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:
doronshadmi said:
You already gave the answer, so what exactly is the purpose of your question?
Click to expand...

You are a master at the sucker punch :wackytwitcy:

Again I thought I might be seeing what some of your words mean or at least a question that might move me in that direction.
Pleaser indulge what must be from your point of view a stupid question. You might have an opportunity to really illuminate how and why your Organic Mathematics is different from ordinary Mathematics.

I repeat with italics:

How much does 4 + 7 using |2| values in two-step equal?
 
Last edited:
doronshadmi said:
You took the wrong direction.
Click to expand...

I certainly did!

So, these mathematicians aren't a step by step lot after all. Well, I didn't think they were, since they don't take all those exhausting steps that can't ever reach the destination. In the case of .99999999.....
"using |N| values in one step the result is 1."

But why am I always getting the impression that you are saying that is wrong, that they are wrong?

They are doing it serially instead of parallely?
Serially: one guy does the whole |N| in one step?
Parallely: |N| guys do the whole |N| in one step?

Serially yields what?
Parallely yields what?

If they both yield the same value .99999.... equals 1,
where are they going wrong?

It seemed to me for years that you insisted that .99999... doesn't really equal 1.

When I suggested that maybe you were saying that it depended upon how the matter was observed, that is from one's perspective, you smacked that down.

So I'm left with puzzle pieces that don't fit together at all, words that are at cross purposes.

I can't even be certain that you allow 4 + 7 to equal 11. It seems it depends upon something other than ordinary addition. And I don't get what that something is.
 
Apathia said:
When I suggested that maybe you were saying that it depended upon how the matter was observed, that is from one's perspective, you smacked that down.
Click to expand...
Your suggestion was after all correct, and recently I have found the way to do it by using greater transfinite cardinality as an observation's point of smaller cardinality of a given series, as explained,for example, in http://www.internationalskeptics.com/forums/showpost.php?p=10354597&postcount=237 and http://www.internationalskeptics.com/forums/showpost.php?p=10355185&postcount=249.

This perspective simply can't be used by a mathematical framework that does not use cardinality as a factor of a given solution, in this case.
 
Last edited:
I'm far from seeing a clear explanation, but I have a hazy sense of how playing with infinities could open a crack of ambiguity.
Since I'm not a mathematician, such a suggestion doesn't horrify me.
But to elucidate that ambiguity in a clear mathematical way is beyond me.
Now that you're not damning every mathematician since Euclid, I hope you could show the clear distinctions of perspective.
I'll still be reading, but I'll not be holding my breath.
 
Apathia said:
II have a hazy sense of how playing with infinities could open a crack of ambiguity.
Click to expand...
Exactly the opposite, using cardinality as a factor of a given solution, is more accurate than a framework that does not use cardinality as a factor of a given solution.
 
In my hazy "thinking" I thought I saw a crack.
This is definitely over my head.
My difficulties are very basic. I don't understand the words being used, especially "cardinality."

And I don't have the background and skill to really follow this conversation.
I'm just curious about how it's supposed to relate to ethics. That's another thing that I don't get. How does using Organic mathematics make anyone a better citizen? How does it work? That's what a non-mathematician wants to know.
 
Apathia said:
In my hazy "thinking" I thought I saw a crack.
This is definitely over my head.
My difficulties are very basic. I don't understand the words being used, especially "cardinality."
Click to expand...

Cardinality is a measure of quantity. It is usually applied to sets. The set {A, B, C} has three members, so its cardinality is 3. Vertical bars are used to symbolize cardinality, as in |{A, B, C}| = 3. (Vertical bars are also used for absolute value, but context usually resolves any ambiguity.)

It gets curious when the sets become infinite. The set of integers is an infinite set; so is the set of even numbers; and so is the set of prime numbers. It is also easy to show that each of those sets are the same size in that a one-to-one correspondence can be established between the elements of one set and another.

The set of real numbers is also infinite, but there is no one-to-one correspondence possible with the set of integers. The cardinality of the set of reals is larger than the set of integers.

There are, in fact, multiple infinities of different size.

The smallest infinity is called aleph-null (usually symbolized by a capital Hebrew aleph and subscript 0). The next smallest is aleph-one, then aleph-two, and so on.

If N denotes the set of whole numbers, then |N| = aleph-null.

What isn't known is just how big an infinity is the cardinality of the set of reals, R. It is suspected that |R| = aleph-one, but this is only a conjecture, the continuum hypothesis.
 
Last edited:
Thank you, jsfisher.

That was a good refresher. The best thing the Doron threads have done for me was diving me to do some reading about Mathematical Infinity. What you've just written is what I found in my reading. It's very fascinating.

Where cardinality becomes incomprehensible to me is whatever Doron means by it. I don't understand what a "higher cardinality" is.
 
Apathia said:
Thank you, jsfisher.

That was a good refresher. The best thing the Doron threads have done for me was diving me to do some reading about Mathematical Infinity. What you've just written is what I found in my reading. It's very fascinating.

Where cardinality becomes incomprehensible to me is whatever Doron means by it. I don't understand what a "higher cardinality" is.
Click to expand...


His "higher" is just a "greater than" relation. The set of reals is of "higher cardinality" than the set of whole numbers. I.e. |R| > |N|.

He also focuses on power sets only. (The power set of a set, S, is the set of all subsets of S, usually denoted by P(S).) Cantor famously proved |P(S)| > |S| for any set S, finite or infinite.

As it turns out, |R| = |P(N)|. In words, the set of reals is the same size as the power set of the whole numbers. But, we don't know if that is the same as aleph-one, so Doron may be skipping some of the infinities in his presentations.
 
Apathia said:
Where cardinality becomes incomprehensible to me is whatever Doron means by it.
Click to expand...

Over time, Doron has presented some contradictory views about infinity. I believe it his process view of things at the root, and so 0.999... never quite "gets to" 1.

His current presentation sharply contradicts his previous ones, though. Before, 0.999... was strictly less than 1 because you can never "get to infinity" by counting. Now, he's pulled a 180 degree turn by allowing us to get to infinity by counting, and then continue to count afterwards.

His argument boils down to if you only count to aleph-null, then, sure, 0.999... is 1, but if you keep going until you reach |R| then you get a different value and 0.999... is < 1.

Doron doesn't express it as counting, but that's the notion he's conveying. Why he has chosen |R| as a stopping point is anyone's guess.

It is all very tangled and completely sidesteps the point that if 0.999... = 1 when you "get to |N|", there is nothing more along the trip to |R| to change that (assuming the whole "getting to" notion had any merit in the first place).

It also completely sidesteps the cold fact that Mathematics has its own way of doing things, and this ain't it. No amount of foot-stomping tantrums changes that.
 
Last edited:
Thank you. That was very helpful.
That's how I would have otherwise taken "higher cardinality." It's just that in context of things Doron said over the years, it seems he has some special cardinality concept.

It seems to me this underlying "counting to" is where the confusion begins.
It's why for the ordinary, mathematically illiterate person, Zeno's paradoxes are still a shocker.

BTW Infinitessimal by Amir Alexander is still sitting unread on my shelf. I'll get around to it. It's about how the Catholic Church and other religious conservatives found the foundation of Calculus morally objectionable and heretical. I'm still trying to understand Doron's moral objections.
If he can have a working mathematics that suits his ideals, who am I to object. Like anyone else, I just want to see how and if it works, scientifically and morally.

His position is shifting, wobbling a bit. It's a disadvantage having read him so many years when things he wrote eight years ago both do and don't represent what he's trying to say now.
 
Apathia said:
If he can have a working mathematics that suits his ideals, who am I to object. Like anyone else, I just want to see how and if it works, scientifically and morally.
Click to expand...

I am with you on this. I have all but abandoned hope, though, that Doron will be able to describe his concepts in any complete sort of way.
 
My framework simply does not ignore cardinality as a factor of a given solution.

In case that the involved entities are numbers, a given series has it least cardinality |2|.

A series has at least 2 numbers, which provide a given number (a given solution) by exactly one-step (no step-by-step precess is involved).

If you read http://www.internationalskeptics.com/forums/showpost.php?p=10354597&postcount=237 by not ignore cardinality as a factor of a given solution, its content is easily understood.
 
Let's provide more details to my previous post.

The investigated object is [0,1].

Let's investigate some finite series:

0+1 is some cardinality |2| series that provides 1 by one-step.

0.910+0.0910+0.00910+0.00110 is some cardinality |4| series that provides 1 by one-step.

etc. ...

-----------------------------

Now let's investigate [0,1] by using countably infinite cardinality |N|.

0.910+0.0910+0.00910+... (which is a |N| cardinality series) provides 1 by one-step.

It is obvious that no series of countably finite cardinality |n| (where n is some natural number > 1) among 0.910+0.0910+0.00910+... (which is a |N| cardinality series) provides 1.

Actually, we are using here (at the level of countably infinite cardinality |N|) the fact that |N|+1=|N| in order to conclude that 0.910+0.0910+0.00910+...=1 (this "+1" provides the number of the interval [1,1] (which is number 1) that enables to claim that 0.910+0.0910+0.00910+...=1, exactly as done by jsfisher here:
jsfisher said:
What about Y = X union {[1,1]}. Isn't Y an infinite set? Do the members of Y not fully cover the interval, [0,1]?
Click to expand...
(http://www.internationalskeptics.com/forums/showpost.php?p=10296055&postcount=4183)

Actually this jsfisher's remark helped me to understand the |N|+1=|N| elegant transfinite trick that enables to claim that, for example, 0.910+0.0910+0.00910+...=1).

-----------------------------

Now let's investigate [0,1] by using uncountably infinite cardinality |P(N)| series.

Again, let's use, for example, 0.910+0.0910+0.00910... (which is a |N| cardinality series).

It is obvious that no |N| cardinality series among uncountably infinite cardinality |P(N)| series provides 1, simply because of the fact that |N|+1=|N|<|P(N)| (and in this framework cardinality is a factor of a given solution).

-----------------------------

So, the general conclusion in this framework is as follows:

A given series provides a given solution (by one-step), where the used cardinal numbers are factors of a given solution.

In this particular example, if a greater cardinality is also used, 1 is inaccessible to the series that is based on smaller cardinality.

-----------------------------

Let's transcend the real-line as follows:

BY using the tower of powers line |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... there are obviously the following numbers:

Between any two irrational numbers along the tower of powers line, there are |P(P(N))| numbers.

Between any two |P(P(N))| numbers along the tower of powers line, there are P(P(P(N)))| numbers.

Between any two P(P(P(N)))| numbers along the tower of powers line, there are |P(P(P(P(N))))| numbers.

...

etc. ... ad infinitum, where the inaccessible limit of the tower of powers line is simply the non-composed 1-dimesional space (which is not necessarily a metric space).

-----------------------------

My theorem is false only if |n>1| < |N| < |P(N)| < |P(P(N))| < |P(P(P(N)))| < |P(P(P(P(N))))| < ... is false.
 
Last edited:
The only thing fit after that chewed out to the bone post is: SFW? Maybe you are right.

Really, Doron, if you can not see that *nobody* is interested in just bickering and kibitzing, then that is where the real problem lies.

In 7 years you have not even inched towards the goal of unity.

You make claims on how other methods do not reach unity. Fair enough, they do not.

But you have yet to use your claims and build up to something. Anything.

All that is happening now is that you are pulling at a thread in a tapestry that is too complex for you to understand and hope that by yanking hard enough it unravels and it's components will be there for all to see.
 
Apathia said:
I'm still trying to understand Doron's moral objections.
If he can have a working mathematics that suits his ideals, who am I to object. Like anyone else, I just want to see how and if it works, scientifically and morally.
Click to expand...
Please, this time, ask more detailed questions about the contents of the following posts:

http://www.internationalskeptics.com/forums/showpost.php?p=10336151&postcount=170

http://www.internationalskeptics.com/forums/showpost.php?p=10336522&postcount=173


Apathia said:
His position is shifting, wobbling a bit. It's a disadvantage having read him so many years when things he wrote eight years ago both do and don't represent what he's trying to say now.
Click to expand...
Dear Apathia,

Lately I have found how to use cardinality in order to support OM's view of the linkage among Unity and multiplicity, where this linkage is the head and heart of the mathematician's work.

If you have questions about http://www.internationalskeptics.com/forums/showpost.php?p=10364025&postcount=277 , I'll do my best in order to answer to those questions.
 
Last edited:
Status
Not open for further replies.

Back
Top Bottom