Deeper than primes - Continuation

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doronshadmi said:
I agree with you that I wrongly assumed that your professional level of mathematics enables you to understand that {a,b,c,...} or {{},{a,b,c,...},...} are expressions that are used "without loss of generality".
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That's the thing about Mathematics. It requires precision of expression, not just hand-waving and assumption.

Why the scary quotes, by the way?

Back to the actually topic of this latest arc, though, the base question is whether you correctly represented Cantor's Theorem. A really good place to start with something like that, yet another thing you passed over, would be which statement of Cantor's Theorem would you be presenting?

If it is the one involving the cardinality of a set and its power set, then there is a whole preface you need to provide about cardinality. Yet another thing you omitted.
 
jsfisher said:
You are having that much trouble figuring out what part of my post you responded to?
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You wrote that I were (mis-)responding to something, so please direct me to the part of your post that were (mis-)responded by me.
 
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doronshadmi said:
You wrote that I were (mis-)responding to something, so please direct me to the part of your post that were (mis-)responded by me.
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As already pointed out, that would be the text that you quoted from my post to which you responded.
 
doronshadmi said:
The claim that philosophy has no impact on the math, is simply your philosophical point of view, so?
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Here we see what is fundamentally wrong with the line of argumentation of Doron Shadmi; he mixes 'objective' and 'subjective' arguments as he sees fit without any logic to it.

This not only invalidates the rest of his work, it also makes it for him to see the error of his ways nigh impossible.

The reader will find the flaws in Doron Shadmi's reasoning obvious and glaring, that much of luck we have.
 
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jsfisher said:
That's the thing about Mathematics.
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{a,b,c,...} or {{},{a,b,c,...},...} are easily understood as A and P(A) expressions without a loss of generality, unless one insists to deal with notations instead of notions, and probably this is your case.

jsfisher said:
If it is the one involving the cardinality of a set and its power set, then there is a whole preface you need to provide about cardinality. Yet another thing you omitted.
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Not at all. It is the one involving P(A) members that is not paired with any A member, such that both P(A) and A members are inaccessible to the platonic common objective (discovered) level of actual infinity (notated the outer "{" and "}" in both sets).
 
doronshadmi said:
It is the one involving P(A) members that is not paired with any A member, such that both P(A) and A members are inaccessible to the platonic common objective (discovered) level of actual infinity (notated the outer "{" and "}" in both sets).
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Edited by full replacement:
So, what is the statement of Cantor's Theorem you had in mind?
 
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doronshadmi said:
{a,b,c,...} or {{},{a,b,c,...},...} are easily understood as A and P(A) expressions without a loss of generality, unless one insists to deal with notations instead of notions, and probably this is your case.
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You need to define your notation rigidly and thoroughly before you can even begin to communicate notions.

So unless one deals with notations, one can not begin to deal with notions.

This is one for the upcoming Doron's errors.
 
jsfisher said:
Edited by full replacement:
So, what is the statement of Cantor's Theorem you had in mind?
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That there is no bijection between P(A) and A members, which is a relative measure that is inaccessible to the absolute level of the platonic common existence of both sets (which is notated by the outer "{" and "}" of {a,b,c,...} or {{},{a,b,c,...},...} that are easily understood as A and P(A) expressions without a loss of generality).

"There exists set X" is only a tautology, where what comes after "such that" is not only a tautology (and in the case of Cantor's Theorem, J's "existence" involved with contradiction, where this contradiction has no impacted on the platonic common level of existence of both P(A) and A, because this common level of existence is only a tautology).
 
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doronshadmi said:
"There exists set X" is only a tautology, where what comes after "such that" is not only a tautology (and in the case of Cantor's Theorem, J's "existence" involved with contradiction, where this contradiction has no impacted on the platonic common level of existence of both P(A) and A, because this common level of existence is only a tautology).
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The irony here being that since tautology has multiple meanings, it fits well. Especially if one notices that Doron's explanations are tautologies...

:big:
 
As long as one does not understand that platonic existence is a discovery that does no need any meaning in order to exist (it is a tautology of existence that does not need any further interpretation or meaning), one wrongly gets this discovery only in terms of subjective invented multiple interpretations and meanings that are not the discovered tautology of existence.
 
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doronshadmi said:
That there is no bijection between P(A) and A members, which is a relative measure that is inaccessible to the absolute level of the platonic common existence of both sets (which is notated by the outer "{" and "}" of {a,b,c,...} or {{},{a,b,c,...},...} that are easily understood as A and P(A) expressions without a loss of generality).
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Wow. That's nothing like the Cantor's Theorem I've ever seen. It just goes all over the place. When you make stuff up, Doron, you really do go all out.

How would you go about proving that, umm, invention?

By the way, Doron, do you realize that a bijection between A and P(A) isn't the same as between the members of A and P(A)? The latter is gibberish.
 
doronshadmi said:
As long as one does not understand that platonic existence is a discovery that does no need any meaning in order to exist (it is a tautology of existence that does not need any further interpretation or meaning), one wrongly gets this discovery only in terms of subjective invented multiple interpretations and meanings that are not the discovered tautology of existence.
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Excellent. You finally admit your invented terminology has no meaning. On this we can all agree.
 
Since there are so many things that doronshamdi misunderstands, let me respond to this post.

doronshadmi said:
The claim that philosophy has no impact on the math, is simply your philosophical point of view, so?
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I believe that you are shifting the burden. You are the one that is claiming philosophy has an impact on math. Too bad you haven't provided any proof of that. Lots of claims, no proof.
 
jsfisher said:
By the way, Doron, do you realize that a bijection between A and P(A) isn't the same as between the members of A and P(A)? The latter is gibberish.
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It is gibberish from your one level reasoning of set, which does not distinguish between the discovered level of A and P(A) (notated by the outer "{" and "}") and the invented level of A and P(A) (notated by their members, and the mapping (which is not bijective, in this case) is done at the invented level of members of these sets).
 
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Little 10 Toes said:
You are the one that is claiming philosophy has an impact on math. Too bad you haven't provided any proof of that. Lots of claims, no proof.
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1. Axioms in the framework of mathematics do not need any proof, otherwise they are not axioms, in the first place.

2. By using philosophical view of these axioms, I explicitly demonstrated that they are derived from using platonic and non-platonic levels, where the existence at the platonic level is only tautology.

Let's take. for example the Axiom Of Infinity:

"There exists a set X (the discovered level of set X) such that (the invented level of set X) the empty set is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X".

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=10024836&postcount=3762.

So we are going even deeper than the mathematical level of axioms (which do not need any proof, otherwise they are not axioms, in the first place).
 
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doronshadmi said:
As long as one does not understand that platonic existence is a discovery that does no need any meaning in order to exist (it is a tautology of existence that does not need any further interpretation or meaning), one wrongly gets this discovery only in terms of subjective invented multiple interpretations and meanings that are not the discovered tautology of existence.
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This is hilarious without disambiguation of 'tautology'.

I think, because Doron uses the word tautology so often of late, that Doron does not know what tautology actually means...
 
doronshadmi said:
2. By using philosophical view of these axioms, I explicitly demonstrated that they are derived from using platonic and non-platonic levels, where the existence at the platonic level is only tautology.
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No such thing was done by you.

The previous sentence is an axiom.
 
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"Invention of the discovered" does not make sense, because the discovered (the platonic objective level of existence) can't be invented by the non-platonic subjective level of existence.

"Discovery of the invented" makes sense, because the invented (the non-platonic subjective level of existence) can be discovered by the platonic objective level of existence.

Again, the subjective level of existence depends on the objective level of existence, but not vice versa.
 
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By using Cantor's theorem it is shown in example 1 of http://www.internationalskeptics.com/forums/showpost.php?p=10030370&postcount=3795 that set {} exists even if according to its definition (as used by Cantor's theorem) any attempt to define some A member as its member, is involved with contradiction (at the level of members of that set).

Actually Cantor's theorem holds (there is a set, which is a member of P(A), that is not paired with any set that is a member of A (which enables to conclude that there is no bijection)) exactly because the level of being a set holds even if the level of being its member is involved with contradiction.
 
doronshadmi said:
By using Cantor's theorem it is shown in example 1 of http://www.internationalskeptics.com/forums/showpost.php?p=10030370&postcount=3795 that set {} exists even if according to its definition (as used by Cantor's theorem) any attempt to define some A member as its member, is involved with contradiction (at the level of members of that set).
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No it has not been shown. It has been claimed.

There is a fundamental difference between the two.

To 'show' something means you have to get the basic definitions and rules airtight, rock-solid and unambiguous. Which you are nowhere near to having.

Next, you need to prove any assertions by following your rules (and not conjuring up any new along the way or modifying the definitions) in a rigorous manner.

So, if you use mathematics, you are bound by the definitions and rules of mathematics.

If what you are using is not mathematics, but for instance philosophy, you need to lay the groundwork first.

There are no two ways about it. And this is the prime reason nobody can take your philosophizing seriously.
 
realpaladin said:
No it has not been shown. It has been claimed.

There is a fundamental difference between the two.
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It is worse than that. He's not even using Cantor's Theorem, and even if Cantor's Theorem were anything like what he imagines Cantor's Theorem to be, he still wouldn't be using Cantor's Theorem.

Cantor's Theorem, always available for anyone to use, is simply |S| < |P(S)| for any set S.

Doron is not using that (or anything like it) at all. Instead, he's convinced the theorem is actually its standard proof method, and it is the proof method he is misconstruing and misapplying (and then fantasizing about addition stuff he'd really like included).

It is fun to watch, though.
 
Little 10 Toes said:
Citation needed. Please show exactly where you got the full quote.
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doronshadmi said:
Please look again at the end of http://www.internationalskeptics.com/forums/showpost.php?p=10024836&postcount=3762.
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Let's see what I was asking to to provide a source on:
doronshadmi said:
Let's take. for example the Axiom Of Infinity:

"There exists a set X (the discovered level of set X) such that (the invented level of set X) the empty set is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X".
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So I go to the bottom of the post he mentioned:
doronshadmi said:
For example, nothing was stripped or "slightly hacked" from ZFC by post http://www.internationalskeptics.com/forums/showpost.php?p=10017698&postcount=3731, only in your one level imagination.


My mistake (I used what is written in http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory without first checking it) the correct one is "the empty set ({})". Thank you for your remark.
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Went to the first link, nothing there which matches with what I was asking a citation for. Went to the second link (Wikipedia). That leads me to the ZFC page. Nothing matches his quote.

Since we are talking about the Axiom of InfinityWP, lets go there.

Nothing matches with what doronshadmi has quoted.

So once again doronshadmi, give me the citation which shows:
doronshadmi said:
"There exists a set X (the discovered level of set X) such that (the invented level of set X) the empty set is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X".
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jsfisher said:
It is worse than that. He's not even using Cantor's Theorem, and even if Cantor's Theorem were anything like what he imagines Cantor's Theorem to be, he still wouldn't be using Cantor's Theorem.

Cantor's Theorem, always available for anyone to use, is simply |S| < |P(S)| for any set S.

Doron is not using that (or anything like it) at all. Instead, he's convinced the theorem is actually its standard proof method, and it is the proof method he is misconstruing and misapplying (and then fantasizing about addition stuff he'd really like included).

It is fun to watch, though.
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I explain Cantor's theorem by using an example without loss of generality (http://en.wikipedia.org/wiki/Without_loss_of_generality), where this kind of explanation is supported by http://en.wikipedia.org/wiki/Cantor...ion_of_the_proof_when_X_is_countably_infinite.

Here is my example without loss of generality:

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{} (which is some P(A) member) is defined as "the set of all A members that are not members of the P(A) members that are paired with them", by the following example:
Code: 
{ a , b , c , ... }
  ↕   ↕   ↕    
{{a},{b},{c}, ... }
Since, by this example, all A members are members of the P(A) members that are paired with them, we get the existing set {} (the empty set) as the P(A) member that is not paired with any A member, exactly because any attempt to define some A member as its member, is involved with contradiction, where this contradiction is only at the level members (the contradiction at the level of members does not have any impact on the level of existence of {}, and by using {} existence without a loss of generality, it can be concluded that there is no bijection (no A member is paired with {}, and this example is used without loss of generality).
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Jsfisher, it is realized that your "Cantor's Theorem, always available for anyone to use, is simply |S| < |P(S)| for any set S" statement is not supported by your understanding that Cantor's Theorem rigorously can be explained by using examples without loss of generality.
 
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jsfisher said:
It is worse than that. He's not even using Cantor's Theorem, and even if Cantor's Theorem were anything like what he imagines Cantor's Theorem to be, he still wouldn't be using Cantor's Theorem.

Cantor's Theorem, always available for anyone to use, is simply |S| < |P(S)| for any set S.

Doron is not using that (or anything like it) at all. Instead, he's convinced the theorem is actually its standard proof method, and it is the proof method he is misconstruing and misapplying (and then fantasizing about addition stuff he'd really like included).
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Well, that is about the Alpha and Omega of just about any mathematics crackpot; claiming mathematics can not deal with little squibly bits hanging on an orange grown on a tree somewhere in the vicinity of Betelgeuse because 'they' forgot to think about 'this and that and such and so'...

And since nobody in their social lives wants to spend so much time kibbitzing, they always think 'I must have something here...'

Ah well...

jsfisher said:
It is fun to watch, though.
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If this were frequented by more Japanese visitors, they would love this thread to death; it resembles those 'watch-it-for-the-humiliation-of-the-contestant' game shows they are so fond of.

And yes, I visit the JREF board exclusively for this thread now (I even donate so I can be sure of entertainment for years to come).
 
doronshadmi said:
Jsfisher, it is realized that your "Cantor's Theorem, always available for anyone to use, is simply |S| < |P(S)| for any set S" statement is not supported by your understanding that Cantor's Theorem rigorously can be explained by using examples without loss of generality.
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Has Doron started talking about himself in the third person neuter form now?

Besides that, the above quote says this:

"It is realized that your statement is not supported by your understanding that it can be explained by using examples without loss of generality."

How much hogwash can someone get into one single sentence?

And 'examples' to prove something? I have heard of 'examples' to disprove something, but afaik it is impossible to prove a general statement 'by example'.

This one is for the Doron Shadmi's error pages.
 
doronshadmi said:
This time please read what is written in http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory about the axiom of infinity.
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7. Axiom of infinity[edit]
Main article: Axiom of infinity
Let S(w)\! abbreviate w \cup \{w\} \!, where w \! is some set (We can see that \{w\} is a valid set by applying the Axiom of Pairing with x=y=w \! so that the set z\! is \{w\} \!). Then there exists a set X such that the empty set \varnothing is a member of X and, whenever a set y is a member of X, then S(y)\! is also a member of X.

\exist X \left [\varnothing \in X \and \forall y (y \in X \Rightarrow S(y) \in X)\right ].
More colloquially, there exists a set X having infinitely many members. The minimal set X satisfying the axiom of infinity is the von Neumann ordinal ω, which can also be thought of as the set of natural numbers \mathbb{N}.
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The above quoted from wikipedia...

I don't see the 'discovered' or 'invented' there... do you, Doron?
 
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jsfisher said:
However, since arithmetic is built on set theory and not the other way around, defining cardinality of sets in terms of numbers isn't appropriate. Instead, we can define it as a relative measure, the comparison of the cardinality of two sets, and thereby avoid the use of numbers (until later).
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This is nothing but your philosophical view, which does not agree with, for example, Poincaré's philosophical view (as appears at the end of http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers#Problem):
These problems with defining number disappear if one takes, as Poincaré did, the concept of number as basic i.e. preliminary to and implicit in any logical thought whatsoever. Note that from such a viewpoint, set theory does not precede number theory.
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jsfisher, you simply unaware of the impact of your philosophical view on your mathematical understanding, and as long as this is the case, there is no communication between us.
 
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doronshadmi said:
This is nothing but your philosophical view, which does not agree with, for example, Poincaré's philosophical view (as appears at the end of http://en.wikipedia.org/wiki/Set-theoretic_definition_of_natural_numbers#Problem)
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You seem to have adopted "philosophical view" as your latest foil to the opposition. It is nice to see you have moved beyond the simplistic "you don't get it" and eschewed the verbal / spatial reasoning montage, but this new expression serves you no better than the prior. If anything it helps underscore your remarkable tendency to get things exactly backwards.

Do carry on, though. Like a blind squirrel, you may eventually find something of value.
 
doronshadmi said:
I explain Cantor's theorem by using an example....
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You go to such amazing lengths to make things seem far more complicated than they are. Cantor's Theorem is not at all hard to explain. For any set S, |S| < |P(S)|. See? Easy.
 
jsfisher said:
You go to such amazing lengths to make things seem far more complicated than they are. Cantor's Theorem is not at all hard to explain. For any set S, |S| < |P(S)|. See? Easy.
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I do believe he's trying to explain it to himself. In that case though, it might not be that easy...
 
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