Re: Question
- From: magidin@xxxxxxxxxxxxxxxxx (Arturo Magidin)
- Date: Tue, 25 Apr 2006 19:23:38 +0000 (UTC)
In article <1145991892.167851.295010@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
zuhair <zaljohar@xxxxxxxxx> wrote:
Arturo Magidin wrote:
In article <1145951829.288518.160480@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
zuhair <zaljohar@xxxxxxxxx> wrote:
[.snip.]
any real number can be represented as a specific sequence of numerals.
using the decimal numeral system for example it can be said that any
real number r can be represented by a string of numerals d_1 d_2 d_3
......... , were d_i = 0 or 1 or 2 or 3 or.... 9 or decimal dot, with
at most one of the d_i being a decimal dot. For example number 33 ,
here we have d_1 = 3 and d_2 =3 and d_3 = .
d_ j = 0 were j>3 .
You once again forgot to exclude the possibility of non-unique
representation (despite the fact that I mentioned it explicitly). Is
there any point in my replying to your posts? Do you read the
responses?
Well you have the right, in order to exclude non-unique representation
I will add a condition to what I have said above that the d_i which is
the decimal point shouldn't be d_1.
Second you think that number 1 can be represented as 0.99999........ (
decimal) , well I think that this is only an approximate representation
that is not correct.
Then you are, quite simply, wrong. 0.9999... represents the infinite
series 9/10 + 9/10^2 + ... + 9/10^n + ... which converges to 1, so
0.9999... is not an "approximate representation that is not correct",
but an EXACT representation that is EXACTLY equal to 1. Period.
The only representation of 1 is 1.0000......... and to me
0.9999........... < 1.
Then you are wrong. If that expression represented a number strictly
smaller than 1, call it x, then (1+x)/2 would be a number strictly
smaller than 1, and strictly larger than x.
What, pray tell, is the decimal representation of this number?
You must also either exclude the possibility that there is some k>0
such that d_i=9 for all i>k, or else the possibility that there is
some k>0 such that d_i=0 for all i>k. Otherwise, your claim about
"specific sequence of numerals" is false. For example, the real number
1 can be represented either as 1.000000..... or as
0.999999999999999999....
wrong see above.
Indeed, wrong. You are just wrong, completely, absolutely, and
totally.
Now the set of all possible permutations of d_1 d_2 d_3 .............
will be the set R.
No, it is not. For example, if the number you started with was "1",
written as 1.000000... then the set of all possible permutations of
these symbols does NOT yield the set of all real numbers. Numbers
whose decimal expansion have exactly k occurrences of the digit 7 can
only be obtained as permutations of each other, and no permutation of
it will yield a number whose decimal expansion has exactly (k+1)
occurrences of the digit 7 (or an infinite number of them).
All of that is gibberish, I didn't say that the set of all real numbers
comes from rearrangment of
the symboles used in representing a single real number.
Nonsense. You wrote: "the set of all possible permutations of
d_1d_2d_3... will be the set R". And previously, you defined
"d_1d_2d_3..." to be a specific string of numerals which represented a
specific real number. So you did, indeed, state that the set of all
real numbers comes from rearranging the symbols used in representing a
single real number. If that is not what you ->MEANT<- to say, well,
that's your problem for not saying what you mean and saying nonsense
instead. However, I strongly suspect that what you meant to say was
also nonsense, just a different kind of nonsense.
and this has the cardinality of c.
Now 2^c is the cardinality of the power set of R.
The power set of R is the set of all combinationas that can come from
R.
No. The power set of R is the set of all SUBSETS OF R. Not the set of
all "combinations of R", a term which you continue to use without
saying just what it means.
Their is no difference between these two terms , they are synonum.
No, they are not. "Combination" has a very specific meaning; "subset"
has a very specific meaning. They are not synonimous. Each subset
corresponds to a "combination without repetition of objects taken form
a set", but not every combination corresponds to a unique subset.
Since any member of R is represented by d_1 d_2 d_3...............
Then the power set of R will have numerals that consists from a
combination of this representation .
No. No. No. No. No.
Is five times enough, or do you need me to repeat it even more times?
Repeat what .
That you are, and continue to, speak nonsense.
what I meant by combinations is what you call all
subsets.
Still nonsense. A subset of R consists of a collection of elements of
R. This is not a rearrangement of a presentation of a number, nor does
it corresopnd to such a rearrangement, nor does it correspond to a
concatenation of two such representations. You are speaking nonsense,
over and over and over and over again.
Here is a subset of R:
{pi^r | r a positive real number}.
Pray tell, what are the "numerals that consist from a combination of
this representation" (->what<- representation?) that represents this
subset and no other?
The power set of R is not the set of all rearrangements of a
sequence. It is the set of all SUBSETS of R; the set of all sets, each
of which has all of its elements as real numbers.
I agree, but you read me wrong.
No, you speak wrong. So far, you either speak falsehoods or nonsense.
In order to describe this number we have 33. followed by Omega of zeros
which is followed by 1. which is followed by Omega of zeros. Now what
that sequence of numerals represent.
It represents nonsense.
Easy answer.
Sometimes the easy answer is correct. What you wrote above is
nonsense, and your assertions about representations of real numbers
are alternating between nonsense and falsehoods.
That is my question.
And the answer is: it represents nonsense. You are utterly confused:
the reals cannot be obtained as the set of all rearrangements of a
single real,
I never said that
You did say that. Exactly and explicitly. You perhaps did not MEAN
that, but you did a rather poor job of saying what you meant in that
case. What you said is exactly what I wrote above.
and the power set of the reals is not the "set of all
combinations of reals".
Yes they are. But you don't understand the world combinations.
I understand it perfectly well. You, on the other hand, don't even
know what a real is, apparently.
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@xxxxxxxxxxxxxxxxx
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