Cont: Deeper than primes - Continuation 2

Status
Not open for further replies.
psionl0 said:
As your diagram in http://www.internationalskeptics.com/forums/showpost.php?p=10334974&postcount=158 showed, this is true only as long as N is some finite subset of the set of natural numbers. You haven't demonstrated this to be true for a countably infinite set of natural numbers.

I'd go as far as to say that if N is the set of all natural numbers, then all of the supersets that you generate are also countably infinite.
Click to expand...
Once again http://www.internationalskeptics.com/forums/showpost.php?p=10339261&postcount=195.
 
psionl0 said:
Just to be technically accurate, the limit of 0.999... stands at 1. No matter how many 9's you add to the string, the sum will never add up to 1.

0.999... is a convergent countably infinite series. Trying to make more of it than that like doronshadmi is doing or making an unprovable assertion that an infinite string of 9's would still add up to < 1 is to just generate meaningless confusing jargon.
Click to expand...
http://www.internationalskeptics.com/forums/showpost.php?p=10354597&postcount=237

More comprehensive view is given in http://www.internationalskeptics.com/forums/showpost.php?p=10354437&postcount=235 and by reading all of it you can realize that 0.999...=1 if only |N| cardinality is used.

psionl0 said:
Trying to make more of it than that like doronshadmi is doing
Click to expand...
I am not trying to make more of it, 0.999... is the result of no more than |N| added values, and it = 1 if only |N| cardinality is used.
 
Last edited:
doronshadmi said:
jsfisher said:
(By the way, your assertion of |N| + 1 terms are involved is just bizarre.
Click to expand...
The bizarre thing here is actually your mathematical framework that excludes the different values of cardinality as an essential factor of a given solution.
Click to expand...

Ah! So, finally you admit you are redefining things. You reject "the bizarre thing" known as mathematics by using terms and concepts differently from their definitions.

....
jsfisher said:
0.9 + 0.09 + 0.009 + ... is an infinite series. The only thing important in determining the value of the series is the limit of the related sequence of partial summations. No infinity is needed to establish that the limit is 1 and thus 0.999... is identical to 1.
Click to expand...
that simply excludes the different values of cardinality (and in this case, transfinite cardinality) as an essential factor of a given solution.
Click to expand...

Why, yes. Yes, it does. That's how mathematics works in this case. It excludes nonsense promulgated by incredulous math cranks.
 
jsfisher said:
Ah! So, finally you admit you are redefining things.
Click to expand...
Wrong jsfisher, I simply wonder how so simple mathematical fact like http://www.internationalskeptics.com/forums/showpost.php?p=10354597&postcount=23 can't be grasped by you.

http://www.internationalskeptics.com/forums/showpost.php?p=10354437&postcount=235 does not redefine anything, it simply uses the fact (something that you don't use) that there are different levels of cardinality that can be used in order to provide mathematical solutions, that can't be provided if each cardinality is taken separately.


jsfisher said:
That's how mathematics works in this case.
Click to expand...
That how mathematics, when done by you, can't be developed beyond a framework that uses only each cardinal number separately from the other cardinal numbers, in order to provide a given solution.
 
Last edited:
jsfisher said:
Why, yes. Yes, it does. That's how mathematics works in this case. It excludes nonsense promulgated by incredulous math cranks.
Click to expand...
Thanks, I forgot about that word... it is now being used in an article, just because I like the 'feel' of it :)

doronshadmi said:
Wrong jsfisher, I simply wonder how so simple mathematical fact like http://www.internationalskeptics.com/forums/showpost.php?p=10354597&postcount=23 can't be grasped by you.

http://www.internationalskeptics.com/forums/showpost.php?p=10354437&postcount=235 does not redefine anything, it simply uses the fact (something that you don't use) that there are different levels of cardinality.


That how mathematics, when done by you, can't be developed beyond a framework that uses only each cardinal number separately from the other cardinal numbers, in order to provide a given solution.
Click to expand...

And here we have the *coughcough* "peace loving" Doron being insulting again with his bog-standard 'you are dumb' tactics.

Now... why would I want to *provide* a *given* solution???
 
psionl0 said:
Just to be technically accurate, the limit of 0.999... stands at 1.
Click to expand...

Not quite. The series represented by 0.999... does converge to a limit of 1, but that limit then is the value of the series. It is not simply something 0.999... gets close to; it is its value.

As a matter of definition, the value of a series is the limit of the corresponding sequence of partial sums.

L = Sum(k=1 to infinity) ak <==> L = lim(N -> infinity) Sum(k=1 to N) ak

0.999... is identically 1 and provably so.

A series can be thought of informally as a summation over infinitely many terms, in fact the sigma notation supports that conceptual view, but it runs into trouble because the informal model comes with a process concept wherein a partial summation gets closer and closer, but never quite reaches its final value. You cannot get to infinity by counting, so the idea of completing the summation conflicts with the apparent step-by-step process.

The formal definition lacks this defect and maintains full consistency.

Doron is mired in a process concept. He muddies the water more with his just use a different infinity approach, too. There are only just so many rational numbers possibly involved in the process view of 0.999.... Giving special consideration to aleph9 (for example) doesn't change that.
 
psionl0 said:
I'd go as far as to say that if N is the set of all natural numbers, then all of the supersets that you generate are also countably infinite.
Click to expand...

Yes, Doron is using N as the set of natural numbers. But Doron is correct with respect to the inequalities. Its power set, P(N), is of a higher cardinality than N.

That is just Cantor's Theorem: For any set S (finite or infinite), |S| < |P(S)|.
 
jsfisher said:
Doron is mired in a process concept.
Click to expand...
Parallel or serial one-step is defiantly not "mired in a process concept", as clearly explained in http://www.internationalskeptics.com/forums/showpost.php?p=10334974&postcount=158/


jsfisher said:
There are only just so many rational numbers possibly involved in the process view of 0.999....
Click to expand...
There are no more than |N| involved Q members here that provide a given value by one-step, which is exactly 1 if only |N| cardinality is involved, OR 0.999...10 < 1 exactly by 0.000...110 if 0.999...10 is among a series of |P(N)| involved values, which are uncountable and therefore symbolized by one notation, as explained, for example, in http://www.internationalskeptics.com/forums/showpost.php?p=10306748&postcount=4282 (the correction of the link in it is found in http://www.internationalskeptics.com/forums/showpost.php?p=10324522&postcount=34).

jsfisher said:
Giving special consideration to aleph9 (for example) doesn't change that.
Click to expand...
http://www.internationalskeptics.com/forums/showpost.php?p=10354774&postcount=244.
 
Last edited:
jsfisher said:
A series is the sum of a sequence of values. It is not, itself, a sequence.
Click to expand...
There are no operations among the values of sequence <0.9,0.09,0.009,...> , so it is trivially clear that sequence <0.9,0.09,0.009,...> in not the series 0.9+0.09+0.009+...

Yet both the series 0.9+0.09+0.009+... and the sequence <0.9,0.09,0.009,...> have at most |N| values.

My theorem (unlike your framework) cares about the cardinality of the participated values in a given series like 0.9+0.09+0.009+... , as explained in http://www.internationalskeptics.com/forums/showpost.php?p=10354774&postcount=244 and http://www.internationalskeptics.com/forums/showpost.php?p=10354597&postcount=237.
 
Last edited:
To use the words "at most" to describe an infinite number of something seems quite odd.
 
ehcks said:
To use the words "at most" to describe an infinite number of something seems quite odd.
Click to expand...

Hey ehcks,

There is more than one infinite number (known as transfinite numbers), for example:

|N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...
 
Last edited:
jsfisher said:
...abandons mathematics in favor of doronetics private terminology and meaning.
Click to expand...
Wrong jsfisher, you are simply exclude cardinality as a factor of given solutions, and in the case of infinity, your framework uses only ∞ for infinity.

Unlike you, my framework uses the well established |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ...

Moreover, since your framework excludes cardinality as a factor of given solutions, it does not have the ability to define a given value by using higher cardinality (stronger resolution) in order to determine a given value that was defined by lower cardinality (by lower resolution).

Generally, the power of resolution that is available in order to determine a given value, is well-defined by |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... mathematical fact, which is not used by jsfisher's framework.
 
Last edited:
doronshadmi said:
Wrong jsfisher, you are simply exclude cardinality as a factor of given solutions, and in the case of infinity, your framework uses only ∞ for infinity.
Click to expand...

Mathematics has its way of doing things; you have yours. The two are not the same.
 
Last edited:
Status
Not open for further replies.

Back
Top Bottom