Deeper than primes - Continuation

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

as if that were some method to be used....
 
doronshadmi said:
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).
Click to expand...


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?
 
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:
Code: 
{ 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(y) 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).
 
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doronshadmi said:
Ok, let's summarize what we have
Click to expand...

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
Click to expand...

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...
Click to expand...

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...
Click to expand...

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

...<latest failure snipped>...
Click to expand...

...and this was no exception.
 
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doronshadmi said:
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>
Click to expand...

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.
 
jsfisher said:
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.
Click to expand...

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.
 
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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.
 
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doronshadmi said:
Too many words that simply show that you have no argument.
Click to expand...

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....
Click to expand...

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?
 
doronshadmi said:
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.
Click to expand...

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.
 
doronshadmi said:
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.
Click to expand...

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.
 
doronshadmi said:
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.
Click to expand...

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.
 
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jsfisher said:
Now, there's a keeper.
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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.


jsfisher said:
Oh, by the way, given your new-found love of intuitionism (coupled with formalism with a gibberish twist)
Click to expand...
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.

jsfisher said:
Proof by contradiction is not constructive. For that matter, you don't get ZFC, either, nor any of the mathematics built on ZFC.
Click to expand...
jsfisher, again you demonstrate your inability to use examples without loss of generality.

jsfisher said:
There are a few ZF-like axiom sets you can start with, though. Which one did you have in mind?
Click to expand...
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.
 
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jsfisher said:
So, are you wandering back to direct perception as you defense for lack of rigor, then?
Click to expand...
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:
Code: 
{ 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(y) 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).
 
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At the end of my previews post, I wrongly wrote

doronshadmi said:
By using Philosophy as meta-view of Mathematics, one enables unable to understand ...
Click to expand...

It has to be:

"By using Philosophy as meta-view of Mathematics, one enables to understand ..."
 
doronshadmi said:
Ho, by the way, I do not use Intuitionism (which rejects the notion of actual infinity).
Click to expand...

"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...
Click to expand...
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...
Click to expand...

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.
 
doronshadmi said:
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 ..."
Click to expand...

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

The fact you are so careless with language says a lot.
 
AdMan said:
No. Look up the word enables. Oh, and previews.

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

On the one hand, since I do not think English is Doron's first language, I cut him some slack; on the other, though, since he often stubbornly adheres to his linguistic nonsense despite correction, the slack is extremely limited.

In any language, his expressions are muddled.

In other words, I love apples.
 
Published anything yet, doron? You could spare us until there's been some actual peer review of your "work", don't you think? You fed us enough gibberish and I personally am fully convinced you're able to do so endlessly. No point in doing it over and over.
 
jsfisher said:
If I recall correctly, both IZF and CZF include intuitionist versions of the Axiom of Infinity, suggesting broad acceptance, not rejection, of actual infinity.
Click to expand...
I agree with you jsfisher, we should not talk about acceptance or rejection of Actual Infinity, but about the understanding of the notion of Actual Infinity.

The first step in order to understand this notion is to ask: "What is Actual Infinity?"

My answer is:

Actual Infinity is the existence that is logically always true (tautological existence).

IZF, CZF or ZF are axiomatic frameworks about sets, so in order to understand Actual Infinity in therms of sets, we first have to understand what is a set in terms of existence, where tautological existence and non-tautological existence are (a) and (b), as follows:

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)).

In other words, as long as IZF, CZF or ZF axioms are used in order to understand Actual Infinity in terms of (b), no understanding of Actual Infinity is established.

Here is, for example, ZF Axiom Of Infinity:

"There exists a set X (this is (a) part of the axiom) such that (this is (b) part of the axiom) {} is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X."

By this axiom, X has two levels of infinity, where (a) is the actual level and (b) is the non-actual level.

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

A request like "Please show me a member of set X that is missing from X"
(where the aim of this request is to support the notion of completeness (and therefore Actual Infinity) 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).
 
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doronshadmi said:
I agree with you jsfisher, we should not talk about acceptance or rejection of Actual Infinity, but about the understanding of the notion of Actual Infinity.
Click to expand...

Great! except that isn't what I said.

The first step in order to understand this notion is to ask: "What is Actual Infinity?"

My answer is:

Actual Infinity is the existence that is logically always true (tautological existence).
Click to expand...

My, isn't that curious. You take a perfectly fine, well-understood concept of completeness and infinitely many as in, for example, the set of integers, and you decide you can redefine it to mean something entirely different.

No. Your answer is horribly wrong.

IZF, CZF or ZF are axiomatic frameworks about sets...
Click to expand...

...each built on a philosophic framework that isn't Doron's to redefine post hoc.
 
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AdMan said:
No. Look up the word enables. Oh, and previews.

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

It is of course previous. English is not my mom's language so I use a speller.

From time to time I write a word in a wrong way and the speller automatically changes it to some valid word, even if it does not fit to what I actually wish to express. If the result of the automatic correction looks like the word that I actually wish to express, there is a chance that I'll miss the mistake.
 
jsfisher said:
You take a perfectly fine, well-understood concept of completeness and infinitely many as in, for example, the set of integers, and you decide you can redefine it to mean something entirely different.[/I].
Click to expand...
It is not entirely different it is simply refined, and you simply do not accept this refining, so?
 
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jsfisher said:
It is entirely different,
Click to expand...
Unlike you, jsfisher, I explicitly define Actual Infinity and Non-actual Infinity in terms of existence, by using Logic:

Actual Infinity is the existence that is logically always true (tautological existence).

From this definition it is easily deduced that Non-actual Infinity is the existence that is logically not always true (it is not tautological existence).

By using this definitions I explicitly demonstrate how they are used by ZF Axiom Of Infinity.

In other words, jsfisher, you have no case (all you have is an orthodox standpoint about my definitions and their use).
 
doronshadmi said:
Unlike you, jsfisher, I explicitly define Actual Infinity and Non-actual Infinity in terms of existence, by using Logic:

Actual Infinity is the existence that is logically always true (tautological existence).
Click to expand...

A singleton is infinite????
 
A singleton has cardinality 1, for example {|{}|} or {|{x}|} or {|x|} (where by ZF x is a set).

Again, given any set, the outer "{" and "}" are used to notate the notion of the level of existence that is logically always true.

This is not the case with the level of members (they may exist (for example {{}}) or not (for example {}).
 
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jsfisher said:
Redefinition of established terminology is not within you purview.
Click to expand...
Your used reasoning is not sufficient in order to distinguish between tautological existence and non-tautological existence, so?
 
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jsfisher said:
You don't get to redefine established notation, either.
Click to expand...
Notations (or their absence) are used in order to express notions.

This is exactly what is done in order to distinguish between tautological and non-tautological levels of existence, which explicitly used by ZF Axiom Of Infinity (more details are given in http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903).

jsfisher, your orthodox reasoning is not useful in order to address the difference between tautological and non-tautological levels of existence, where these levels are explicitly used by ZF Axiom Of Infinity.

In other words, you have no case (except some fetish attitude about notations).
 
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jsfisher said:
You take a perfectly fine, well-understood concept of completeness and infinitely many as in, for example, the set of integers, and you decide you can redefine it to mean something entirely different.[/I].
Click to expand...
Any set, including the set of integers, is the result of the linkage between two level of existence, where one level is logically tautological existence, and the other level is logically non-tautological existence.

The use of these two levels of existence is clearly shown in ZF axiomatic framework (http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903), where your orthodox reasoning is insufficient in order to state any useful thing about it.
 
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doronshadmi said:
Any set, including the set of integers, is the result of the linkage between two level of existence, where one level is logically tautological existence, and the other level is logically non-tautological existence.
Click to expand...

Or so you claim. Corrupting language is not an acceptable method for establishing your claim.

The use of these two levels of existence is clearly shown in ZF axiomatic framework
Click to expand...

So clearly, in fact, that you are completely unable to show where these levels exist in ZF. Instead, you simply reiterate your claim, or re-post links to previous claim reiterations, or both. Never do you actually show anything other than a baseless claim infused with gibberish.
 
jsfisher said:
Or so you claim. Corrupting language is not an acceptable method for establishing your claim.



So clearly, in fact, that you are completely unable to show where these levels exist in ZF. Instead, you simply reiterate your claim, or re-post links to previous claim reiterations, or both. Never do you actually show anything other than a baseless claim infused with gibberish.
Click to expand...
jsfisher, from your orthodox standpoint the best you can get about http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903 is indeed gibberish.

I wish you the best.
 
doronshadmi said:
"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(y) is also a member of X."
Click to expand...

doronshadmi said:
"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(y) is also a member of X."
Click to expand...

Still waiting for the citation that shows exactly your quote.
 
Little 10 Toes said:
Still waiting for the citation that shows exactly your quote.
Click to expand...
This is my last post to you on this subject.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=10039964&postcount=3903.

Now take

doronshadmi said:
"There exists a set X (this is (a) part of the axiom) such that (this is (b) part of the axiom) {} is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X."
Click to expand...


and omit what is written in italic letters within the brackets

( http://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory#7._Axiom_of_infinity )
... there exists a set X such that the empty set Ø is a member of X and, whenever a set y is a member of X, then S(y) is also a member of X.
Click to expand...

and here it is.
 
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