Deeper than primes - Continuation

[realpaladin]

@JSFisher I guess he means that you are not 'using Philosophy'...

as if that were some method to be used....

 

[doronshadmi]

How many repetitions are required? Cantor's Theorem: For any set, S, |S| < |P(S)|.

You want a link for that? Ok, links.

So, after all jsfisher can't support his argument by using detailed reply to http://www.internationalskeptics.com/forums/showpost.php?p=10034959&postcount=3854 (including its links).

 

[jsfisher]

So, after all jsfisher can't support his argument by using detailed reply to http://www.internationalskeptics.com/forums/showpost.php?p=10034959&postcount=3854 (including its links).

Which detail in for any set, S, |S| < |P(S)| was unclear?

Since nothing like that appears in the posts you allege to explain Cantor's Theorem, it should be obvious (in every detail) that your posts do not explain Cantor's Theorem. It is almost as if you think Cantor's Theorem be something else entirely.

Here, let me explain Cantor's Theorem again to you: For any set, S, |S| < |P(S)|.

Can you find any parallels to this explanation in your posts?

 

[doronshadmi]

Ok, let's summarize what we have:

By using Philosophy (meta-view of Mathematics) and Mathematics, it is shown that the axioms of a formal theory like ZFC are expressions that combine at least two levels of existence, which are:

a) The discovered platonic (and therefore objective) level of existence, where this level of existence is logically a tautology (existence as always true) (this notion is symbolized by the outer "{" and "}" of any set, whether it is empty or non-empty).

b) The invented non-platonic (and therefore subjective) level of existence, where this level of existence is logically not a tautology (existence is not always true) (this notion is defined between the outer "{" and "}" of any set, and its non-tautological existence is symbolized (in the case of non-empty set) or not symbolized (in the case of the empty set)).

Cardinality at the level of ZFC is a relative measure that uses, at least, the notion of pairs (where a notion like pair is undefined unless the notion of numbers is defined (and in this case, at least number 2 and 1 are used)) in order to determine the adjustment (= (there is bijection) or < (there is no bijection)) between sets, without explicitly use numbers (yet, as shown above, numbers are used as hidden assumption by relative measure, and this hidden assumption is exposed by using Philosophical meta-view of a formal mathematical framework like ZFC).

Cantor's theorem uses relative measure between S and P(S) and proves (by using poof by contradiction) that S and P(S) members are not paired with each other (where this result is notated by |S|<|P(S)|, where S is a placeholder for any set (S is not any particular set)).

We can use an example Cantor's theorem, without loss of generality, in order to demonstrate the philosophical use of (a) and (b) by this theorem, as follows:

{} (which is some P(S) member) is defined as "the set of all S members that are not members of the P(S) members that are paired with them", by the following example:

{ a , b , c , ... }
  ↕   ↕   ↕    
{{a},{b},{c}, ... }

Since, by this example, all S members are members of the P(S) members that are paired with them, we get the existing set {} (the empty set) as the P(S) member that is not paired with any S member, exactly because any attempt to define some S member as its member, is involved with contradiction, where this contradiction is only at the level members (the contradiction is at the non-platonic level of existence of members and does not have any impact on the platonic level of existence of {}).

By using {} platonic existence (without a loss of generality) it can be concluded that there is no bijection (no S member is paired with {}, and this example is used without loss of generality).

Let's examine the notion of Cardinal numbers by (a) and (b) philosophical meta-view:

{||}=0 , {|{}|}=1 , {|{},{{}}|}=2 etc. , or {|{}, {{}}, {{},{{}}}, ... |} < aleph0, where aleph0 is a measurement at platonic level of existence, which is inaccessible to the non-platonic level of existence of members.

Let's understand, for example, ZFC Axiom Of Infinity by using (a) and (b) philosophical meta-view:

"There exists a set X (this is the part that uses set's platonic level of existence) such that (this is the part that uses set's non-platonic level of existence (the level of members, which defines set's identity, but not set's platonic level of existence)) {} is a member of X and, whenever a set y is a member of X, then S is also a member of X."

A question like "Please show me a member of set X that is missing from X" is irrelevant if one understands the different levels of existence of set X, by using Philosophy (meta-view of Mathematics) and Mathematics (the non-platonic level of existence of the members of set X is inaccessible to the platonic level of existence of set X).

 

[jsfisher]

Ok, let's summarize what we have

You mean what you made up, invented out of thin air without any basis other than a multi-year fool's errand hell-bent on proving Mathematics wrong.

By using Philosophy (meta-view of Mathematics) and Mathematics, it is shown that the axioms of a formal theory like ZFC are expressions that combine at least two levels of existence

You didn't show any such thing. You simple alleged it to be true without any proof or even basic definitions for your unique terminology usage.

...<some pipe dream stuff snipped>...

Cantor's theorem uses relative measure between S and P(S) and proves...

No and no. The theorem does not prove anything. The theorem simply states that for any set, S, |S|<|P(S)|.

Please stop conflating the theorem itself with a common proof method for the theorem.

...<more snippage>...
We can use an example Cantor's theorem...

Well, you haven't been able to so far.

...<latest failure snipped>...

...and this was no exception.

 

[realpaladin]

Ok, let's summarize what we have:

By using Philosophy (meta-view of Mathematics) and Mathematics, it is shown that the axioms of a formal theory like ZFC are expressions that combine at least two levels of existence, which are:

<snip>

Aaaand we can stop at the first paragraph, right there.

Philosophy is not a meta-view of mathematics. There is no such thing.

And *no* single proof is *ever* accepted by *anyone* if it says:

By using <school of thought X> it is shown that <Y>.

You could say that if and only if you point to a rigidly defined methodology.

Like:

By using <method X> it is shown that <Y>.

But since Doron just handwaves a bit in some unspecified direction, it is not even necessary to read any further.

The pillars under the rest of his discourse are faltering, so the whole of his discourse can be discounted.

Q.E.D.

 

[doronshadmi]

You mean what you made up, invented out of thin air without any basis other than a multi-year fool's errand hell-bent on proving Mathematics wrong.



You didn't show any such thing. You simple alleged it to be true without any proof or even basic definitions for your unique terminology usage.



No and no. The theorem does not prove anything. The theorem simply states that for any set, S, |S|<|P(S)|.

Please stop conflating the theorem itself with a common proof method for the theorem.



Well, you haven't been able to so far.



...and this was no exception.

Too many words that simply show that you have no argument.

As for Cantor's Theorem, it is Cantor's statement about S and P(S), where by using proof by contradiction it is proven that |S|<|P(S)|, so jsfisher you simply have no argument about anything of what is written in http://www.internationalskeptics.com/forums/showpost.php?p=10038541&postcount=3884, mathematically or philosophically.

Generally, a theorem is a mathematical statement that was proven, therefore (unlike mathematical hypothesis) a theorem indeed proves things.

 

[Little 10 Toes]

Too many words that simply show that you have no argument.

Pot, meet thyself.

Or ...

Look into a mirror lately?

 

[doronshadmi]

No matter how many times it will be explained, there are people that simply can't understand that Philosophy (by being used as meta-view of Mathematics) can provide new notions about mathematical axioms.

These new notions defiantly do not need any proof, because they go deeper than the axioms that are clarified by them, and it is well known that even at the level of axioms no proof is needed.

In other words, Philosophy AND Mathematics is a framework that simply can't be grasped by such people.

 

[doronshadmi]

Look into a mirror lately?

Yes, and what about you?

 

[jsfisher]

Too many words that simply show that you have no argument.

Now, there's a keeper.

As for Cantor's Theorem, it is Cantor's statement about S and P(S), where by using proof by contradiction....

Oh, by the way, given your new-found love of intuitionism (coupled with formalism with a gibberish twist), do keep in mind that only constructive proof methods are admissible. Proof by contradiction is not constructive. For that matter, you don't get ZFC, either, nor any of the mathematics built on ZFC.

There are a few ZF-like axiom sets you can start with, though. Which one did you have in mind?

 

[jsfisher]

These new notions defiantly do not need any proof, because they go deeper than the axioms that are clarified by them, and it is well known that even at the level of axioms no proof is needed.

So, are you wandering back to direct perception as you defense for lack of rigor, then?

Intuitionism tempered by formalism confounded by gibberish with a hardy dose of direct perception. The entertainment value of doronetics is ever increasing even though its utility holds constant.

 

[realpaladin]

No matter how many times it will be explained, there are people that simply can't understand that Philosophy (by being used as meta-view of Mathematics) can provide new notions about mathematical axioms.

Correct! Except for the meta-view nonsense, since no such thing exists.

And since philosophy (only people without education would write a capital there) ranges from contemplating the number of angels dancing on a pinhead to whether or not 'good' and 'evil' really exist, only ignorants would use a broad claim like 'using philosophy for something'.

If you philosophize, then, by the *rules* of philosophy (of which I have taken courses, since real science education also includes philosophy on science) you first need to specify your assumptions, your groundrules and your logic rules.

Since Doron has done none of this and uses the term philosophy as a blanket term for voodoo which he is unable to explain, you may discount anything where he uses the term philosophy.

 

[realpaladin]

These new notions defiantly do not need any proof, because they go deeper than the axioms that are clarified by them, and it is well known that even at the level of axioms no proof is needed.

And of course, this begs the question... if no proof is needed, why do you bend backwards to provide it?

If no proof is needed, get on with the show and let's move on to where you can explain your errors on the two islands again.

 

[doronshadmi]

Now, there's a keeper.

Your keepers do not improve your notions, because thay are not refer also to yourself for better self criticism. You should try it form time to time.

Oh, by the way, given your new-found love of intuitionism (coupled with formalism with a gibberish twist)

Ho, by the way, I do not use Intuitionism (which rejects the notion of actual infinity). In other words, I do use Platonism AND Formalism, such that actual infinity is at the platonic level of existence of a given formal axiom of sets, where potential infinity is at the non-platonic level of existence of the members of a given formal axiom of sets, where the non-platonic level of existence is inaccessible to the platonic level of existence.

Proof by contradiction is not constructive. For that matter, you don't get ZFC, either, nor any of the mathematics built on ZFC.

jsfisher, again you demonstrate your inability to use examples without loss of generality.

There are a few ZF-like axiom sets you can start with, though. Which one did you have in mind?

I deal with ZFC axioms. Since you do not use philosophical meta-view of Mathematics, you are unable to understand that Existential quantification is at the platonic level of existence (the level of actual infinity), where Universal quantification is at the non-platonic level of existence (Universal quantification dependents of the platonic level of existence of Existential quantification, but not vice versa).

So, once again, you have no argument, but this time please try to understand exactly why you have no argument. Maybe you can learn something be doing that.

 

[doronshadmi]

So, are you wandering back to direct perception as you defense for lack of rigor, then?

No, I use platonic and non-platonic levels of existence as fundamental terms of Philosophy AND Mathematics framework.

Your replies do not even scratch this framework (where this framework is introduced in http://www.internationalskeptics.com/forums/showpost.php?p=10038541&postcount=3884).

Here is an improved version of this link:

I wish to share with you my suggested framework, which combines Philosophy
and Mathematics, such that Philosophy is used as meta-view of Mathematics.

By using Philosophy (as meta-view of Mathematics) and Mathematics, it is
shown that the axioms of a formal theory like ZFC (* an example is given at
the end of this post) are expressions that combine at least two levels of
existence, which are:

a) The discovered platonic (and therefore objective) level of existence,
where this level of existence is logically a tautology (existence is always
true) (this notion is symbolized by the outer "{" and "}" of any given set,
whether it is empty or non-empty).

b) The invented non-platonic (and therefore subjective) level of existence,
where this level of existence is logically not a tautology (existence is not
always true) (this notion is defined between the outer "{" and "}" of any
given set, and its non-tautological existence is symbolized (in the case of
non-empty set) or not symbolized (in the case of the empty set)).

Cardinality at the level of ZFC is a relative measure that uses, at least,
the notion of pairs (where a notion like pair is undefined unless the notion
of numbers is defined (and in this case, at least number 2 and 1 are used))
in order to determine the adjustment (= (there is bijection) or < (there is
no bijection)) between sets, without explicitly use numbers (yet, as shown
above, numbers are used as hidden assumption by relative measure, and this
hidden assumption is exposed by using Philosophical meta-view of a formal
mathematical framework like ZFC).

Cantor's theorem uses relative measure between S and P(S) which enables to
prove (by using poof by contradiction) that S and P(S) members are not paired
with each other (where this result is notated by |S|<|P(S)|, where S is a
placeholder for any set (S is not any particular set)).

We can use an example of Cantor's theorem, without loss of generality, in
order to demonstrate the philosophical use of (a) and (b) by this theorem, as
follows:

{} (which is some P(S) member) is defined as "the set of all S members that
are not members of the P(S) members that are paired with them" by the
following example:

{ a , b , c , ... }
  ↕   ↕   ↕
{{a},{b},{c}, ... }

Since, by this example, all S members are members of the P(S) members that
are paired with them, we get the existing set {} (the empty set) as the P(S)
member that is not paired with any S member, exactly because any attempt to
define some S member as its member, is involved with contradiction, where
this contradiction is only at the level members (the contradiction is at the
non-platonic level of existence of members and does not have any impact on
the platonic level of existence of {}).

By using {} platonic existence (without a loss of generality) it can be
concluded that there is no bijection (no S member is paired with {}, and this
example is used without loss of generality).


Let's examine the notion of Cardinal numbers by (a) and (b) philosophical
meta-view:

{||}=0 , {|{}|}=1 , {|{},{{}}|}=2 etc. , or {|{}, {{}}, {{},{{}}}, ... |} <
aleph0, where aleph0 is a measurement at platonic level of existence (notated
here by the outer "{" and "}" of any given set, empty or non-empty), which is
inaccessible to the non-platonic level of existence of members.

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

* Let's understand, for example, ZFC Axiom Of Infinity, by using (a) and (b)
philosophical meta-view:

"There exists a set X (this is the part that uses set's platonic level of existence) such that (this is the part that uses set's non-platonic level of existence (the level of members, which defines set's identity, but not set's platonic level of existence)) {} is a member of X and, whenever a set y is a member of X, then S is also a member of X."

A question like "Please show me a member of set X that is missing from X"
(where the aim of this question is to support the notion of completeness at
the non-platonic level of existence of members) is irrelevant if one
understands the different levels of existence of set X, by using Philosophy
(as meta-view of Mathematics) and Mathematics (the non-platonic level of
existence of the members of set X is inaccessible to the platonic level of
existence of set X).

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

By using Philosophy as meta-view of Mathematics, one enables unable to understand that Existential quantification is about the platonic level of existence (which also the level of actual infinity), where Universal quantification is about the non-platonic level of existence (where the non-platonic level of existence is inaccessible to the platonic level of existence).

 

[doronshadmi]

At the end of my previews post, I wrongly wrote

By using Philosophy as meta-view of Mathematics, one enables unable to understand ...

It has to be:

"By using Philosophy as meta-view of Mathematics, one enables to understand ..."

 

[jsfisher]

Ho, by the way, I do not use Intuitionism (which rejects the notion of actual infinity).

"Ho" is what Santa says. The interjection you need is "oh".

As for intuitionism, if you are going to reject it, you should at least understand it better. Intuitionism does not categorically reject actual infinity. It is not one thing, a monolith, and there are differing formulations of intuitionism having differing views on infinity.

If I recall correctly, both IZF and CZF include intuitionist versions of the Axiom of Infinity, suggesting broad acceptance, not rejection, of actual infinity.

In other words...

You continually misuse this phrase. It is equivalent to, "Let me restate that same thing only differently." You use it as a connector of two orthogonal ideas. "I hate bananas. In other words, I love apples." No.

In other words, your constructions involving "in other words" are gibberish.

I do use Platonism AND Formalism...

No, you do not. If you want to delve into a philosophy of mathematics, have at it. Philosophy provides a foundation, a background logic (sometimes called the meta-logic) upon which to build the mathematics.

It does not, as you continue to insist, give you leave to post hoc re-interpret mathematics established under one philosophy to claim it really means something completely different under a different, and in your case poorly formed philosophy.

What you are doing is ass-backwards.

If you want to adopt some Shadmized philosophy melded from formalism and platonism, knock yourself out. However, you'll need to build up your own mathematics from it, and you get no entry to attack ZFC, et al., which are rooted in classic logic.

 

[Little 10 Toes]

The subtly of English escapes doronshadmi.

 

[AdMan]

At the end of my previews post, I wrongly wrote



It has to be:

"By using Philosophy as meta-view of Mathematics, one enables to understand ..."

No. Look up the word enables. Oh, and previews.

The fact you are so careless with language says a lot.