Deeper than primes - Continuation

[doronshadmi]

let's start with what you got right.

Let's start with what you still do not get.

Any given set exists as an abstract mathematical object, whether it does or does not have members.

The notion of this abstract existence (which is inaccessible to the abstract existence of its members) is notated by the outer "{" and "}".

In other words, no matter how many members a given set has, they are inaccessible to the abstract level of existence of the set (where this inaccessible level is notated by the outer "{" and "}").

If you are able to understand what is written above, then you have no problems to understand the following:

|{}|=0 , |{{}}|=1 |{{},{{}}}|=2 etc. , or |{{}, {{}}, {{},{{}}}, ... }| < aleph0 exactly because the level members is inaccessible to the level of being a set, where the abstract notion of being a set is notated by the outer "{" and "}" (where aleph0 is at the level of being a set).

So, the level of (abstractly) existing members is no more than potential infinity w.r.t (abstractly) actual infinity at the level of being a set, where "|{{}, {{}}, {{},{{}}}, ... }| < aleph0" is the exact abstract mathematical expression of this notion (the abstract notion of the difference between potential infinity and actual infinity).

By using only verbal_symbolic expressions for sets (for example: Ø , S , N etc.) one simply misses the abstract notion of the difference between potential infinity and actual infinity.

 

[jsfisher]

Please explain it in more details (do you mean that something like Hilbert's paradox of the Grand Hotel is used? (in that case please see http://www.internationalskeptics.com/forums/showpost.php?p=10015020&postcount=3717)).

No, Doron, I do not mean something like Hilbert's Hotel. I mean just what I said. ZFC Set Theory has no numbers, and so set cardinality (at the set theoretic level) cannot possibly involve numbers.

...
if you really wish to discuss about what you call doronetics
...

Doronetics is boring. It is not well-defined, it is not consistent, and it is not at all useful. Why would I want to discuss doronetics?

On the other hand, you don't present your bare assertions with a "in doronetics..." qualifier. If you did, your posts would pass mostly unchallenged and almost entirely ignored. Instead, you attack real Mathematics with out basis or logic.

 

[doronshadmi]

No, Doron, I do not mean something like Hilbert's Hotel. I mean just what I said. ZFC Set Theory has no numbers, and so set cardinality (at the set theoretic level) cannot possibly involve numbers.

You wrote:

At the set-theoretic level, cardinality is a relative measure.

Please explain in more details what exactly do you mean by "cardinality is a relative measure" ?

 

[jsfisher]

You wrote:

Please explain in more details what exactly do you mean by "cardinality is a relative measure" ?

The statement still has the same meaning it had in all the previous times it was made. Some examples:

http://www.internationalskeptics.co...768789&highlight=relative+measure#post9768789
http://www.internationalskeptics.co...761429&highlight=relative+measure#post9761429
http://www.internationalskeptics.co...742620&highlight=relative+measure#post9742620
http://www.internationalskeptics.co...455142&highlight=relative+measure#post8455142

 

[doronshadmi]

As Cantor observed in his work, cardinality in its basic form in Set Theory is a relative measure. Less than, greater than, and equal to. The cardinality of {a} and {a,a,a,a,a}, those being indistinguishable sets, are equal.

Again, set theory provides no way to determine how many of an element are in a set, only whether it is in the set.

What is written above does not change my argument, which is:

Any given set exists as an abstract mathematical object, whether it does or does not have members (for example {} exists even if the level of its members is empty of existing members).

The notion of this abstract existence (which is inaccessible to the abstract existence of its members) is notated by the outer "{" and "}".

In other words, no matter "what is in the set" (your words), it is inaccessible to the abstract level of existence of the set (where this inaccessible level is notated by the outer "{" and "}").

This inaccessibility is shown also when we use set theory as the basis of cardinal numbers, for example:

|{}|=0 , |{{}}|=1 |{{},{{}}}|=2 etc. , or |{{}, {{}}, {{},{{}}}, ... }| < aleph0 exactly because the level members is inaccessible to the level of being a set, where the abstract notion of being a set is notated by the outer "{" and "}" (where aleph0 is at the level of being a set).

So, the level of (abstractly) existing members is no more than potential infinity w.r.t (abstractly) actual infinity at the level of being a set, where "|{{}, {{}}, {{},{{}}}, ... }| < aleph0" is the exact abstract mathematical expression (in terms of cardinal numbers) of this notion (the abstract notion of the difference between potential infinity and actual infinity).

By using only verbal_symbolic expressions for sets (for example: Ø , S , N etc.) one simply misses the abstract notion of the difference between potential infinity and actual infinity.

 

[jsfisher]

What is written above does not change my argument

Really? What about the part where you referred to numbers? Besides, your "argument" seems to be that for any element X in I (where I is the induction set), |X| < |I|. I am unimpressed by your revelation.

...which is:...
The notion of this abstract existence (which is inaccessible to the abstract existence of its members) is notated by the outer "{" and "}".

You made that up. Unless you can establish this personal invention of yours in actual mathematics, your argument dies on the vine right here.

 

[doronshadmi]

Really? What about the part where you referred to numbers?

The part that is referred to numbers, comes after the part that is referred to the set theory level.

Moreover, at the set theory level I show that the abstract level of set's of existence , is independent of the abstracted level of members existence.

It can be shown right form the empty set, and goes further to non-empty sets.

In other words, the identity of a given set is derived from the level of members, but this identity can't be expressed unless it is related to the independent already (abstract) existing level of being a set.

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

There exists a set X (set X already exists) such that (and here comes the part that defines the identity of the already abstract exiting level of being a set) the empty set {} is a member of X and, whenever a set y is a member of X, then S is also a member of X.

 

[Little 10 Toes]

Let's start with what you still do not get.

Ok then.

Any given set exists as an abstract mathematical object, whether it does or does not have members.

No. I have a set of objects that are on my computer desk. They are not an abstract mathematical object. Perhaps you don't get what a set is.

The notion of this abstract existence (which is inaccessible to the abstract existence of its members) is notated by the outer "{" and "}".

Perhaps, again, you don't understand what a set is. Why would a set's members want to have access to the set's existence, abstract or otherwise? { and } are not notions, they are what has been agreed upon what will show the start and end of set notation. For example, the comma that is used to separate items in a list is not itself an item of the list (milk, cookies, bed). Same thing with sets and { }.

The rest of the post is based on the above misunderstandings.

 

[jsfisher]

The part that is referred to numbers, comes after the part that is referred to the set theory level.

This statement does not parse. But it doesn't matter much since bringing in numbers as you have done moves things beyond ZFC.

Moreover, at the set theory level I show that the abstract level of set's of existence , is independent of the abstracted level of members existence.

Showed? You didn't show anything. You alleged. You also invented. You didn't show.

It can be shown right form the empty set, and goes further to non-empty sets.

Do your "showing" within ZFC. That gives you a handful of axioms to work with and not much else. Show us what accessibility means and how it works within the confines of those axioms.

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

Great! You are actually going back to the basics....

There exists a set X (set X already exists) such that (and here comes the part that defines the identity of the already abstract exiting level of being a set) the empty set {} is a member of X and, whenever a set y is a member of X, then S is also a member of X.

Yes, and...?

A slightly hacked restatement of the Axiom doesn't get you to any conclusion. Where did you want to go from here?

 

[doronshadmi]

Ok then.

No. I have a set of objects that are on my computer desk. They are not an abstract mathematical object. Perhaps you don't get what a set is.

Little 10 Toes, the physical aspect of set's existence is the trivial level of this notion, where at this trivial level one can't fully develop the notion of set.

Therefore Mathematics deals with the notion of set, without being restricted only to its physical (non-abstract) aspect.

 

[doronshadmi]

A slightly hacked restatement of the Axiom doesn't get you to any conclusion. Where did you want to go from here?

Set's (abstract or non-abstract) existence, is independent on its (abstract or non-abstract) identity.

Set's (abstract or non-abstract) identity, is dependent on its (abstract or non-abstract) existence.

Since Set's identity is derived from the level of its members, anything that is related to this level, is inaccessible to the level of Set's (abstract or non-abstract) existence.

Set's existence is the objective aspect of it.

Set's identity is the subjective aspect of it.

The subjective depends on the objective, but not vice versa.

 

[doronshadmi]

EDIT:

Let's understand the following expressions:

x can't be related to A (for example: "is a member of" , "is not a member of" (as shown above), unless A already exists (where this existence is accurately notated by the outer "{" and "}", whether it has members, or not).

The expressions "is a member of" or "is not a member of" determine the (abstract\non-abstract) identity of A by x (or its absence), but not its (abstract\non-abstract) existence.

As long as one does not understand the difference between A's (abstract\non-abstract) existence and A's (abstract\non-abstract) identity, one does not understand the difference between being a set and being a member of a given set (which are two levels of existence, where being a set is the (abstract\non-abstract) objective level of existence and being a member is the (abstract\non-abstract) subjective level of existence, which is inaccessible to the objective level of existence (that is accurately notated by the outer "{" and "}")).

By following this notion as the basis of cardinal numbers (which are based on ZFC set-theory level, but they are not at ZFC set-theory level), the right way to express cardinal numbers is as follows:

{||}=0 , {|{}|}=1 {|{},{{}}|}=2 etc. , or {|{}, {{}}, {{},{{}}}, ... |} < aleph0 (where aleph0 is at the level of the outer "{" and "}", where no number of members is accessible to it).

 

[jsfisher]

Set's (abstract or non-abstract) existence, is independent on its (abstract or non-abstract) identity.

Are you making up things again/still? If not, express what you mean within the constructs of ZFC.

 

[jsfisher]

EDIT:

Let's understand the following expressions:

[qimg]http://upload.wikimedia.org/math/3/3/8/3380195a8703c35a0552323381e606ef.png[/qimg]

[qimg]http://upload.wikimedia.org/math/c/3/1/c313514351bfc97bf7ff8da1f74018ae.png[/qimg]

x can't be related to A (for example: "is a member of" , "is not a member of" (as shown above), unless A already exists (where this existence is accurately notated by the outer "{" and "}", whether it has members, or not).

So much wrong in so little space. "Already exists"? Are you now imposing some time relationship for sets? How does that square with, you know, actual set theory?

Moreover, if something could exist as a set, then it does exist as a set. There is no set that does not exist in formal set theory, ZFC or otherwise.

On the other hand, were we to somehow admit the existence of non-existence sets, one could still argue that the membership relationship still has meaning. X cannot be a member of A if A does not exist.

And finally, where oh where does notation in ZFC have the significance you claim?

 

[doronshadmi]

So much wrong in so little space. "Already exists"? Are you now imposing some time relationship for sets?

Time is not involved here.

"Already exists" means that the objective level of (abstract\non-abstract) set's existence, is independent on the subjective level of its members that define its identity, but not its (abstract\non-abstract) existence.

And finally, where oh where does notation in ZFC have the significance you claim?

Are you making up things again/still? If not, express what you mean within the constructs of ZFC.

There exists a set X (set's objective level) such that (the subjective level of members that defines the identity of the objective level) the empty set {} is a member of X and, whenever a set y is a member of X, then S is also a member of X.

X cannot be a member of A if A does not exist.

"There exists a set A" whether X can or can't be its member. X defines A's identity, but not A's (abstract\non-abstract) existence.

Set' existence is a platonic discovery (objectivity). Set's identity is a non-platonic invention (subjectivity).

So, by using set as a building-block of Mathematics, Mathematics is discovered\invented.

 

[doronshadmi]

EDIT:

Let's refine the explanation of the following expressions (by trying to help person's like jsfisher not to be confused with time):

[qimg]http://upload.wikimedia.org/math/3/3/8/3380195a8703c35a0552323381e606ef.png[/qimg]

[qimg]http://upload.wikimedia.org/math/c/3/1/c313514351bfc97bf7ff8da1f74018ae.png[/qimg]


"x is member of" or "x is not a member of" are (in this example) expressions at the subjective level of existence (where inventions, like identity, are used) of set.

A (in this example) is an expression at the objective level of existence (where discoveries, are independent of any subjective level of existence, like identity) of set.

By following these notions as the basis of cardinal numbers (which are based on ZFC set-theory level, but they are not at ZFC set-theory level), the right way to express cardinal numbers is shown in the following example:

{||}=0 , {|{}|}=1 , {|{},{{}}|}=2 etc. , or {|{}, {{}}, {{},{{}}}, ... |} < aleph0 (where aleph0 is at the level of the outer "{" and "}", where no number of members is accessible to it).

As can be seen, the subjective level of members is inaccessible to the objective level of a given set (where this objectivity is expressed by the outer "{" and "}", whether a given set is empty or non-empty).

In other words, given set with infinitely many members, they are inherently under construction (for example: "whenever a set y is a member of X, then S is also a member of X") since the subjective level of members is inaccessible to the objective level of set (where this objectivity is expressed by the outer "{" and "}", whether a given set is empty or non-empty).

 

[jsfisher]

Time is not involved here.

In that case, don't use the word, already. It has a meaning, and you do not get to change it.

...
There exists a set X (set's objective level)...

"Objective level"? You continue to make stuff up. What does this mean and how does it fit into ZFC?

 

[doronshadmi]

EDIT:

Here is Russell's paradox formal symbolic expression:

By understanding the difference between the objective and subjective level of a given set, one easily deduces that R set's objective level is not R set's subjective level, or in other words "R is a member of R" does not lead to any paradox.

Moreover, "R is a member of R" provides infinitely many nested subjective levels, which are inaccessible to R set's objective level (notated by the outer "{" and "}"), for example:

{ …{…{…{…}…}…}… }

By using this detailed expression, one enables to understand that "R is a member of R" does not lead to any paradox.

 

[doronshadmi]

EDIT:

In that case, don't use the word, already. It has a meaning, and you do not get to change it.

"Already" does not involved with time, if one gets this word in terms of objective existence that is independent of some beginning in time.

Unfortunately, this is not how you get the word "already".

"Objective level"? You continue to make stuff up. What does this mean and how does it fit into ZFC?

It is answered in the post, but you ignored it:

"There exists a set A" whether X can or can't be its member. X defines A's identity, but not A's (abstract\non-abstract) existence.

Set's existence is a platonic discovery (objectivity). Set's identity is a non-platonic invention (subjectivity).

So, by using set as a building-block of Mathematics, Mathematics is discovered\invented.

It seems that you are quit ignorant about the profound discussion about the foundations of Mathematics (are they discovered or invented?).

I provide a simple solution to this profound discussion, which is demonstrated at ZFC level.

You continue to not make any stuff, up or down, by ignoring, for example, also http://www.internationalskeptics.com/forums/showpost.php?p=10019806&postcount=3736.

 

[jsfisher]

EDIT:

Here is Russell's paradox formal symbolic expression:

[qimg]http://upload.wikimedia.org/math/b/e/0/be0bd5b59ed09618db939d03dbd6b22f.png[/qimg]

That is not a set in any formal set theory.