Infinite Binary Strings: A Question
leonstreet
one can ask which half of AB P lies on, left or right, then having selected
the half interval P lies on we can ask which half of that interval P lies
upon, and so on repeatedly. If we happen to have chosen a point P such that
AP is incommensurable with AB, the point P will never lie exactly at the
end of any half interval. (It will never lie at the end of any fractional
interval of the line segment.) So the point P produces an infinite, and
aperiodic, infinite string eg LRRLLLR......
Is the converse true? That is, does an infinite, aperiodic binary
string pick out a precise point on AB? Comon sense, perhaps, would tell us
that you cannot get to a point by this repeated narrowing down -- it's
intervals all the way down. Mathematics seems to be telling us that, by
somehow treating the infinite sequence of narrowings down as a whole, an
infinite binary string would indeed determine a precise point on AB. My
question is: which bit of mathematics is it, exactly, that is telling us
that this is so?
Leon
victor_me...@yahoo.co.uk
> Given a line segment AB, and a point P arbitrarily chosen upon it,
> one can ask which half of AB P lies on, left or right, then having selected
> the half interval P lies on we can ask which half of that interval P lies
> upon, and so on repeatedly. If we happen to have chosen a point P such that
> AP is incommensurable with AB, the point P will never lie exactly at the
> end of any half interval. (It will never lie at the end of any fractional
> interval of the line segment.) So the point P produces an infinite, and
> aperiodic, infinite string eg LRRLLLR......
>
> Is the converse true? That is, does an infinite, aperiodic binary
> string pick out a precise point on AB?
Such a process will pick out a real number.
> Comon sense, perhaps, would tell us
> that you cannot get to a point by this repeated narrowing down
That's not what my common sense tells me.
> Mathematics seems to be telling us that, by
> somehow treating the infinite sequence of narrowings down as a whole, an
> infinite binary string would indeed determine a precise point on AB.
What do you mean by a line? If you agree that line segments are
parameterized by intervals in the real numbers, then it would.
If you don't agree this, then you may have a different conception of
line, which may or may not admit a mathematical model. (Maybe you
think
that Conway's surreal numbers better model your concept?) Your
citation
of "common sense" would indicate that you do have some other notion of
line. But if your concept of line cannot be mathematically modelled,
then it is useless asking mathematicians about it.
Victor Meldrew
"I don't believe it!"
Herman Jurjus
To put it a bit simplistically:
At the end of the 19th century, it was Dedekind that wanted this 'to be
so', and he was thrilled when he heard about a guy called Cantor, who
had a theory providing just that: set theory.
Nowadays, it's a necessary consequence of the way the real number line
is -defined- using set theory (either via Dedekind cuts or via Cauchy
sequences, see for example
http://en.wikipedia.org/wiki/Construction_of_real_numbers).
In short, the answer to your question is: the standard -definition- of
the real number set is the bit of mathematics that you're looking for.
--
Cheers,
Herman Jurjus
LudovicoVan
> Leon Street wrote:
> > Given a line segment AB, and a point P arbitrarily chosen upon it,
> > one can ask which half of AB P lies on, left or right, then having selected
> > the half interval P lies on we can ask which half of that interval P lies
> > upon, and so on repeatedly. If we happen to have chosen a point P such that
> > AP is incommensurable with AB, the point P will never lie exactly at the
> > end of any half interval. (It will never lie at the end of any fractional
> > interval of the line segment.) So the point P produces an infinite, and
> > aperiodic, infinite string eg LRRLLLR......
>
> > Is the converse true? That is, does an infinite, aperiodic binary
> > string pick out a precise point on AB? Comon sense, perhaps, would tell us
> > that you cannot get to a point by this repeated narrowing down -- it's
> > intervals all the way down. Mathematics seems to be telling us that, by
> > somehow treating the infinite sequence of narrowings down as a whole, an
> > infinite binary string would indeed determine a precise point on AB. My
> > question is: which bit of mathematics is it, exactly, that is telling us
> > that this is so?
>
> To put it a bit simplistically:
> At the end of the 19th century, it was Dedekind that wanted this 'to be
> so', and he was thrilled when he heard about a guy called Cantor, who
> had a theory providing just that: set theory.
>
> Nowadays, it's a necessary consequence of the way the real number line
> is -defined- using set theory (either via Dedekind cuts or via Cauchy
>
> In short, the answer to your question is: the standard -definition- of
> the real number set is the bit of mathematics that you're looking for.
So, to each point corresponds one, and only one, (infinite) string,
and -- conversely -- to each (infinite) string corresponds one, and
only one, point.
Correct?
-LV
Mariano Suárez-Alvarez
To some points there corresponds *two* infinite sequences.
-- m
LudovicoVan
Two *sequences* converging to a _unique_ limit that is the above
infinite *string*.
And so, again: to each *point* corresponds one, and only one,
(infinite) *string*; and viceversa.
Correct?
-LV
LudovicoVan
I guess, in a strict sense, the point just IS the string.
-LV
Hendrik Boom
For example, to the midpoint correspond the two strings
011111111111111111111111111111111111.....
and
100000000000000000000000000000000000.....
-- hendrik
Mariano Suárez-Alvarez
I honestly have no idea what you wrote.
-- m
LudovicoVan
Ah, then it's correct!
-LV
LudovicoVan
Which point in the middle of what?
If those are the two bounding sequences, they still converge to a
unique limit (string), namely the very 1.0000000000000000...
Correct?
-LV
leonstreet
wrote:
I'm surprised that I need a particular conception of the line to
frame this question. I would have thought any common or garden conception
would do. And all I need to divide a line segment in two is a concept of
rational number.
My half-hearted appeal to common sense is meant to suggest that
there is an issue here, which I believe to be an issue about infinite
strings. The point of framing the question this way was to try and avoid
getting bogged down with questions about real numbers. But to try and
satisfy your scruples, I can put the question another way. What reason is
there to believe that an infinite binary string as in, eg. 0.0110101....,
has a precise value, to believe that no half plus one quarter plus one
eighth plus no sixteenth etc has a precise sum, or in other words to
believe that the sequence of progressive sums here (.0, .01, .011 etc) has
a Least Upper Bound? There is a perfectly good reason, in that we want the
real numbers to serve as a metric for continuously variable media (space,
time and the like). But is it inherently plausible? I gather there are
constructivist positions in which the LUB axiom would be rejected. I have
also read that the LUB axiom can be proved in set theory. Do you know where
I can find the proof? Is it hard? Does it involve the Axiom of Choice?
These are all questions I am interested in. (Though whether I would be able
to follow the answers is another matter.) But as I say, I think there is an
issue here about infinite strings, in all generality, though obviously real
numbers are a massive application of the concept.
Many thanks,
Leon
Virgil
<fc443151-a42c-48d7...@f36g2000hsa.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
Wrong!
That uniqueness of representation is true for exactly those points for
which there are infinitely many left intervals and infinitely many right
intervals in the sequence of nested intervals.
I.e., an infinite string is unique for points which are INTERIOR to
every interval in its sequence of narrowings (not the endpoint of any
such interval), but there are dual strings for any point which is an
endpoint of any such interval.
For example, using julio's own notataion, 0(1) and 1(0) are different
infinite strings representing the same point.
LudovicoVan
Typo, should of course read:
1000000000000000000000000000000....
the (unique) middle point/string.
-LV
Virgil
<f1332f38-58d8-4ab6...@f36g2000hsa.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
WRONG!
In julio¹s own notation, the two different infinite strings 0(1) and
1(0) represent the same point ( or number)
Virgil
<5e14be95-b4f6-40d2...@z66g2000hsc.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
Then some points must be like siamese twins, like the point having (in
julio's own notation) both 0(1) and 1(0) as its twin infinite strings.
LudovicoVan
> In article
> <fc443151-a42c-48d7-8e9a-689ffa7d1...@f36g2000hsa.googlegroups.com>,
Of course that's not what I said. I am making a distinction between
the *sequences* and the infinite *strings* which you have missed.
That might indeed be a delicate question.
Again: the two *sequences* converge to a unique *point*/*string*. IOW,
there is a distinction between the sequences of digits, as -say- may
be output by a TM, which are finite although unbounded, and the
infinite strings corresponding to "points". In fact, we pass from
sequences to infinite strings through limits: they _are_ distinct
entities.
Back to our example, the two *sequences* can be written, to avoid
ambiguity:
lower seq. := 0,1,1,1,1,1,1,1,1,1,1,1,1,...
upper seq. := 1,0,0,0,0,0,0,0,0,0,0,0,0,...
Still the _unique_ "midpoint" here corresponds to the *string*:
midpoint == "10000000000000000000000...0"
(There is a reason why I am putting a final "0" there. In any case, we
can always rewrite "1(0)".)
-LV
leonstreet
wrote:
Many thanks for that, and I understand (I think) your answer.
But isn't what's at stake here more general than the issue of the
definition of the real number? And I'm still struck by the feeling that the
idea of this binary sequence of lefts and rights determining a point is
unconvincing. Of course I need to say why.
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?
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. In the present context
that is the arbitrary point P. If we had chosen the particular point P such
that AB/AP = pi, then the string we produce for this ratio by applying a
fromula for pi has its anchorage in the definite position this point has on
the line, so that geometry serves a kind of GPS for this string. (It is not
necessary to believe that pi is a geometrical notion -- it is sufficient
that it is believed to be a precise ratio, which can then certainly be
represented as a point on a line.) In short, I don't believe that the
infinite binary string associated with an arbitrary point on the line
segment has an independent existence apart from that point. On that basis,
it cannot determine anything. If that's not complete bollocks, I'd be glad
to say more. In any case, again many thanks for a thoughtful response.
Leon
Virgil
<49fd711d-229c-4249...@b1g2000hsg.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
So what is this critical difference you make between an infinite
sequence of 0's and 1's and an infinite string of 0's and 1's?
>
> That might indeed be a delicate question.
>
> Again: the two *sequences* converge to a unique *point*/*string*. IOW,
> there is a distinction between the sequences of digits, as -say- may
> be output by a TM, which are finite although unbounded, and the
> infinite strings corresponding to "points". In fact, we pass from
> sequences to infinite strings through limits: they _are_ distinct
> entities.
To mathematicians, sequences need not be finite unless explicitely
declared as such, and, as you failed to make any such declaration, they
need not be finite.
>
> Back to our example, the two *sequences* can be written, to avoid
> ambiguity:
>
> lower seq. := 0,1,1,1,1,1,1,1,1,1,1,1,1,...
> upper seq. := 1,0,0,0,0,0,0,0,0,0,0,0,0,...
>
> Still the _unique_ "midpoint" here corresponds to the *string*:
>
> midpoint == "10000000000000000000000...0"
>
> (There is a reason why I am putting a final "0" there. In any case, we
> can always rewrite "1(0)".)
(
If you are going to do that, why not simply write it as "1" (or as
"1()" to indicate infinitely many empty strings follow the "1") to
eliminate all those unnecessary 0's.
To write 'midpoint == "10000000000000000000000...0"' falsely indicates
that the process of bisecting intervals must have an end while the
corresponding "011111...." need not have an end.
LudovicoVan
> LudovicoVan <ju...@diegidio.name> wrote:
<snip>
> > Of course that's not what I said. I am making a distinction between
> > the *sequences* and the infinite *strings* which you have missed.
>
> So what is this critical difference you make between an infinite
> sequence of 0's and 1's and an infinite string of 0's and 1's?
The sequences are *not* infinite, they are finite although unbounded.
That's critical.
-LV
Virgil
wrote:
If there is some rule, however complex, by which, for any given positive
natural n, the nth digit can be determined to be a 0 or a 1, then the
expansion is "computable', which is satisfactory for the existence of
the the number represented by that string for all mathematics, including
constructionist mathematics.
It can be proved indirectly that there "are" infinite sequences of 0's
and 1's for which no such rule can exist. Constructionists reject the
"existence" of such numbers, but in standard mathematics they are
regarded as existing but inaccessible.
Virgil
<4215c0ea-5fbc-43bf...@25g2000hsx.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
No single sequence, or even finite set of sequences, can be finite but
unbounded, it requires an infinite sets of sequences to be unbounded.
So that what is critical for you is also impossible.
Cenny Wenner
The term 'unbounded' is sometimes used to describe that there is no
upper bound on the elements of e.g. a, possibly implicit, set,
sequence or function. However, here we speak only of particular
strings, like 0111... and 1000... . Could you describe what you mean
by these strings being unbounded? Are you saying that we should
instead consider constructs such as these to represent (the limit of?)
the sequences 0, 01, 011, ..., and 1, 10, 100, ..., respectively?
Achava Nakhash, the Loving Snake
> On Fri, 12 Sep 2008 16:39:14 +0200, Herman Jurjus <hjur...@hetnet.nl>
>
> - Show quoted text -
My own intuition of the line is that it has no "holes" and that it is
obviously possible to chop it into rational number sized bits. Had
Pythagoras not made his unfortunate discovery, that would have been
enough for a long time. Now, suppose we start chopping the line into
halves - that is chop the original line into halves, then pick one of
two created halves and iterate. My personal intuition is that there
cannot be hole that we are chopping toward. In terms of the infinite
descending sequence of closed intervals, it means that there must be
some point in their intersection. This is, of course, assured by
compactness, which you don't have until you make some formal
definition of the meaning of the points on a line, but my personal
intuition says that the point must be there no matter how we chop to
get it.
As other posters have mentioned, you can study the formal definition
of the real numbers, either the Dedekind constructions (which I don't
prefer) or the Cantor construction by equivalence classes of Cauchy
sequences - which, with a minor twist, also gets you the p-adic
numbers and is more akin to what is done by analysts.
Your real problem seems to be with the logical foundations. For you,
I think, sets must be built by construction; they can't just exist.
My personal intution has no problem with sets just existing, and so I
have essentially no insight into why many people have a problem with
this. This is also known as the problem of actual infinity versus
potential infinity. For me, the axiom of choice is obviously true.
In fact, when I was an undergraduate, the common saying among my math
major friends was, "The axiom of choice is obviously true, the well-
ordering theorem is obviously false, and Zorn's lemma was so weird
that there was available intuition about it." I agree with the first
and last statements. I have no intuition either for or against the
well-ordering theorem.
My own approach is practical. Zorn's lemma, equivalent to the
obviously true axiom of choice, is essential in many parts of
mathematics. I mostly know about the one's in algebra where its use
originally surprised me a great deal. Since it is so useful, and
since no contradiction has every come from using it, and also since
the axiom of choice is obviously true, why not use it? The truth or
falsity of axioms is, of course, in a particularly weird philosophical
place. It is an axiom, so what is meant by either true or false?
Here we must be guided by our intuition and by our experience. Logic
is helpful but not directly helpful.
Finally, I was told that the introduction to some book on foundations
had an introduction or preface which told the tale of a great castle
which had stood for centuries. In the basement of the castle were a
large number of spiders very actively weaving webs all through the
basement. One day a property manager was hired and he had the entire
basement cleaned up. The spiders survived, but their webs did not.
The became frantic, spinning webs as quickly as they could, because
they were convinced that if they did not, the entire structure would
collapse.
Regards,
Achava
Herman Jurjus
> If there is some rule, however complex, by which, for any given positive
> natural n, the nth digit can be determined to be a 0 or a 1, then the
> expansion is "computable', which is satisfactory for the existence of
> the the number represented by that string for all mathematics, including
> constructionist mathematics.
>
> It can be proved indirectly that there "are" infinite sequences of 0's
> and 1's for which no such rule can exist. Constructionists
Do you mean constructivists?
> reject the
> "existence" of such numbers
FYI, there are flavours of intuitionism that don't.
> , but in standard mathematics they are
> regarded as existing but inaccessible.
--
Cheers,
Herman Jurjus
David R Tribble
>> To some points there corresponds *two* infinite sequences.
>>
>> 011111111111111111111111111111111111.....
>> and
>> 100000000000000000000000000000000000.....
>
LudovicoVan wrote:
> Which point in the middle of what?
>
> If those are the two bounding sequences, they still converge to a
> unique limit (string), namely the very 1.0000000000000000...
No, the two infinite strings 0.111... and 1.000... are two
different digit strings representing exactly the same real.
Peter Webb
"Leon Street" wrote in message
news:ljokc4dtvuje2kcv5...@4ax.com...
> Given a line segment AB, and a point P arbitrarily chosen upon it,
> one can ask which half of AB P lies on, left or right, then having
> selected
> the half interval P lies on we can ask which half of that interval P lies
> upon, and so on repeatedly. If we happen to have chosen a point P such
> that
> AP is incommensurable with AB, the point P will never lie exactly at the
> end of any half interval. (It will never lie at the end of any fractional
> interval of the line segment.) So the point P produces an infinite, and
> aperiodic, infinite string eg LRRLLLR......
>
> Is the converse true? That is, does an infinite, aperiodic binary
> string pick out a precise point on AB?
Yes.
> Comon sense, perhaps, would tell us
> that you cannot get to a point by this repeated narrowing down -- it's
> intervals all the way down. Mathematics seems to be telling us that, by
> somehow treating the infinite sequence of narrowings down as a whole, an
> infinite binary string would indeed determine a precise point on AB. My
> question is: which bit of mathematics is it, exactly, that is telling us
> that this is so?
Each truncation of the infinite aperiodic string to n digits produces a
rational number, The sequence of numbers produced as n goes from 1 to
infinity defines a Cauchy sequence, and hence defines a Real number.
>
> Leon
K_h
<leon street> wrote in message
news:jlelc4h9uqr26tvio...@4ax.com...
There really is no problem with arbitrary infinite binary strings. For any
given line segment, you can use such a string to reach many points on the
line segment. In the case of a line segment like [0,1], the infinite binary
string for 1/pi is completely determined but there are 2^ALEPH_0 locations
on [0,1] whose infinite binary strings cannot be completely determined in
any finite way. Those strings still exist but they have no finite
representation and no physical instantiation. The total string for 1/pi has
no physical instantiation but the formula for 1/pi can serve as an algorithm
for computing it. Using the string for 1/pi, the location 1/pi can be
reached by going left or light, in each successive half-size segment, at
time t=1-1/k, for the kth character in 1/pi's string. At time t=1 you will
be at the location 1/pi. ALEPH_0 is so huge that it breaks out of the
unending successive approximations to 1/pi and takes you to 1/pi exactly.
The way to get to the end of an unending process is to go through every step
of the process. This whole line of reasoning is also true for an arbitrary
infinite binary string.
k
LudovicoVan
You too, please note that I am stressing a distinction in meaning
between the _words_: *SEQUENCES* vs. *STRINGS*/"POINTS"/"NUMBERS".
That's crucial for any progress in the discussion, even if just to
show such approach invalid.
-LV
LudovicoVan
> On Fri, 12 Sep 2008 05:54:25 -0700 (PDT), victor_meldrew_...@yahoo.co.uk
I too would be very interested in more insights on this.
My guess, for the sake, is that rejecting a LUB axiom amounts to
rejecting an axiom of infinity, and an axiomatization of induction
with it, so much more than just rejecting "continuity"; I'd expect
only strict finitist to do such a thing. But I am more or less
guessing...
-LV
LudovicoVan
wrote:
You give a straight "yes" above, but here all we have is a definition
by a limit, so that the real number here has still not been "strongly
constructed", as we keep missing an "extended" case. In simpler and
hopefully less improper words: such a definition does not make justice
to the *existence* of "infinite strings", which is the culprit of the
OP. FWIW, it is my intuition that, to overcome the empasse, our
infinite strings, the "points on the line", should rather be taken as
primitive objects on their own right.
-LV
LudovicoVan
> <leon street> wrote in message
>
> news:jlelc4h9uqr26tvio...@4ax.com...
>
>
>
>
>
Could you please elaborate a little bit? Which strings/numbers are you
hinting at here? Uncomputables? Why exactly 2^ALEPH_0 of them?
Shouldn't 2^ALEPH_0 rather be the "length of the segment" [0,1]
itself?
> The total string for 1/pi has
> no physical instantiation but the formula for 1/pi can serve as an algorithm
> for computing it. Using the string for 1/pi, the location 1/pi can be
> reached by going left or light, in each successive half-size segment, at
> time t=1-1/k, for the kth character in 1/pi's string. At time t=1 you will
> be at the location 1/pi.
You mean (by "de-mapping") at time T = k = oo, that is, in *exactly*
ALEPH_0 steps, right?
(This too is interesting because we have an external time, and the
external environment is the locus of continuity, and where the
infinite strings come from at all... hmm, wandering...)
-LV
David C. Ullrich
At least here you admit you're just guessing, although it's
not clear why you think your guess will be of general interest.
In fact you're totally wrong. For example, if the only numbers
we admit are rationals then the LUB property fails - this
has nothing to do with the axiom of infinity or with induction.
>-LV
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
David C. Ullrich
<ju...@diegidio.name> wrote:
>On 12 Sep, 16:47, Mariano Suárez-Alvarez
><mariano.suarezalva...@gmail.com> wrote:
>> On Sep 12, 12:23 pm, LudovicoVan <ju...@diegidio.name> wrote:
>>
>>
>>
>>
>>
>> > On 12 Sep, 15:39, Herman Jurjus <hjur...@hetnet.nl> wrote:
>>
>> > > Leon Street wrote:
>> > > > Given a line segment AB, and a point P arbitrarily chosen upon it,
>> > > > one can ask which half of AB P lies on, left or right, then having selected
>> > > > the half interval P lies on we can ask which half of that interval P lies
>> > > > upon, and so on repeatedly. If we happen to have chosen a point P such that
>> > > > AP is incommensurable with AB, the point P will never lie exactly at the
>> > > > end of any half interval. (It will never lie at the end of any fractional
>> > > > interval of the line segment.) So the point P produces an infinite, and
>> > > > aperiodic, infinite string eg LRRLLLR......
>>
>> > > > Is the converse true? That is, does an infinite, aperiodic binary
>> > > > string pick out a precise point on AB? Comon sense, perhaps, would tell us
>> > > > that you cannot get to a point by this repeated narrowing down -- it's
>> > > > intervals all the way down. Mathematics seems to be telling us that, by
>> > > > somehow treating the infinite sequence of narrowings down as a whole, an
>> > > > infinite binary string would indeed determine a precise point on AB. My
>> > > > question is: which bit of mathematics is it, exactly, that is telling us
>> > > > that this is so?
>>
>> > > To put it a bit simplistically:
>> > > At the end of the 19th century, it was Dedekind that wanted this 'to be
>> > > so', and he was thrilled when he heard about a guy called Cantor, who
>> > > had a theory providing just that: set theory.
>>
>> > > Nowadays, it's a necessary consequence of the way the real number line
>> > > is -defined- using set theory (either via Dedekind cuts or via Cauchy
>> > > sequences, see for examplehttp://en.wikipedia.org/wiki/Construction_of_real_numbers).
>>
>> > > In short, the answer to your question is: the standard -definition- of
>> > > the real number set is the bit of mathematics that you're looking for.
>>
>> > So, to each point corresponds one, and only one, (infinite) string,
>> > and -- conversely -- to each (infinite) string corresponds one, and
>> > only one, point.
>>
>> > Correct?
>>
>> To some points there corresponds *two* infinite sequences.
>
>Two *sequences* converging to a _unique_ limit that is the above
>infinite *string*.
>
>And so, again: to each *point* corresponds one, and only one,
>(infinite) *string*; and viceversa.
>
>Correct?
No. It's curious how you misunderstand clearly
stated things. The statement "some points
correspond to two infinite sequences" means
exactly that some points correspond to two
infinite sequences.
This is exactly the fact that some reals have
two different binary expansions. The string
LRRRRRRR... leads to the same point as
the string RLLLLLL... .
David C. Ullrich
What's crucial is that when "we" say something about strings
you understand what we mean by the term. If you want to
use the same words to mean something else it's crucial that
you use different words instead.
In fact 011111... and 1000.... are two different infinite
binary strings. If you want to say they're the same string
then you mean something different by "infinite binary
string" than we do.
David C. Ullrich
Why in the world do you think that anyone cares about
your feelings or intutions about any of this? You simply
don't know what you're talking about.
LudovicoVan
What you reiterate above is of course clear. I am *adding* to the
discussion.
You just confirm youself a pernicious troll, and overall simply an
idiot.
-LV
LudovicoVan
> On Sat, 13 Sep 2008 02:38:32 -0700 (PDT), LudovicoVan
> <ju...@diegidio.name> wrote:
> >On 13 Sep, 01:50, David R Tribble <da...@tribble.com> wrote:
> >> Hendrik Boom wrote:
> >> >> To some points there corresponds *two* infinite sequences.
> >> >> For example, to the midpoint correspond the two strings
>
> >> >> 011111111111111111111111111111111111.....
> >> >> and
> >> >> 100000000000000000000000000000000000.....
>
> >> LudovicoVan wrote:
> >> > Which point in the middle of what?
>
> >> > If those are the two bounding sequences, they still converge to a
> >> > unique limit (string), namely the very 1.0000000000000000...
>
> >> No, the two infinite strings 0.111... and 1.000... are two
> >> different digit strings representing exactly the same real.
>
> >You too, please note that I am stressing a distinction in meaning
> >between the _words_: *SEQUENCES* vs. *STRINGS*/"POINTS"/"NUMBERS".
> >That's crucial for any progress in the discussion, even if just to
> >show such approach invalid.
>
> What's crucial is that when "we" say something about strings
> you understand what we mean by the term. If you want to
> use the same words to mean something else it's crucial that
> you use different words instead.
You have no monopoly on reasoning or language.
LudovicoVan
You just confirm youself a pernicious troll, and overall simply an
idiot.
As an idiot, of course you won't get it until it's finished.
-LV
LudovicoVan
> On Sat, 13 Sep 2008 02:47:12 -0700 (PDT), LudovicoVan
<snip>
> >My guess, for the sake, is that rejecting a LUB axiom amounts to
> >rejecting an axiom of infinity, and an axiomatization of induction
> >with it, so much more than just rejecting "continuity"; I'd expect
> >only strict finitist to do such a thing. But I am more or less
> >guessing...
>
> At least here you admit you're just guessing, although it's
> not clear why you think your guess will be of general interest.
The "at least" is completely unwarranted, as this reiterated ad
hominem.
> In fact you're totally wrong. For example, if the only numbers
> we admit are rationals then the LUB property fails - this
> has nothing to do with the axiom of infinity or with induction.
Then you are just after something dirrent than the OP.
Actually, you just confirm yourself a pernicious troll, and overall
simply an idiot and a waste of time.
Bye David.
-LV
Denis Feldmann
Wrong : he is a troll and know his craft well
Virgil
<1f3a8d75-4205-4a81...@8g2000hse.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
> On 13 Sep, 01:50, David R Tribble <da...@tribble.com> wrote:
> > Hendrik Boom wrote:
> > >> To some points there corresponds *two* infinite sequences.
> > >> For example, to the midpoint correspond the two strings
> >
> > >> 011111111111111111111111111111111111.....
> > >> and
> > >> 100000000000000000000000000000000000.....
> >
> > LudovicoVan wrote:
> > > Which point in the middle of what?
> >
> > > If those are the two bounding sequences, they still converge to a
> > > unique limit (string), namely the very 1.0000000000000000...
> >
> > No, the two infinite strings 0.111... and 1.000... are two
> > different digit strings representing exactly the same real.
>
> You too, please note that I am stressing a distinction in meaning
> between the _words_: *SEQUENCES* vs. *STRINGS*
A string is a sequence of characters, and all sequences require that
between any two of their members there are only finitely many other
members.
Since julio's *STRINGS* do not behave this way, they are garbage.
Virgil
<3761fd3e-2d96-4594...@y21g2000hsf.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
> But I am more or less
> guessing...
>
> -LV
More!
Virgil
<8b012a7c-83d3-4a28...@73g2000hsx.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
Even for the most strict of constructionists the constructibility of a
real it is only needed that given some positive constructible rational
epsilon one can construct a rational within epsilon of the real.
No "extended" case needed.
In simpler and
> hopefully less improper words: such a definition does not make justice
> to the *existence* of "infinite strings", which is the culprit of the
> OP. FWIW, it is my intuition that, to overcome the empasse, our
> infinite strings, the "points on the line", should rather be taken as
> primitive objects on their own right.
In pure geometry, those real points ARE taken as primitive, but in
modeling the real line by the set of real numbers, the real numbers are
taken as primitive.
Virgil
<3d5f9ab9-712e-417f...@p25g2000hsf.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
What you are "adding" lessens the content.
>
> You just confirm youself a pernicious troll, and overall simply an
> idiot.
Whenever anyone points out some obvious foolishness or idiocy in julio's
postings, he responds with insults, as above.
Weighing the relevance and quality of julio's posts against David's
gives a ration of infinitesimal value.
Virgil
<3c259766-2a53-4473...@w7g2000hsa.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
He does in comparison to julio's reasoning and language.
>
> You just confirm youself a pernicious troll, and overall simply an
> idiot.
Returning such insults when bested, as julio has been, are the mark of a
poor but habitual loser.
Balthasar
>
> ... he is a troll and know his craft well
>
troll := asshole full of shit ?
B.
Keith Ramsay
On Sep 12, 3:42 pm, Herman Jurjus <hjur...@hetnet.nl> wrote:
|Virgil wrote:
|> If there is some rule, however complex, by which, for any given
positive
|> natural n, the nth digit can be determined to be a 0 or a 1, then
the
|> expansion is "computable', which is satisfactory for the existence
of
|> the the number represented by that string for all mathematics,
including
|> constructionist mathematics.
|
|> It can be proved indirectly that there "are" infinite sequences of
0's
|> and 1's for which no such rule can exist. Constructionists
|
|Do you mean constructivists?
|
|> reject the
|> "existence" of such numbers
|
|FYI, there are flavours of intuitionism that don't.
I think essentially anything that you could call
intuitionism has a theory of "free choice sequences". The
notion of free choice sequences was introduced by Brouwer.
A free choice sequence is given term by term without at any
point committing to a rule. Intuitionism is essentially
Brouwer's name for his own philosophy, and while people
who continue to develop it may not always agree completely,
to say that such a notion isn't valid probably means that
one isn't really developing his philosophy further, but
developing something else.
In order to say that a real number r given by a free choice
sequence is equal to 0, say, the generator of the sequence
would have to commit to following a certain rule, hence it's
not the case that r=0. Likewise for the number being equal
to a given computable number c. This is distinct from saying
that for each computable number c, there exists an positive
integer n such that |r-c|>1/n, because that would mean
committing to some scheme for ensuring that one is departing
from the rule. In classical logic the two notions of
uncomputable are of course the same thing, since the
absurdity of r being equal to c is considered sufficient
reason to say that there exists such an n. But constructive
existence of such an n is not guaranteed.
Brouwer also had a more general notion of a sequence given
freely, but with the option of committing to a rule at some
point, but he seems to have decided it was less worth
pursuing.
You might suppose that constructists who aren't intuitonists
would be "unhappy" with all this somehow, but I'm not
convinced that you would be right. Bishop has some remarks
on Brouwer's real analysis in _Foundations of Constructive
Analysis_. My impression is that non-intuitionist
constructivists often just don't consider the study of
free-choice sequences to be worth spending much effort on.
In my opinion, I think we have to accept it as at least
being coherent and meaningful.
Constructivists in the Markov school are supposed to have
explicitly assumed that each sequence is computable.
Constructive mathematics done neither by intuitionist nor
Markov-school constructivists usually just omits to assume
anything special about how a sequence is generated. One
omits the law of excluded middle or any of its equivalents,
that implies the existence of uncomputable sequences. This
is less a matter of denying the existence of uncomputable
sequences as a matter of just remaining neutral. As long as
there's nothing to be gained by making these extra
assumptions, why not just leave them out? You might, for
example, want to apply your reasoning to a sequence
generated by some random natural process, and normally
this is perfectly fine.
|> , but in standard mathematics they are
|> regarded as existing but inaccessible.
Inaccessible only in the sense of not being computable.
Keith Ramsay
Keith Ramsay
On Sep 12, 6:54 am, Leon Street wrote:
| Is the converse true? That is, does an infinite, aperiodic
binary
|string pick out a precise point on AB? Comon sense, perhaps, would
tell us
|that you cannot get to a point by this repeated narrowing down --
it's
|intervals all the way down.
I asked a pre-calc class essentially this question (though
not assuming that the sequence of nested intervals whose
lengths goes to 0 was necessarily this kind of sequence of
"binary" intervals). I wanted them to think a bit about it
and explore their own intuition on it. I finally got one of
my students (one of the "A" students) to hazard a guess,
and her impression was that it wasn't enough to specify a
single point.
I am, like a lot of philosophers, more interested in
intuitions than most mathematicians are. I think that in
this case you have identified a certain counterintuitive
quality in the standard construction of the real line.
Counterintuitive elements are at least worth spelling out
more specifically. Taking seriously that something is
counterintuitive doesn't mean of course assuming that
there's something wrong with it.
I find the interplay between reasoning and intuition
interesting. There's a sense in which an intuition can
be dispelled by further examination. But even in some
cases where it's very clear that the thing which seemed
counterintuitive is correct, a certain air of
counterintuitiveness remains. The counterintuitive
quality of space-filling curves, for example, seems
very stubborn for me. I think I understand how they're
constructed pretty well, and it's very nearly dispelled
the perception that something is strange about them,
but not 100% somehow.
In 1800 there were serious discussions among some of the
most prominent mathematicians of the time (like Gauss)
about some of the basic concepts in real analysis. It was
quite a long time before the current paradigm (to use a
somewhat overused word) became settled. It was a very
interesting process, not to be underestimated.
This intuition that functions should be given by formulas,
for example, was attractive to some for a long time. There
was a perception by some that functions given by a formula
and curves given "mechanically" should be treated as
distinct concepts. Fourier series tested these intuitions,
since it seemed to say that even some very arbitrarily
given function, even one just given "mechanically" by
drawing a curve, could be expressed in a sense by an
infinite series.
|Mathematics seems to be telling us that, by
|somehow treating the infinite sequence of narrowings down as a whole,
an
|infinite binary string would indeed determine a precise point on AB.
My
|question is: which bit of mathematics is it, exactly, that is telling
us
|that this is so?
There are two ingredients. The existence of a number in the
middle of the nested sequence is due to one of the
completeness axioms for the real line. The uniqueness is
due to Archimedes' principle (also called Eudoxus'
principle). Whether these are treated as axioms or as
theorems depends on how you develop the theory. There are
various ways of developing it, but they all give you the
same end result, so it's not necessary to choose just one.
Archimedes principle says that if x>0 and y is some real,
then n*x > y for some integer n. Here we define n*x
inductively to be x+...+x where there are n x's. If two
distinct reals a,b were in all of the nested intervals,
say a<b, then it would follow that for some n, n(b-a)>1,
or b-a>1/n. But past a certain point all of the intervals
are shorter than 1/n, so this is impossible.
The completeness principles implying that there is a real
lying inside all of the nested intervals are perhaps more
fundamental. For these I think I'll just point you to a
couple of Wikipedia entries. The first is for order
completeness, and the other is for completeness as a
metric space:
http://en.wikipedia.org/wiki/Completeness_(order_theory)
http://en.wikipedia.org/wiki/Complete_(topology)
If you digest the reasoning showing that the existence of
a unique real lying in the intersection of the intervals
follows from one of these sets of axioms, it should help
to dispel the sense of counterintuitiveness about it.
One of the ways of developing the theory that is maybe one
of the simplest is just to define a real number to be a
compatible collection of rational intervals containing ones
of arbitrarily short length, where two reals are considered
equal if their intervals are mutually compatible. Compatible
means that any finite intersection is nonempty. That way
the fact that a nested collection of intervals like yours
defines a real follows relatively directly from the
definition. (And this highlights the fact that it is a
matter of definition.)
On the other hand, I think for some students a more
axiomatic approach may do more to make the theory seem
intuitive. The completeness axioms (any of them) can be
informally described as saying that the real line doesn't
have holes in it, and the Archimedean principle says that
it doesn't include infinitesimals. Despite what some
people say, I don't think most people's intuitions about
space or numbers usually lean toward active denial of
either property.
This can be introduced in the context of Euclidean geometry.
It's one of the aspects of the theory that took the longest
to appreciate. For centuries it wasn't realized that it
was needed. The points on the plane that can be constructed
by straightedge-and-compass from a given line segment
satisfy the rest of the axioms, though, so you need something
more to show that 20 degree angles exist and so on. If we
have a ray XY and draw a circle around X, the points Z on
the circle for which the angle YXZ is less than 20 degrees
and the points Z for which YXZ is greater than 20 degrees
would disconnect the circle if there weren't also points
which form an angle of exactly 20 degrees.
Keith Ramsay
Peter Webb
>
> Why in the world do you think that anyone cares about
> your feelings or intutions about any of this? You simply
> don't know what you're talking about.
You just confirm youself a pernicious troll, and overall simply an
idiot.
As an idiot, of course you won't get it until it's finished.
-LV
*********************
If you are referring to Ullrich, he is an anti-troll; his posts are short,
sharp and correct rather than long, vague and wrong as are most trolls. When
he makes a mistake, he acknowledges it, not trollish at all. He is certainly
not an idiot. He can be rude, but that's probably because he doesn't give a
flying fuck about whether the people he responds to like him or not.
leonstreet
>In article <jlelc4h9uqr26tvio...@4ax.com>, leon street
>wrote:
>
>> Many thanks for that, and I understand (I think) your answer.
>> But isn't what's at stake here more general than the issue of the
>> definition of the real number? And I'm still struck by the feeling that the
>> idea of this binary sequence of lefts and rights determining a point is
>> unconvincing. Of course I need to say why.
>>
>> 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?
>
>If there is some rule, however complex, by which, for any given positive
>natural n, the nth digit can be determined to be a 0 or a 1, then the
>expansion is "computable', which is satisfactory for the existence of
>the the number represented by that string for all mathematics, including
>constructionist mathematics.
>
>It can be proved indirectly that there "are" infinite sequences of 0's
>and 1's for which no such rule can exist. Constructionists reject the
>"existence" of such numbers, but in standard mathematics they are
>regarded as existing but inaccessible.
>>
Thanks for the clarification. I want to address another poster's
remarks about arbitrary infinite binary strings, which may have a bearing
on this.
leon
leonstreet
Snake" <ach...@hotmail.com> wrote:
>On Sep 12, 12:37 pm, leon street wrote:
>> On Fri, 12 Sep 2008 16:39:14 +0200, Herman Jurjus <hjur...@hetnet.nl>
>> wrote:
>>
>>
>>
>>
>>
>> >Leon Street wrote:
>> >> Given a line segment AB, and a point P arbitrarily chosen upon it,
>> >> one can ask which half of AB P lies on, left or right, then having selected
>> >> the half interval P lies on we can ask which half of that interval P lies
>> >> upon, and so on repeatedly. If we happen to have chosen a point P such that
>> >> AP is incommensurable with AB, the point P will never lie exactly at the
>> >> end of any half interval. (It will never lie at the end of any fractional
>> >> interval of the line segment.) So the point P produces an infinite, and
>> >> aperiodic, infinite string eg LRRLLLR......
>>
>> >> Is the converse true? That is, does an infinite, aperiodic binary
>> >> string pick out a precise point on AB? Comon sense, perhaps, would tell us
>> >> that you cannot get to a point by this repeated narrowing down -- it's
>> >> intervals all the way down. Mathematics seems to be telling us that, by
>> >> somehow treating the infinite sequence of narrowings down as a whole, an
>> >> infinite binary string would indeed determine a precise point on AB. My
>> >> question is: which bit of mathematics is it, exactly, that is telling us
>> >> that this is so?
>>
>> >To put it a bit simplistically:
>> >At the end of the 19th century, it was Dedekind that wanted this 'to be
>> >so', and he was thrilled when he heard about a guy called Cantor, who
>> >had a theory providing just that: set theory.
>>
>> >Nowadays, it's a necessary consequence of the way the real number line
>> >is -defined- using set theory (either via Dedekind cuts or via Cauchy
>> >sequences, see for example
>> >http://en.wikipedia.org/wiki/Construction_of_real_numbers).
>>
>> >In short, the answer to your question is: the standard -definition- of
>> >the real number set is the bit of mathematics that you're looking for.
>>
>> Many thanks for that, and I understand (I think) your answer.
>> But isn't what's at stake here more general than the issue of the
>> definition of the real number? And I'm still struck by the feeling that the
>> idea of this binary sequence of lefts and rights determining a point is
>> unconvincing. Of course I need to say why.
>>
>> 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?
>>
>> 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. In the present context
>> that is the arbitrary point P. If we had chosen the particular point P such
>> that AB/AP = pi, then the string we produce for this ratio by applying a
>> fromula for pi has its anchorage in the definite position this point has on
>> the line, so that geometry serves a kind of GPS for this string. (It is not
>> necessary to believe that pi is a geometrical notion -- it is sufficient
>> that it is believed to be a precise ratio, which can then certainly be
>> represented as a point on a line.) In short, I don't believe that the
>> infinite binary string associated with an arbitrary point on the line
>> segment has an independent existence apart from that point. On that basis,
>> it cannot determine anything. If that's not complete bollocks, I'd be glad
>> to say more. In any case, again many thanks for a thoughtful response.
>>
>> Leon- Hide quoted text -
>>
>> - Show quoted text -
>
>My own intuition of the line is that it has no "holes" and that it is
>obviously possible to chop it into rational number sized bits. Had
>Pythagoras not made his unfortunate discovery, that would have been
>enough for a long time. Now, suppose we start chopping the line into
>halves - that is chop the original line into halves, then pick one of
>two created halves and iterate. My personal intuition is that there
>cannot be hole that we are chopping toward. In terms of the infinite
>descending sequence of closed intervals, it means that there must be
>some point in their intersection. This is, of course, assured by
>compactness, which you don't have until you make some formal
>definition of the meaning of the points on a line, but my personal
>intuition says that the point must be there no matter how we chop to
>get it.
My intuition of the spatial line is that it has no holes. My
intuition of the number line is that it has had its holes filled in. But in
the light of the replies I've had, here from you and from others,
particularly Ramsay, I've got a lot to think about. My original question
was perhaps slightly faux naif, but not that faux! I hadn't really
considered the analytic geometry aspects at all, for example.
>As other posters have mentioned, you can study the formal definition
>of the real numbers, either the Dedekind constructions (which I don't
>prefer) or the Cantor construction by equivalence classes of Cauchy
>sequences - which, with a minor twist, also gets you the p-adic
>numbers and is more akin to what is done by analysts.
>
>Your real problem seems to be with the logical foundations.
Perhaps fundamentals, rather than foundations.
>For you,
>I think, sets must be built by construction; they can't just exist.
>My personal intution has no problem with sets just existing, and so I
>have essentially no insight into why many people have a problem with
>this. This is also known as the problem of actual infinity versus
>potential infinity. For me, the axiom of choice is obviously true.
>In fact, when I was an undergraduate, the common saying among my math
>major friends was, "The axiom of choice is obviously true, the well-
>ordering theorem is obviously false, and Zorn's lemma was so weird
>that there was available intuition about it." I agree with the first
>and last statements. I have no intuition either for or against the
>well-ordering theorem.
I've come across that before. I was reading something by a
mathematician about the axiom of choice recently to the effect that the
point of the independence of the axiom is that it is true (or 'true', at
least) in some contexts and false in others. He gave as an example of the
latter case something to do with Ramsey theory, which I couldn't follow
(though I've since just started to read up on Ramsey theory), in that to
accept the axiom would lead to markedly artificial and counter-intuitive
results, and as an example of the former something I think to do with
infinite variable spaces, the point being I think that we clearly need to
be able to think of an arbitrary point, one with some arbitrary value in
each of the infinitely many dimensions.
>
>My own approach is practical. Zorn's lemma, equivalent to the
>obviously true axiom of choice, is essential in many parts of
>mathematics. I mostly know about the one's in algebra where its use
>originally surprised me a great deal. Since it is so useful, and
>since no contradiction has every come from using it, and also since
>the axiom of choice is obviously true, why not use it? The truth or
>falsity of axioms is, of course, in a particularly weird philosophical
>place. It is an axiom, so what is meant by either true or false?
>Here we must be guided by our intuition and by our experience. Logic
>is helpful but not directly helpful.
>
>Finally, I was told that the introduction to some book on foundations
>had an introduction or preface which told the tale of a great castle
>which had stood for centuries. In the basement of the castle were a
>large number of spiders very actively weaving webs all through the
>basement. One day a property manager was hired and he had the entire
>basement cleaned up. The spiders survived, but their webs did not.
>The became frantic, spinning webs as quickly as they could, because
>they were convinced that if they did not, the entire structure would
>collapse.
I can appreciate that an obsession with foundations may be more of
historical interest than a pressing modern concern.
>Regards,
>Achava
Many thanks. That was interesting and helpful.
leon
leonstreet
Let me try and explain why I think there is a problem with
arbitrary infinite strings. This will be more of a rough sketch of an
argument, might be completely potty, but if it is on the wrong track you
will probably be able to tell me how.
If I want to make an arbitrary choice from a set of finite strings,
say binary strings of length n, there are various ways I can do this. I can
write it out. I can toss a coin n times. I can pick one from a list,
perhaps by picking a number between 1 and 2^n. None of these methods is
available to me if I wish to choose an arbitrary infinite string. We should
also note that if a finite string arbitrarily chosen happens to have some
'structure', for example if it happens to consist of alternating 1s and 0s
beginning with 1, this is incidental to its arbitrary status. This would be
a way of identifying the string alternative to its position as say, the
k-th string in the list. Now consider the infinite binary string which
consists of alternating 1s and 0s, beginning with 1. The description I have
just given is no longer a dispensable way of identifying the string, but
rather the string is just that string as I have described it. This is its
identity. This is how such periodic strings are made, one might say. Start
with 10 and repeat.
When we make a list of finite strings of length n, it is natural to
start with all 0s, or alternatively all 1s, and then, starting at either
end, systematically build up the possible strings by incremental changes.
We could of course start with an arbitrary string, and list all the
possible transformations of that string. We could write a transformation,
acting upon say a nine digit string, as follows: CLLCLCCCL
which means: Change the first digit in the string to be transformed (from 1
to 0 or 0 to 1), Leave the second digit as it is (0 -> 0; 1 -> 1), and so
on. But then the list of transformations is in effect the same binary list
that we enumerated in the first place, with Cs and Ls instead of 1s and 0s.
How would we go about making a list of all the systematic changes that can
be made to an infinite binary string? Such a list must be a list of all the
infinite periodic binary strings. All those strings which consist of an
initial (possibly null) finite string followed by a repeating string of a
set length, which can indeed be put into a single list. So if we were to
try and make a list of all infinite binary strings and began with the pi
string (the binary equivalent of pi, dropping the radix point), the strings
produced would be periodic transformations of the pi string. In other
words, the only strings that could be produced would be predictable
variations of the pi string. But since in general an aperiodic string is
quite unpredictable, is explicated by computation with no short-cut pattern
in the digits, it is impossible for say pi and some quite other irrational
quantity to appear in the same list.
Why make such a fuss about this when there is the very fine and
well known argument that the power set of an infinite set 'exceeds' the set
(which since select/deselect is a binary structure is equivalent to an
argument about infinite binary strings)? But the argument is too dazzling,
and prevents us from seeing whats involved in the notion of a list beyond
the fact that it's 1:1 with the natural numbers. All the aperiodic strings
cannot be in the same list because, at bottom, the idea of just two
aperodic strings being in the same list is oxymoronic. It's not about there
being some mysteriously supernumerous quantity of infinite strings.
Infinite strings always come with some means of production, either
some rule of composition (typically producing a periodic string, but there
are more subtle rules eg for Champernowne numbers and the like), or some
specified computation, or in the case of the original question, a point on
a line segment. Of course we can think of the string in isolation from its
production method, just as we can think of a painting in isolation from the
artist. For example, Constable's Haywain hangs in the National Gallery, but
John Constable himself has long since fused with the earth in the
churchyard of St John's at Hampstead. And just as two distinct paintings
can be distinguished by their material differences, two infinite binary
strings can always be distinguished by some variation at a finite place in
their development. But all infinite strings are produceable, just as all
paintings have to be painted. This is not a restriction on what can be an
infinite string, or a real number. (Any result of an arithmetical operation
on a number or numbers is automatically another produceable number.
Completeness isn't a worry.) It's an analysis of what an infinite binary
string is. The notion of a completely arbitrary infinite binary strings
seems to me to be empty.
Thanks,
leon
K_h
<leon street> wrote in message
There are methods but they cannot be physically instantiated. You can do
your kth coin toss at time t=1-1/k and at time t=1 you will have an
arbitrary infinite string. Again, the way to get to the end of an endless
task is to do each step of the task.
This is a deep question in set theory. If Godel's axiom of construction,
V=L, is true then all infinite binary strings (of the kind you are
describing) could be well ordered by their descriptions.
> Such a list must be a list of all the
> infinite periodic binary strings. All those strings which consist of an
> initial (possibly null) finite string followed by a repeating string of a
> set length, which can indeed be put into a single list. So if we were to
> try and make a list of all infinite binary strings and began with the pi
> string (the binary equivalent of pi, dropping the radix point), the
> strings
> produced would be periodic transformations of the pi string. In other
> words, the only strings that could be produced would be predictable
> variations of the pi string. But since in general an aperiodic string is
> quite unpredictable, is explicated by computation with no short-cut
> pattern
> in the digits, it is impossible for say pi and some quite other irrational
> quantity to appear in the same list.
Those infinite strings that cannot be encapsulated in any finite way
require, in your case, ALEPH_0 bits of information. A given string of this
kind is only `unpredictable' in the sense that it cannot be characterized in
any finite way.
> Why make such a fuss about this when there is the very fine and
> well known argument that the power set of an infinite set 'exceeds' the
> set
> (which since select/deselect is a binary structure is equivalent to an
> argument about infinite binary strings)? But the argument is too dazzling,
> and prevents us from seeing whats involved in the notion of a list beyond
> the fact that it's 1:1 with the natural numbers. All the aperiodic strings
> cannot be in the same list because, at bottom, the idea of just two
> aperodic strings being in the same list is oxymoronic. It's not about
> there
> being some mysteriously supernumerous quantity of infinite strings.
There are 2^ALEPH_0 infinite strings of the kind this thread has been
discussing.
> Infinite strings always come with some means of production, either
> some rule of composition (typically producing a periodic string, but there
> are more subtle rules eg for Champernowne numbers and the like), or some
> specified computation, or in the case of the original question, a point on
> a line segment.
Many infinite strings do not have finite rules that fully characterize them,
but these strings do have infinite rules that do fully characterize them.
An infinite string that can only be specified by ALEPH_0 bits of information
are computable via all ALEPH_0 bits of information and this computation can
be completed at time t=1, where the kth character of the string is computed
at time t=1-1/k..
Thanks,
k
leonstreet
wrote:
>
>On Sep 12, 6:54 am, Leon Street wrote:
>| Is the converse true? That is, does an infinite, aperiodic
>binary
>|string pick out a precise point on AB? Comon sense, perhaps, would
>tell us
>|that you cannot get to a point by this repeated narrowing down --
>it's
>|intervals all the way down.
Wow. That probably covers it. After I first read this, I was going
to say this is my stop, go away and absorb the lesson. This is still
probably what I should still say -- certainly there is plenty for me to
think about. After a few comments in the text, I might try and say express
the niggling feeling that it doesn't quite cover it.
>I asked a pre-calc class essentially this question (though
>not assuming that the sequence of nested intervals whose
>lengths goes to 0 was necessarily this kind of sequence of
>"binary" intervals). I wanted them to think a bit about it
>and explore their own intuition on it. I finally got one of
>my students (one of the "A" students) to hazard a guess,
>and her impression was that it wasn't enough to specify a
>single point.
>
>I am, like a lot of philosophers, more interested in
>intuitions than most mathematicians are. I think that in
>this case you have identified a certain counterintuitive
>quality in the standard construction of the real line.
>Counterintuitive elements are at least worth spelling out
>more specifically. Taking seriously that something is
>counterintuitive doesn't mean of course assuming that
>there's something wrong with it.
Understood.
>I find the interplay between reasoning and intuition
>interesting. There's a sense in which an intuition can
>be dispelled by further examination. But even in some
>cases where it's very clear that the thing which seemed
>counterintuitive is correct, a certain air of
>counterintuitiveness remains. The counterintuitive
>quality of space-filling curves, for example, seems
>very stubborn for me. I think I understand how they're
>constructed pretty well, and it's very nearly dispelled
>the perception that something is strange about them,
>but not 100% somehow.
That's interesting.
>In 1800 there were serious discussions among some of the
>most prominent mathematicians of the time (like Gauss)
>about some of the basic concepts in real analysis. It was
>quite a long time before the current paradigm (to use a
>somewhat overused word) became settled. It was a very
>interesting process, not to be underestimated.
>
>This intuition that functions should be given by formulas,
>for example, was attractive to some for a long time. There
>was a perception by some that functions given by a formula
>and curves given "mechanically" should be treated as
>distinct concepts. Fourier series tested these intuitions,
>since it seemed to say that even some very arbitrarily
>given function, even one just given "mechanically" by
>drawing a curve, could be expressed in a sense by an
>infinite series.
I can recall trying to wade through some of the mathematical parts
of Penrose's Road to Reality. There's an extended section about the
continuity of functions, especially in relation to complex number. So I can
roughly follow you here.
OK. To do
>One of the ways of developing the theory that is maybe one
>of the simplest is just to define a real number to be a
>compatible collection of rational intervals containing ones
>of arbitrarily short length, where two reals are considered
>equal if their intervals are mutually compatible. Compatible
>means that any finite intersection is nonempty. That way
>the fact that a nested collection of intervals like yours
>defines a real follows relatively directly from the
>definition. (And this highlights the fact that it is a
>matter of definition.)
>
OK. Slightly disturbed that I'm talking about spatial intervals, whereas
this is about numerical intervals, apparently. It struck me that we define
the real numbers so as to be space-like, then we turn round and analyze
space with notions of compactness and the like and show that it's number
like, which seems slightly Irish, but will probably all come out in the
wash when I think about these things properly.
>On the other hand, I think for some students a more
>axiomatic approach may do more to make the theory seem
>intuitive. The completeness axioms (any of them) can be
>informally described as saying that the real line doesn't
>have holes in it, and the Archimedean principle says that
>it doesn't include infinitesimals. Despite what some
>people say, I don't think most people's intuitions about
>space or numbers usually lean toward active denial of
>either property.
Certainly not for space.
>This can be introduced in the context of Euclidean geometry.
>It's one of the aspects of the theory that took the longest
>to appreciate. For centuries it wasn't realized that it
>was needed. The points on the plane that can be constructed
>by straightedge-and-compass from a given line segment
>satisfy the rest of the axioms, though, so you need something
>more to show that 20 degree angles exist and so on. If we
>have a ray XY and draw a circle around X, the points Z on
>the circle for which the angle YXZ is less than 20 degrees
>and the points Z for which YXZ is greater than 20 degrees
>would disconnect the circle if there weren't also points
>which form an angle of exactly 20 degrees.
I confess I hadn't of the impossibility of trisecting 60 degrees as
casting doubt on the existence of the point at 20 degrees, but that's
probably me not thinking through the consequences of the axiomatic method.
>Keith Ramsay
There are two, I think connected, worries I still have, that I'm
not sure are squarely addressed. I'm not sure they are cogent, and they may
indeed dissipate by the appropiate study as you've suggested. But I may as
well express them. They suggest to me an asymmetry between the infinite
binary strings on the one hand and the points on the line segment on the
other, almost regardless of how the reals are defined. One concern, which
I've expressed to another poster, is with the idea of an arbitrary infinite
binary string. Whether those concerns are real or not, I suppose I'm taking
it for granted that there's nothing problematic about selecting an
arbitrary point on a line segment. I suppose I think of this in terms of
just plumping for one, with an infinitely thin pencil point.
The second concern is harder to express. If the irrationals are to
sit alongside the reals on the number line, it seems reasonable to expect
that they should have, in common with the rationals, a precise numerical
magnitude, by which I mean that if we take a line segment as a
REPRESENTATION of the number line (in the unit interval, say), then an
irrational number should have upon it an exact length which ought to be
derivable from the number, i.e. from the infinite, aperiodic binary string
in the case of an irrational number.
Suppose we inscribe an infinite binary tree in a triangle as
follows. We let the distance between successive node levels be half the
distance between the previous node levels. Instead of left sloping path
steps we will have downward paths, and right-sloping path steps will be at
45 degrees. The root node is at O, with A at unit distance vertically
below. And at right anles to OA we have a unit length AB going off to the
right. The first 0-step goes to the midpoint of OA, the 1-step to the
midpoint of OB. The next steps go half that distance again leading to 4
equidistant nodes, and so on. Every infinite binary string has a
representation as a point P on AB, with a precise length AP. As before. an
irrational point P will generate an infinite binary string since a line
drawn from O to P will produce an infinite sequence of minima as it
traverses through adjacent nodes at each level of the tree. The question
is, how does a string determine a particular point on AB? I still have the
feeling that a string being able to do this should not be a matter of
definition. In the case of periodic strings this is easy to do. After
following any initial finite path, we then join up the beginnings and ends
of the periodic blocks and project them in a straight line to AB.
(Equivalent to summing a geometic series.) How can an aperiodic string
achieve this? Is there some non-linear way of projecting to AB? How can the
binary string for pi - 3, say, be thought of as picking out that precise
length without our already knowing where that point is
(non-representationally, so to speak)?
If this is rubbish, just tell me. I'll go back to the books and the Wiki.
Huge thanks,
leon.
leonstreet
<webbf...@DIESPAMDIEoptusnet.com.au> wrote:
>
>"Leon Street" wrote in message
>news:ljokc4dtvuje2kcv5...@4ax.com...
>> Given a line segment AB, and a point P arbitrarily chosen upon it,
>> one can ask which half of AB P lies on, left or right, then having
>> selected
>> the half interval P lies on we can ask which half of that interval P lies
>> upon, and so on repeatedly. If we happen to have chosen a point P such
>> that
>> AP is incommensurable with AB, the point P will never lie exactly at the
>> end of any half interval. (It will never lie at the end of any fractional
>> interval of the line segment.) So the point P produces an infinite, and
>> aperiodic, infinite string eg LRRLLLR......
>>
>> Is the converse true? That is, does an infinite, aperiodic binary
>> string pick out a precise point on AB?
>
>Yes.
>
>> Comon sense, perhaps, would tell us
>> that you cannot get to a point by this repeated narrowing down -- it's
>> intervals all the way down. Mathematics seems to be telling us that, by
>> somehow treating the infinite sequence of narrowings down as a whole, an
>> infinite binary string would indeed determine a precise point on AB. My
>> question is: which bit of mathematics is it, exactly, that is telling us
>> that this is so?
>
>Each truncation of the infinite aperiodic string to n digits produces a
>rational number, The sequence of numbers produced as n goes from 1 to
>infinity defines a Cauchy sequence, and hence defines a Real number.
>
>
Thanks. It is more than likely that the muddles are all in my head.
Still, one cannot help projecting them.
leon
Mariano Suárez-Alvarez
Well, if you are going to allow "infinite rules" you
can simply put the actual digits in the string in the rules...
Mariano Suárez-Alvarez
>
> [yet another argument elided]
>
> Infinite strings always come with some means of production, [...]
Well, here is were you are wrong.
-- m
fishfry
<ed0851db-54ec-4e65...@73g2000hsx.googlegroups.com>,
Mariano Suárez-Alvarez <mariano.su...@gmail.com> wrote:
Karl Marx says the workers should control the means of production! I can
see it now, a socialist paradise of workers hammering out sequences of
ones and zeros all day long, singing songs of solidarity with the
oppressed masses of the world.
victor_me...@yahoo.co.uk
Of course, obsession with the "means of production" is a classic
Marxist shibboleth.
Victor Meldrew
"I don't believe it!"
Herman Jurjus
> The question
> is, how does a string determine a particular point on AB? I still have the
> feeling that a string being able to do this should not be a matter of
> definition.
The thing that is a matter of definition is not so much 'the string
being able to determine a point'. It's the fact that there /is/ a point
for it to determine that's a matter of definition (namely of the real
number set). The real number set is defined so as to include a limit for
every sequence that 'geometrically seems to converge' (Cauchy sequence),
so that every seemingly convergent sequence has something to converge to.
> In the case of periodic strings this is easy to do. After
> following any initial finite path, we then join up the beginnings and ends
> of the periodic blocks and project them in a straight line to AB.
> (Equivalent to summing a geometic series.) How can an aperiodic string
> achieve this? Is there some non-linear way of projecting to AB? How can the
> binary string for pi - 3, say, be thought of as picking out that precise
> length without our already knowing where that point is
> (non-representationally, so to speak)?
The string is /not/ thought of as picking out a precise length - at
least not in one stroke. The precise length is 'a sensible something' to
the same extent that the string is.
What /is/ a length? a real number - ok. How do you specify such a real
number? giving a sequence of rationals converging to the number is one
way to specify such a number.
Real numbers are in many respects quite similar to infinite strings.
> If this is rubbish, just tell me. I'll go back to the books and the Wiki.
No, it's not rubbish, but books and wiki won't harm you anyway.
--
Cheers,
Herman Jurjus
Herman Jurjus
> 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.
> 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.
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
Peter Webb
"Herman Jurjus" <hju...@hetnet.nl> wrote in message
news:48ce2837$0$27211$ba62...@text.nova.planet.nl...
> 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.
Herman Jurjus
>
> "Herman Jurjus" <hju...@hetnet.nl> wrote in message
> news:48ce2837$0$27211$ba62...@text.nova.planet.nl...
>> 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
Mariano Suárez-Alvarez
<webbfam...@DIESPAMDIEoptusnet.com.au> wrote:
> "Herman Jurjus" <hjur...@hetnet.nl> wrote in message
What do you mean by 'produce', exactly?
-- m
K_h
"Mariano Suárez-Alvarez" <mariano.su...@gmail.com> wrote in message
news:c2457c52-cfb8-4e3a...@f36g2000hsa.googlegroups.com...
The infinite rules need to be parsed out into finite indexed segments that
can be distributed through the cumulative hierarchy of V=L.
k
Mariano Suárez-Alvarez
> "Mariano Suárez-Alvarez" <mariano.suarezalva...@gmail.com> wrote in message
I am sorry, but I have no idea what you are saying here.
-- m
LudovicoVan
wrote:
In that, I guess we are all on the same line.
-LV
amy666
> Given a line segment AB, and a point P arbitrarily
> y chosen upon it,
> one can ask which half of AB P lies on, left or
> right, then having selected
> the half interval P lies on we can ask which half of
> that interval P lies
> upon, and so on repeatedly. If we happen to have
> chosen a point P such that
> AP is incommensurable with AB, the point P will never
> lie exactly at the
> end of any half interval. (It will never lie at the
> end of any fractional
> interval of the line segment.) So the point P
> produces an infinite, and
> aperiodic, infinite string eg LRRLLLR......
>
> Is the converse true? That is, does an infinite,
> , aperiodic binary
> string pick out a precise point on AB? Comon sense,
> perhaps, would tell us
> that you cannot get to a point by this repeated
> narrowing down -- it's
> intervals all the way down. Mathematics seems to be
> telling us that, by
> somehow treating the infinite sequence of narrowings
> down as a whole, an
> infinite binary string would indeed determine a
> precise point on AB. My
> question is: which bit of mathematics is it, exactly,
> that is telling us
> that this is so?
>
> Leon
a typical sci.math discussion.
the answer ?
-> 3 valued logic !
since the question of the OP is similar to :
is 0 negative or positive ?
and the answer is neither ;
thus the question + or - for every real has 3 possible outcomes.
binary representations clearly " fail " but 3-valued logic works very fine !!
regards
tommy1729
David C. Ullrich
wrote:
Right. And if he'd asked whether the Earth was round
that would be similar to the question of whether gerbils
are edible.
>and the answer is neither ;
It's certainly true that 0 is neither positive nor negative.
The idea that this somehow leads to or requires 3-valued
logic is simply stupid.
>thus the question + or - for every real has 3 possible outcomes.
>
>binary representations clearly " fail " but 3-valued logic works very fine !!
Exactly how do binary representations "fail"?
>regards
>
>tommy1729
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
leonstreet
wrote:
>leon street wrote:
>> The question
>> is, how does a string determine a particular point on AB? I still have the
>> feeling that a string being able to do this should not be a matter of
>> definition.
>
>The thing that is a matter of definition is not so much 'the string
>being able to determine a point'. It's the fact that there /is/ a point
>for it to determine that's a matter of definition (namely of the real
>number set). The real number set is defined so as to include a limit for
>every sequence that 'geometrically seems to converge' (Cauchy sequence),
>so that every seemingly convergent sequence has something to converge to.
>
>> In the case of periodic strings this is easy to do. After
>> following any initial finite path, we then join up the beginnings and ends
>> of the periodic blocks and project them in a straight line to AB.
>> (Equivalent to summing a geometic series.) How can an aperiodic string
>> achieve this? Is there some non-linear way of projecting to AB? How can the
>> binary string for pi - 3, say, be thought of as picking out that precise
>> length without our already knowing where that point is
>> (non-representationally, so to speak)?
>
>The string is /not/ thought of as picking out a precise length - at
>least not in one stroke. The precise length is 'a sensible something' to
>the same extent that the string is.
OK. I follow you so far, I think.
>What /is/ a length? a real number - ok.
But irrational lengths, even constructable irrational lengths exist
in Euclid, do they not, without any developed notion of real number? On the
other hand, perhaps a line segment with its rational landmarks and a few
constructable irrational points is not sufficient for the notion of a
continuum of precise lengths. Certainly I can see that the weakly logical
sense in which the limit of a decreasing sequence of spatial intervals is a
point, and the limit of a decreasing sequence ot time intervals is a point,
and so on, may not be sufficient for the notion of a bullseye, which the
original question suggested, not sufficient to force a dichotomy between
hitting the bullseye and approaching it ever more nearly. Sometimes I can
see my concerns disappearing in a smoke-puff of finitistic prejudice. But
then I start to have thoughts like: Why does a point need to be
constructable anyway? Doesn't a precise length exist (in proportion to a
given unit length) merely be the demonstrable act of picking a point on a
line segment, without knowing what the length is? Wouldn't it make more
sense to define the reals as lengths, in fact? (More precisely, as ratios
of magnitudes in any continuous medium.)
Then we do not have to explain why this numerical string, say for
1/sq.rt.2, when construed metrically, happens to converge to this
(independently constructable) point in the unit line segment.
Anyway, more thought and study required on my part.
Thanks a bunch,
leon
leonstreet
I don't see the barrier here as being merely a physical one, and
this appeal to a supertask seems unconvincing.
OK. I'll have to look that up. Thanks.
>> Such a list must be a list of all the
>> infinite periodic binary strings. All those strings which consist of an
>> initial (possibly null) finite string followed by a repeating string of a
>> set length, which can indeed be put into a single list. So if we were to
>> try and make a list of all infinite binary strings and began with the pi
>> string (the binary equivalent of pi, dropping the radix point), the
>> strings
>> produced would be periodic transformations of the pi string. In other
>> words, the only strings that could be produced would be predictable
>> variations of the pi string. But since in general an aperiodic string is
>> quite unpredictable, is explicated by computation with no short-cut
>> pattern
>> in the digits, it is impossible for say pi and some quite other irrational
>> quantity to appear in the same list.
>
>Those infinite strings that cannot be encapsulated in any finite way
>require, in your case, ALEPH_0 bits of information. A given string of this
>kind is only `unpredictable' in the sense that it cannot be characterized in
>any finite way.
The suggestion is that it's a consequence of what's involved in the
notion of aperiodic which makes it impossible for two unrelated aperiodic
strings to exist in the same list, and whether 'unpredictable' is a good
description of this aperiodicity is not crucial. However I am far from
believing that my arguments are totally convincing, and I will certainly
think about them again in the light of what you have said.
>> Why make such a fuss about this when there is the very fine and
>> well known argument that the power set of an infinite set 'exceeds' the
>> set
>> (which since select/deselect is a binary structure is equivalent to an
>> argument about infinite binary strings)? But the argument is too dazzling,
>> and prevents us from seeing whats involved in the notion of a list beyond
>> the fact that it's 1:1 with the natural numbers. All the aperiodic strings
>> cannot be in the same list because, at bottom, the idea of just two
>> aperodic strings being in the same list is oxymoronic. It's not about
>> there
>> being some mysteriously supernumerous quantity of infinite strings.
>
>There are 2^ALEPH_0 infinite strings of the kind this thread has been
>discussing.
Whether infinite strings can be understood in the purely
combinatorical way this implies is the point at issue, however poorly I
have presented it.
>> Infinite strings always come with some means of production, either
>> some rule of composition (typically producing a periodic string, but there
>> are more subtle rules eg for Champernowne numbers and the like), or some
>> specified computation, or in the case of the original question, a point on
>> a line segment.
>
>Many infinite strings do not have finite rules that fully characterize them,
>but these strings do have infinite rules that do fully characterize them.
>An infinite string that can only be specified by ALEPH_0 bits of information
>are computable via all ALEPH_0 bits of information and this computation can
>be completed at time t=1, where the kth character of the string is computed
>at time t=1-1/k..
>
>
>Thanks,
>k
Many thanks,
leon
leonstreet
wrote:
>Peter Webb wrote:
I'm not sure that I follow exactly your discussion of set theory,
but the following might be pertinent.
My notion of strings being produceable has not really anything
directly to do wtih the usual notions of construction or computation. It's
merely the positive spin of an essentially negative point, namely that
there is no such thing as an arbitrary infinite string in a purely
combinatorial sense. For example, if there are an uncountable infinity of
points in a line segment, then there are also an uncountable infinity of
binary strings, since any point produces a distinct string (sometimes there
will be two strings per point, but this doesn't matter).
1001001011.......
does not refer to a particular infinite binary string without, so to speak,
a propulsion system, something that dictates deterministically how the
string is to be continued.
Do we not have the idea, for some string S1 S2 S3 S4......
that the value of Si, for each i = 1, 2, 3, 4..., is arbitrarily 0 or 1?
Yes, sort of. But we cannot give an example of such an arbitrary infinite
binary string, any more than we can complete the decimal expansion of pi.
For example, the string:
1010101010.....
is not an example of such a string, for in an arbitrary string the value of
Si should be chosen independently of the value of Sk. This is not the case
here, for we know that, for example, every Si for even i is 0.
For some purposes it may be useful to think of an infinity of
arbitrary values. Suppose that the 'position' of the state of some system
is determined by an infinite number of independent variables. It seems
reasonable to think that an arbitrary state of the system consists of an
arbitrary assignment of values in each of the variables. But if we are
thinking of this situation spatially, there is an extra layer of
arbitrariness here, namely that in space there is (without an observer) no
preferred direction or axis, no natural order in which to take this
infinitley many axes. As soon as we imagine instead that there is some
natural and fixed order in which to take these axes, rather than a mere
labelling, so that they are, immutably, axis 1, 2, 3 etc., we are back to
the problems of specifying an example of an arbitrary infinite string.
For many purposes also, I'm sure it does not matter that infinite
strings do not intrinsically have arbitrariness. The secondary randomness
which a string has in virtue of the fact that it could have been produced
by selecting an arbitrary point in a line segment (along the lines of the
original question) will be quite sufficient.
But if what we are dealing with is the nature of infinite strings
themselves, then it seems to me that this lack of combinatorical
arbirariness becomes quite important. And it seems to me that for example
the power set proof of uncountablility, while not wrong as such,
presupposes this notion of combinatorial arbitrariness, and begs all the
questions about lists and aperiodicity that need to be answered.
Thanks again,
leon
amy666
so we agree.
> The idea that this somehow leads to or requires
> 3-valued
> logic is simply stupid.
first you admit , then you take that back to say im stupid ?
three possible values => positive , negative , neither.
thus three valued logic.
surely even you can understand that ?!?
if you agree that the question is A positive or negative has 3 potential answers for any real A , but only one correct for A = 0 ...
>
> >thus the question + or - for every real has 3
> possible outcomes.
> >
> >binary representations clearly " fail " but 3-valued
> logic works very fine !!
>
> Exactly how do binary representations "fail"?
>
> >regards
> >
> >tommy1729
>
> David C. Ullrich
>
> "Understanding Godel isn't about following his formal
> proof.
> That would make a mockery of everything Godel was up
> to."
> (John Jones, "My talk about Godel to the post-grads."
> in sci.logic.)
tommy1729
LudovicoVan
No, it isn't. The OP was after a necessary and sufficient condition.
> and the answer is neither ;
Neither what? What is an infinite binary string? And/or what is a
point on a line segment?
> thus the question + or - for every real has 3 possible outcomes.
No, there are at least 5 possible outcomes to your question: yes, no,
neither, both, nonsense. The Chinese have been (un-)teaching it for 6
thousend years now. We managed to lose an element along the way.
-LV
fishfry
wrote:
<snip>
> My notion of strings being produceable has not really anything
> directly to do wtih the usual notions of construction or computation. It's
> merely the positive spin of an essentially negative point, namely that
> there is no such thing as an arbitrary infinite string in a purely
> combinatorial sense. For example, if there are an uncountable infinity of
> points in a line segment, then there are also an uncountable infinity of
> binary strings, since any point produces a distinct string (sometimes there
> will be two strings per point, but this doesn't matter).
>
<snip>
Leon -- would the following model help? You have a countable set of fair
coins, numbered 1, 2, 3, ...
You flip all the coins simultaneously. That defines an infinite binary
string -- heads for one, tails for zero, say.
What exactly are you saying, now? That the only strings that could
possibly result are the ones that can be described by algorithms or
short text descriptions? Are you denying the existence of coin tosses
that result in random strings?
A string is random if there is no shorter description of the string than
the string itself. In other words we can't describe it as, "alternate 1
and 0," or "pi in base 2" or anything like that. To describe the string
we have to write out the entire string. That's what randomness means.
You seem to be denying the existence of randomness.
I assume you understand that there are only countably many finite-length
algorithms or descriptions (the strings written in some countable
alphabet); but that there are uncountably many infinite binary strings.
So you seem to be denying the existence of randomness.
Is that what you're getting at?
Virgil
<16114eb2-7ce9-49ab...@m45g2000hsb.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
Both only by Bourbaki.
Neither by everyone else who is sensible.
And nonsense only by the nonsensical.
LudovicoVan
> In article
> <16114eb2-7ce9-49ab-8dd8-1169822af...@m45g2000hsb.googlegroups.com>,
The dumb gardians of their own enslavement.
-LV
Virgil
<ecf02b1e-7e49-45d1...@k37g2000hsf.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
It is LV's own ignorance that keeps him enslaved.
The knowledge that he chooses to remain ignorant of does not enslave
anyone.
LudovicoVan
You're like the kid who keeps saying "I win, I win".
You surely win a kick in your dumb ass, dumbass.
-LV
Virgil
<835decdb-4024-414e...@k7g2000hsd.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
If there is any "winner", it is mathematics as a whole which wins when
those seriously into it agree on what things mean.
So that if you want to be seriously into it, you should learn what those
who are serious about it find it useful to learn.
LudovicoVan
Idiot and liar: you and mathematics have nothing to do.
You and your kind are just the scum of the world.
-LV
David C. Ullrich
wrote:
No. A three-valued logic is a logic where _statement_
have three possible _truth values_. The fact that
a number can be positive, negative or neither does
not lead to any such statements. "2 is positive" is
true. "0 is positive" is false. "0 is negative" is false.
"0 is neither positive nor negative" is true. There's
no need for any truth values other than the standard
two.
>surely even you can understand that ?!?
>
>if you agree that the question is A positive or negative has 3 potential answers for any real A , but only one correct for A = 0 ...
"A is positive or negative" has _two_ possible values.
For most values of A it is true. For A = 0 it is _false_.
"0 is positive or negative" is simple _false_.
The _sign_ of 0 is neither positive nor negative.
But signs are not truth values.
Regarding whether "stupid" is the right word for your
assertions here, you don't seem to have noticed that
if we happen to be talking about five things then
your reasoning would say we need a five-valued
logic. Your reasoning indicates that since we talk
about infinite sets all the time we need an infinite-
valued logic. Thinking this is silly - thinking that
it's clearly true but you're the first person to have
noticed it is simply stupid.
When you notice something that seems to contradict
accepted mathematical facts you should consider the
possibility that you're simply misunderstanding
something. Especially when what you've "noticed"
is so basic and elementary that it's not reasonable
to assume that it's simply been overlooked for
centuries.
amy666
> On 13 Sep, 12:34, David C. Ullrich
> <dullr...@sprynet.com> wrote:
> > On Sat, 13 Sep 2008 02:47:12 -0700 (PDT),
> LudovicoVan
>
> <snip>
>
> > >My guess, for the sake, is that rejecting a LUB
> axiom amounts to
> > >rejecting an axiom of infinity, and an
> axiomatization of induction
> > >with it, so much more than just rejecting
> "continuity"; I'd expect
> > >only strict finitist to do such a thing. But I am
> more or less
> > >guessing...
> >
> > At least here you admit you're just guessing,
> although it's
> > not clear why you think your guess will be of
> general interest.
>
> The "at least" is completely unwarranted, as this
> reiterated ad
> hominem.
an ad hominem reply is david ullrich's trademark !
>
> > In fact you're totally wrong. For example, if the
> only numbers
> > we admit are rationals then the LUB property fails
> - this
> > has nothing to do with the axiom of infinity or
> with induction.
>
> Then you are just after something dirrent than the
> OP.
>
> Actually, you just confirm yourself a pernicious
> troll, and overall
> simply an idiot and a waste of time.
>
> Bye David.
>
> -LV
hahaha !!!
David is called a troll , an idiot and a waste of time :)
you rock Julio :p
regards
tommy1729
Virgil
<ecb20965-09dd-4683...@z66g2000hsc.googlegroups.com>,
LudovicoVan <ju...@diegidio.name> wrote:
> Idiot and liar: you and mathematics have nothing to do.
>
> You and your kind are just the scum of the world.
The mathematical content of the above is zero.
Its contribution to sci.math is negative.
amy666
> In article
> <fc443151-a42c-48d7...@f36g2000hsa.goog
> legroups.com>,
> LudovicoVan <ju...@diegidio.name> wrote:
>
> > On 12 Sep, 15:39, Herman Jurjus <hjur...@hetnet.nl>
> wrote:
> > > Leon Street wrote:
> > > > Given a line segment AB, and a point P
> arbitrarily chosen upon it,
> > > > one can ask which half of AB P lies on, left or
> right, then having
> > > > selected
> > > > the half interval P lies on we can ask which
> half of that interval P lies
> > > > upon, and so on repeatedly. If we happen to
> have chosen a point P such
> > > > that
> > > > AP is incommensurable with AB, the point P will
> never lie exactly at the
> > > > end of any half interval. (It will never lie at
> the end of any fractional
> > > > interval of the line segment.) So the point P
> produces an infinite, and
> > > > aperiodic, infinite string eg LRRLLLR......
> > >
> > > > Is the converse true? That is, does an
> infinite, aperiodic binary
> > > > string pick out a precise point on AB? Comon
> sense, perhaps, would tell
> > > > us
> > > > that you cannot get to a point by this repeated
> narrowing down -- it's
> > > > intervals all the way down. Mathematics seems
> to be telling us that, by
> > > > somehow treating the infinite sequence of
> narrowings down as a whole, an
> > > > infinite binary string would indeed determine a
> precise point on AB. My
> > > > question is: which bit of mathematics is it,
> exactly, that is telling us
> > > > that this is so?
> > >
> > > To put it a bit simplistically:
> > > At the end of the 19th century, it was Dedekind
> that wanted this 'to be
> > > so', and he was thrilled when he heard about a
> guy called Cantor, who
> > > had a theory providing just that: set theory.
> > >
> > > Nowadays, it's a necessary consequence of the way
> the real number line
> > > is -defined- using set theory (either via
> Dedekind cuts or via Cauchy
> > > sequences, see for
> > >
> examplehttp://en.wikipedia.org/wiki/Construction_of_re
> al_numbers).
> > >
> > > In short, the answer to your question is: the
> standard -definition- of
> > > the real number set is the bit of mathematics
> that you're looking for.
> >
> > So, to each point corresponds one, and only one,
> (infinite) string,
> > and -- conversely -- to each (infinite) string
> corresponds one, and
> > only one, point.
> >
> > Correct?
>
> Wrong!
>
> That uniqueness of representation is true for exactly
> those points for
> which there are infinitely many left intervals and
> infinitely many right
> intervals in the sequence of nested intervals.
>
> I.e., an infinite string is unique for points which
> are INTERIOR to
> every interval in its sequence of narrowings (not the
> endpoint of any
> such interval), but there are dual strings for any
> point which is an
> endpoint of any such interval.
>
> For example, using julio's own notataion, 0(1) and
> 1(0) are different
> infinite strings representing the same point.
.. unless we consider that an infinitesimally small difference ...
regards
tommy1729
Virgil
<32754468.1221766679...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
Calling David what he patently is not does not make him into what he is
not, regardless of how often it is done.
Virgil
<20259354.1221766853...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
> > For example, using julio's own notation, 0(1) and
> > 1(0) are different
> > infinite strings representing the same point.
>
> .. unless we consider that an infinitesimally small difference ...
For which julio has no string, so in julio's notation there are no such
things.
>
> regards
>
> tommy1729
amy666
real x = positive , negative or neither
=> three possible truth values.
"2 is positive" is
> true. "0 is positive" is false. "0 is negative" is
> false.
> "0 is neither positive nor negative" is true. There's
> no need for any truth values other than the standard
> two.
but 3-valued logic is better in this case !!
let 1 be true and 0 be false.
is x positive (1 or 0) negative (1 or 0) or neither (1 or 0)
FIRST there are 3 potential answers , thus 2 valued logic fails in the sense of a single 2-logic gate.
SECOND the answer can be
100 ( positive )
010 ( negative )
001 ( neither )
but not 101 or 111 or 110 or 011
thus you might recode the 3-valued with ( binary ) 2-valued logic , this is not so valid in the sense that not all 2-valued logic outputs are valid !
and since 3/2 is no integer you cannot rewrite 3 in base 2 where every bit can be 0 or 1 without restriction.
thus we end up in , after decoding into base 1 ( unair ), all possible values :
111 positive
011 negative
001 neither
which is just 1 2 3 -> 3-valued logic !!
nothing wrong with 3 valued logic david !!!
nothing STUPID about it !!
what is wrong is assuming positive / negative / neither are 3 2-valued logic questions / gates , since one cannot answer
positive true
negative true
neither false
this mistake ( of 2-valued logic !! ) is not possible in 3-valued logic !!
>
> >surely even you can understand that ?!?
> >
> >if you agree that the question is A positive or
> negative has 3 potential answers for any real A , but
> only one correct for A = 0 ...
>
> "A is positive or negative" has _two_ possible
> values.
> For most values of A it is true. For A = 0 it is
> _false_.
> "0 is positive or negative" is simple _false_.
>
> The _sign_ of 0 is neither positive nor negative.
> But signs are not truth values.
see my explaination above ...
not that you would understand , read or accept it ...
oh well.
>
> Regarding whether "stupid" is the right word for your
> assertions here, you don't seem to have noticed that
> if we happen to be talking about five things then
> your reasoning would say we need a five-valued
> logic.
indeed , " stupid " is not the right word.
its a word you use too often david !
note that you mention five-valued logic , not me !
i cant think of any intresting question in math that has exactly 5 potential answers ...
with the possible exception of a group with 5 elements of course.
Your reasoning indicates that since we talk
> about infinite sets all the time we need an infinite-
> valued logic. Thinking this is silly - thinking that
> it's clearly true but you're the first person to have
> noticed it is simply stupid.
hmm what would an infinite valued logic look like ?
indeed , a real number.
and that is far from new.
or any integer ( countable infinite-valued logic ? )
also not new.
plz stick with the finite possibilities in logic as is standard.
( btw you might bump into AC otherwise )
>
> When you notice something that seems to contradict
> accepted mathematical facts you should consider the
> possibility that you're simply misunderstanding
> something.
may i remind you that 3-valued logic is , despite not so popular , an accepted mathematical concept ??!!!??
3-valued logic -> 26.000 hits on google with a lot of math pdf's , standard resourses , university links , references to mathematicians and logicians (like Bochvar) etc.
even a patent :
and wikipedia :
http://en.wikipedia.org/wiki/Multi-valued_logic
also mathplanet , mathworld , arxiv etc etc
here is a mathematician who studied 3-valued logic :
http://en.wikipedia.org/wiki/Jan_%C5%81ukasiewicz
and those are just the free sources !!
even your ( NOT MINE ) infinite-valued logic has a name :
fuzzy logic.
which you should have known if you were a real mathematician !
really david .... tss .....
so , who is the fool now hmmm ??
Especially when what you've "noticed"
> is so basic and elementary that it's not reasonable
> to assume that it's simply been overlooked for
> centuries.
considering the refs i gave above , its not totally overlooked.
but its neither popular or well known.
but it should be !!!
( even JSH and galathaea have investigated it )
regards
tommy1729
amy666
> In article
> <16114eb2-7ce9-49ab...@m45g2000hsb.goog
Very well said Virgil.
And finally someone who has at least heard of multiple valued logic and ALSO understood it.
( unlike david )
high regards
tommy1729
Virgil
<952077.12217731458...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
Thew existence of three diverse types of numerical values does not
require that "x = positive" has 3 possible logical values.
There are many more than three types of numerical values, so do you also
wish to claim a separate truth value for each of them?
>
>
> "2 is positive" is
> > true. "0 is positive" is false. "0 is negative" is
> > false.
> > "0 is neither positive nor negative" is true. There's
> > no need for any truth values other than the standard
> > two.
>
> but 3-valued logic is better in this case !!
>
> let 1 be true and 0 be false.
>
> is x positive (1 or 0) negative (1 or 0) or neither (1 or 0)
>
> FIRST there are 3 potential answers , thus 2 valued logic fails in the sense
> of a single 2-logic gate.
For the value of an unknown member of the naturals there are infinitely
many possible answers, so you better make your logic have infinitely
many truth values.
K_h
<leon street> wrote in message
It can be a difficult subject so don't beat yourself up too much. Every
binary string of the form 1011010011100... is in the family
{0,1}^{0,1,2,3,4,5,....}. In this family, there will be many strings that
cannot be obtained from other each other in any finite way. But with
infinite sets, combinatorial manipulations are not only finite in character.
For example, let J(S) be the set of all bijections from S to itself. That
is, J(S) is the set of all permutations of S. The factorial of a cardinal S
is S!:
S! = |J(S)|
Using the axiom of choice, it is possible to show that S!=2^S for any
infinite set S. So combinatorial analysis is not limited to finite sets nor
is it limited to finite manipulations of infinite sets. Hope this helps.
k
David R Tribble
>> For example, using julio's own notataion, 0(1) and
>> 1(0) are different infinite strings representing the same point.
>
Tommy/amy666 wrote:
> .. unless we consider that an infinitesimally small difference ...
Julio was talking about real numbers, so why do you bring up
infinitesimals?
And what infinitesimal difference did you have in mind, anyway?
David R Tribble
>> three possible values => positive , negative , neither.
>> thus three valued logic.
>
David C. Ullrich wrote :
>> No. A three-valued logic is a logic where _statement_
>> have three possible _truth values_. The fact that
>> a number can be positive, negative or neither does
>> not lead to any such statements.
>
Exactly. Consider:
Q. 0 is positive.
A. false.
Q. 0 is negative.
A. false.
Q. 0 is positive or negative.
A. false.
Tommy/amy666 wrote:
> real x = positive , negative or neither
> => three possible truth values.
Of course you left out the fourth value, "both".
Q. 7 is odd or prime.
(Wink, wink, nudge, nudge.)
K_h
"K_h" <KHo...@SX729.com> wrote in message
news:9dGdnSi_6LuFmE7V...@comcast.com...
Yo, I meant to use small case s for the cardinal and typed it in wrong!
Sorry if this caused any confusion. The fix is below:
Let J(S) be the set of all bijections from S to itself: J(S) is the set of
all permutations of S. Let s=|S| and the factorial of the cardinal s is:
s! = |J(S)|
Using the axiom of choice, it is possible to show that s!=2^s for any
infinite set S with s=|S|.
k
David C. Ullrich
wrote:
You're simply not paying attention. positive, negative
and neither are not truth values. Every statement
that's been made in this thread is either true or
false.
I didn't say it wasn't.
>3-valued logic -> 26.000 hits on google with a lot of math pdf's , standard resourses , university links , references to mathematicians and logicians (like Bochvar) etc.
>
>even a patent :
>
>
>http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO1&Sect2=HITOFF&d=PALL&p=1&u=%2Fnetahtml%2FPTO%2Fsrchnum.htm&r=1&f=G&l=50&s1=5644253.PN.&OS=PN/5644253&RS=PN/5644253
>
>and wikipedia :
>
>http://en.wikipedia.org/wiki/Multi-valued_logic
>
>
>also mathplanet , mathworld , arxiv etc etc
>
>here is a mathematician who studied 3-valued logic :
>
>http://en.wikipedia.org/wiki/Jan_%C5%81ukasiewicz
>
>and those are just the free sources !!
>
>
>even your ( NOT MINE ) infinite-valued logic has a name :
>
>fuzzy logic.
>
>which you should have known if you were a real mathematician !
>
>
>really david .... tss .....
>
>so , who is the fool now hmmm ??
You, unless you're just trolling.
>
>
>
> Especially when what you've "noticed"
>> is so basic and elementary that it's not reasonable
>> to assume that it's simply been overlooked for
>> centuries.
>
>considering the refs i gave above , its not totally overlooked.
The fact that you can give references that mention 3-valued
logic doesn't show that you're right about anything you've
said here.
amy666
> In article
> <20259354.1221766853...@nitrogen.math
> forum.org>,
> amy666 <tomm...@hotmail.com> wrote:
>
> > Virgil wrote :
> >
> > > In article
> > >
> <fc443151-a42c-48d7...@f36g2000hsa.goog
> > > legroups.com>,
> > > LudovicoVan <ju...@diegidio.name> wrote:
> > >
> > > > On 12 Sep, 15:39, Herman Jurjus
> <hjur...@hetnet.nl>
> > > wrote:
> > > > > Leon Street wrote:
> > > > > > Â Â Given a line segment AB, and a point P
> > > arbitrarily chosen upon it,
> > > > > > one can ask which half of AB P lies on,
> left or
> > > right, then having
> > > > > > selected
> > > > > > the half interval P lies on we can ask
> which
> > > half of that interval P lies
> > > > > > upon, and so on repeatedly. If we happen to
> > > have chosen a point P such
> > > > > > that
> > > > > > AP is incommensurable with AB, the point P
> will
> > > never lie exactly at the
> > > > > > end of any half interval. (It will never
> lie at
> > > the end of any fractional
> > > > > > interval of the line segment.) Â So the
> point P
> > > produces an infinite, and
> > > > > > aperiodic, infinite string eg LRRLLLR......
> > > > >
> > > > > > Â Â Is the converse true? That is, does an
which is why I , and perhaps many others - perhaps even you - , dont support julio's notation.
regards
tommy1729
amy666
> Tommy/amy666 wrote:
> >> three possible values => positive , negative ,
> neither.
> >> thus three valued logic.
> >
>
> David C. Ullrich wrote :
> >> No. A three-valued logic is a logic where
> _statement_
> >> have three possible _truth values_. The fact that
> >> a number can be positive, negative or neither does
> >> not lead to any such statements.
> >
>
> Exactly. Consider:
> Q. 0 is positive.
> A. false.
>
> Q. 0 is negative.
> A. false.
>
> Q. 0 is positive or negative.
> A. false.
assume the answers in order : false , true , false.
that is inconsistant !!
thus 2-valued logic does not apply to your example.
3-valued logic does not have that fallacy.
>
>
> Tommy/amy666 wrote:
> > real x = positive , negative or neither
> > => three possible truth values.
>
> Of course you left out the fourth value, "both".
because that is inconsistant.
which is the reason 3-valued is better than 2-valued.
( apart from bourbaki who does accept "both" most do not )
>
> Q. 7 is odd or prime.
>
> (Wink, wink, nudge, nudge.)
???
regards
tommy1729
Virgil
<4154431.12218228425...@nitrogen.mathforum.org>,
amy666 <tomm...@hotmail.com> wrote:
> David R Tribble wrote :
>
> > Tommy/amy666 wrote:
> > >> three possible values => positive , negative ,
> > neither.
> > >> thus three valued logic.
> > >
> >
> > David C. Ullrich wrote :
> > >> No. A three-valued logic is a logic where
> > _statement_
> > >> have three possible _truth values_. The fact that
> > >> a number can be positive, negative or neither does
> > >> not lead to any such statements.
> > >
> >
> > Exactly. Consider:
> > Q. 0 is positive.
> > A. false.
> >
> > Q. 0 is negative.
> > A. false.
> >
> > Q. 0 is positive or negative.
> > A. false.
>
> assume the answers in order : false , true , false.
>
> that is inconsistant !!
>
> thus 2-valued logic does not apply to your example.
>
> 3-valued logic does not have that fallacy.
The assumption of answers you require above is impossible with two
values logic since the third in two valued logic is true whenever either
the first or the second ( or both) are true.
So YOUR logic, of however many values, is inconsistent.
amy666
how many kinds of infinitesimals on the real line do you know about ?
regards
tommy1729