffaw. Have you tried counting the number of assertions you've
>> made here that turned out to be false, usually obviously false?
>> I think my favorite is the current one about Diophantine equations...
>
>Which is?
>
>> Ok, let's try to find a definition of "Solovay random". Searching
>> google should be easier than searching this thread, also more
>> reliable...
>>
>> Hmm. The first hit I got on "Solovay random" was this:
>>
>> http://www.umcs.maine.edu/~chaitin/ait10.html
>>
>> Unfortunately the definition of "Solovay random" there refers
>> back to previous pages without any links to them. One could
>> try to find the previous pages. No, wait: On the page above
>> Chaitin says
>>
>> " Now I'll present Solovay's proof that Solovay randomness is
>actually
>> equivalent to Martin-Löf randomness."
>>
>> So unless Chaitin is lying Solovay randomness is indeed equivalent
>> to Martin-Löf randomness. And _you_ gave a definition of Martin-Löf
>> in a previous post here. It was your definition of Martin-Löf
>> randomness that I used in the proof I gave that yes,
>> P(x is Solovay random) = 1.
>
>No, Chaitin isn't lying on that page.
>
>> So there we have it. Assuming (giggle) that the definitions you've
>> stated in this thread are correct then the proof I gave in answer
>> to your question about whether P(x is Solovay random) = 1 is
>> also precisely correct. And not only correct, but incredibly
>> obvious - if it's not obvious that P(x is Solovay random) = 1
>> it must not be obvious that a countable union of null sets
>> is a null set.
>
>Obvious, yes.
Giggle. If it's obvious then why did you say you were
experiencing difficulty with it?
No, I'm not putting words in your mouth. A quote from a few
posts up:
>>Secondly, regardless of previous discussions, I am experiencing a
>>"difficulty" with a concept which we were examining. I would be more
>>than glad if you could explain this. Now, assuming uniform dist. in
>>(0,1), I have a random variable X. What is the probability that it is a
>>Solovay random real? P(X : X a solovay real)? Is it "certain" that it's
>>going to turn out to be that?
(I replied with a simple proof that yes, the probability is 1.
You replied to that saying I'd become boring and didn't know
what I was talking about. Now it turns out that the obvious
proof I gave was precisely correct. Even though you had difficulty
with it...)
>> And if it's not obvious that a countable union of null
>> sets is a null set then we have _no_ understanding of
>> mathematical probability theory whatever - this is like
>> an awesomely basic result. So we conclude that you were
>> fibbing when you said you were familiar with the
>> sigma-algebra approach to probability (as though this
>> was not obvious already from your comments on how
>> probability = 1 implies certainty).
>
>I can still demand a good clarification of the word "certainty" in
>mathematical talk. Regardless of the formal way that is taken for
>granted. Sigma-algebra approach wasn't taught to us mere computer
>students, I had had to learn it during my self-study.
Guffaw. If you'd learned this then you wouldn't have said so
many stupid things about it.
There's nothing shameful about not knowing something. But
when you dimiss someone's comments on something, saying you
understand that but don't want to talk about it, when in
fact it's clear that you have no understanding of the topic
whatever, you look foolish.
When I say you look foolish please don't take that as a
complaint, by the way.
>> *************
>>
>> A truly remarkable number of the things you've said turn out
>> to be false - when you have the sense to ask a question instead
>> of making an assertion it happens really a lot that the answer
>> is obvious to anyone who understands the definitions and
>> basic results. This is why your often repeated complaints
>> that people should shut up if they don't know what they're
>> talking about are so hilarious.
>
>You should still shut up, if you will not read my posts carefully.
>
>You have said that I'm confusing many senses of randomness that are not
>compatible. This is wrong. I was clearly referring to which value would
>come about in the probability theoretical sense when I said a "random
>real". Since, we can show that Martin-Lof, Solovay and Chaitin
>randomness is the same thing, and they also convey some necessary
>conditions for such a value in probability theory (!)
Huh?
>I felt free to
>use these terms interchangeably. What else than probability can I talk
>about, when I say let's choose a random real with uniform probability
>distribution in (0,1) interval?
My gosh, you still haven't got this point, even though it's been
explained tens of times!
Look. This all started when you asked me whether a random real
could be computable, and in spite of my conviction that you'd
give a very confused answer (which conviction has been amply
borne out, you're _still_ showing confusion on this topic)
I asked you what a random real was. What else can you talk
about, you ask? If you mean Solovay random or whatever you
should have replied with "Solovay random", with the definition.
Instead you said that you were talking about a real uniformly
distributed on [0,1].
Let's invent some terminology - the whole problem is that
the same words are used for different concepts. Let's at
least temporarily say that if we mean "real, uniformly
distributed on [0,1]}, ie "random real, as in probability
theory", we will say "p-random real" - let's use
"random real" to mean Solovay or whatever random real.
To the point: asking whether a p-random real can be
computable makes no sense. Or it makes sense but the
answer is awesomely obvious, so much so that someone
asking the question must be missing something. Yes,
a p-random real can be computable. Because "p-random"
simply does _not_ distinguish some reals from other
reals! Regardless of the way the English appears,
the meaning of "p-random real" is _not_ an adjective
applied to the noun "real" - p-random makes no
distinction between Omega and 1/2. Each of those
two numbers is as p-random as the other (which is
to say not at all, it's simply not true that some
numbers are p-random and some are not.)
>But you said that the term "random real" is something specific to AIT.
>I said no such thing, and I still think that's nonsense. Is "random
>number", too, a term that is specific to AIT? What about "random
>string"? What about "infinite random string"?
Depends on exactly what you mean. If I was reading something in
probability and I saw the phrase "random string" I would have
a pretty good idea what was meant. But what was meant would
_not_ imply that some strings are random and some are not;
it would _not_ imply that some strings are more random
than others. _In_ the sense in which we seem to be taking
those phrases here, no, there's no such thing as a "random
string" in probability - that sequence of words could well
appear, but it doesn't mean what it appears to mean at first glance.
>In my opinion, if randomness means ANYTHING, it must have an objective
>and absolute definition. I am not going to buy that the senses of
>randomness in probability theory and information theory are different!
That's what makes you so unique. It's true whether you buy it or
not - your unwillingness to buy it in spite of such a large number
of perfectly clear explanations from so many people is very funny.
(Again, it also makes your insistence that you do understand
mathematical probability theory very funny, because every time
you say you don't buy this distinction you _show_ that you
don't.)
I'll let you in on a little secret: In the sense in which we're
using the term, as an adjective that applies to certain objects
but not to others, the concept of randomness simply does not
appear in mathematical probability theory. Just as what probability
really means is not specified. It's an _axiomatic_ theory -
beginning with certain assumptions about probabilities
and axioms about how probability works we derive statements
about other probabilities. But nowhere does the theory
say anything about what randomness really means.
Putting hypothetical words in your mouth - feel free to
remove the ones that don't belong:
"What? That would make it totally useless!" you say?
Not so. Geometry says nothing whatever about what a
point is - that doesn't make it useless.
"What? That would make mathematical probability theory
totally irrelevant!" Yes, it _does_ make probability
theory irrlevant to some sorts of things, for example
it makes it irrelevant to the sort of question that
you seem to be interested in, what the phrase
"random real" should mean. That irrelevance is
exactly what people have been asserting. Congratulations,
you finally got it.
>What you say I think boils down to this: an artificial distinction.
Huh? What does "artificial" have to do with anything? Supposing that
mathematical probability theory is "artificial", whatever that means,
that doesn't change the _fact_ that it says nothing about which
reals are random and which are not.
>Yes, Martin-Lof randomness is equivalent to Solovay randomness as
>Chaitin summarizes in that page. I've then made the intuitive mistake
>that by definition weak Chaitin random should not be equivalent to
>strong Chaitin randomness. This is an error on my part.
Of course it was an error on your part. We're used to that. Some
of us remain silent on topics we know nothing about and usually
say things true regarding topics that we claim to understand.
Others of us rarely get things right regarding topics that we
insist on claiming to be experts about, on topics where we say
everyone else should shut up because they don't know what they're
talking about.
>But this is merely the definitions that are at the start of Turing
>hierarchy.
>
>Have a look at the page where Chaitin discusses the hierarchy of Omegas
>and corresponding hierarchy of randomness definitions.
>
>Having made a mistake is not important if you can correct them later.
Huh? It's not clear what you think you're refuting here. I didn't
say your mistakes were "important", I just said that the fact that
you make so _many_ mistakes (regarding things that you know nothing
about and also regarding things you insist you do understand) makes
your complaints that others should shut up because they don't
know what they're talking about hilarious.
That fact _does_ make that phenomenon hilarious - whether a mistake
is "important" doesn't change that.
>On the other hand, my hatebuddy Torkel Franzel is allowed to make any
>number of fundamental errors in his posts, because perhaps you like his
>style of trolling and redundancy?
What mistakes has he made? It seems to me that the vast majority of
his posts consist of nothing but _questions_ about things you've
said, in particular requests for clarification that you usually
decline to honor. I don't recall seeing a place where you pointed
out he'd made a mistake - what's an example?
(No, I didn't say he never made mistakes, just that I don't recall
any. When you assert that he's allowed to get things wrong you
really _should_ clarify with "any number of" examples.)
>Regards,
************************
David C. Ullrich
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