Deeper than primes - Continuation

[jsfisher]

"Already" does not involved with time

Already, adj., before this time : before now : before that time : so soon : so early.

 

[jsfisher]

It is answered in the post

Pointing out where you previously used a term is not the same as pointing out what you meant by the term. So, what did you mean by the term? How does that meaning relate to ZFC (as in in a way that is actually tied to a ZFC axiom instead of just what you'd like it to be).

If, on the other hand, you'd really like to abandon Mathematics in favor of discussing doronetics or Doron's philosophy about Mathematics, just say so. I'll ignore any such post, but as long as you continue to attack Mathematics itself with misconception, gibberish, faulty logic, and undefined terms, expect some push-back.

 

[doronshadmi]

Already, adj., before this time : before now : before that time : so soon : so early.

No jsfisher, "Already", as I use it, means that something exists, independently of any temporal observation of it (these is no temporal before, now or after to its existence)

If you insist to force your temporal observation on it, you simply miss the platonic point of view of set's objective level, and how this objective level is independent on the subjective level, which provide its identity by using members (or their absence).

Given a set, the outer "{" and "}" are always there as the invariant aspect of it, where its members are the variant aspect of it, which provide its identity, but not its (abstract\non-abstract) existence.

 

[doronshadmi]

That is not a set in any formal set theory.

That is not a set in any theory that does not distinguish between the objective and subjective levels of a given set, so?

 

[doronshadmi]

Pointing out where you previously used a term is not the same as pointing out what you meant by the term. So, what did you mean by the term? How does that meaning relate to ZFC (as in in a way that is actually tied to a ZFC axiom instead of just what you'd like it to be).

If, on the other hand, you'd really like to abandon Mathematics in favor of discussing doronetics or Doron's philosophy about Mathematics, just say so. I'll ignore any such post, but as long as you continue to attack Mathematics itself with misconception, gibberish, faulty logic, and undefined terms, expect some push-back.

As long as you simply ignore my detailed answers to to you, there is no communication, what so ever, between us.

but as long as you continue to attack Mathematics itself

What do you know, jsfisher uses philosophical decelerations about Mathematics, without even being aware of doing it.

This blindness, explains a lot about the dis-communication between us, even if I rewrite my posts according to some of his remarks.

A long as he is not aware of his own philosophical foundations about Mathematics, there is no use to reply to him.

The notion of the invariant aspect of existence of a given set, is notated by the outer "{" and "}", where this invariant level of existence is independent on the variant level of members (they may exist, or not), which provide the identity of sets.

 

[jsfisher]

No jsfisher, "Already", as I use it....

You don't get to make arbitrary adjustments to the meaning of things. Not for mathematical terms; not for English words.

As long as you continue to try to do so, you continue to fail.

 

[jsfisher]

That is not a set in any theory that does not distinguish between the objective and subjective levels of a given set, so?

Where are you asking me? You are the one making a big deal about it, yet you are unwilling and unable to actually explain your objection. Fanciful concepts of "objective level" and "subjective level" may be important in your imagination, but you have failed to point out any particular defect attributable to your fantasy concept in ZFC.

 

[jsfisher]

As long as you simply ignore my detailed answers to to you, there is no communication, what so ever, between us.

You mistake "discount" for "ignore". Your posts are not ignored.

They also fail to meet any standard of being detailed answers. As you yourself have reminded everyone just a few posts up, you use words in arbitrary ways without regard to established meaning, so whatever lack of communication exists (and there is plenty) is entirely your fault.

Now, would you care to define what you mean by identity of a set, or is that yet another a bit of Doronese you'd prefer to leave flexible.

 

[doronshadmi]

Now, would you care to define what you mean by identity of a set, or is that yet another a bit of Doronese you'd prefer to leave flexible.

At the moment that I'll see that you are aware of your philosophical foundations about Mathematics, I will continue the discussion with you.

This moment did not came yet.

 

[jsfisher]

At the moment that I'll see that you are aware of your philosophical foundations about Mathematics, I will continue the discussion with you.

This moment did not came yet.

So, as to the actual question, "would you care to define what you mean by identity of a set," we'll take your post as a, "no."

Well, at least with respect to defining your terms, Doron, you are consistent.

 

[doronshadmi]

You don't get to make arbitrary adjustments to the meaning of things. Not for mathematical terms; not for English words.

As long as you continue to try to do so, you continue to fail.

What can we say to a person that has no abilities to distinguish between the platonic objective level of existence, which is always discovered (and therefore it is already exists independently of the moment of its discovery), and the non-platonic subjective level of existence, which is always invented (and therefore it is not already exists, or in other words, inventions are always depend on some moment in time)?

 

[Little 10 Toes]

And this deals with sets how?

 

[doronshadmi]

Any fundamental mathematical theory like ZFC at least defines existence that may or may not identified, for example:

"There exists" X (whether X is identified or not).

The identity of X is the definition of its properties, where in the case of sets, the most fundamental properties is being empty or non-empty (the existing set X may be empty or non-empty, but in both cases these fundamental properties can't be defined, unless X existence is defined).

So, there is an hierarchy of dependency, where X identity depends on X existence, but X existence does not depend on X identity.

Let's examine the notions above more carefully, according to the following expression:

"There does not exist" X with identity Y.

Some claims that according to this expression, identity Y is independent on X existence.

These is true about X, but still "there exists" non-X with identity Y, or in other words, there is an hierarchy of dependency, where Y identity depends on non-X existence, but non-X existence does not depend on Y identity.

By using philosophical terms, "there exists" X (or non-X) is a discovery (an invariant objective foundation) that enables inventions (variant subjective expressions, where one of them is some identity of the discovered), where inventions depend on the discovered, but not vice versa.

Now, 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 objective level (notated here by the outer "{" and "}"), where no number of members (where members are at the subjective level) is accessible to it).

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

More generally, Mathematics (abstract or non-abstract) is useful exactly because it is discovered and invented, where the invented depends on the discovered, but not vice versa.

 

[jsfisher]

Any fundamental mathematical theory like ZFC at least defines existence that may or may not identified, for example:

"There exists" X (whether X is identified or not).

It would be useless to stipulate the existence of a set without identifying it (as you seem to now be using the word, identify). That would be where that "such that" part of the proposition comes into play.

So, no. Set Theory is not founded on set existence as separate from set identity.

The identity of X is the definition of its properties, where in the case of sets, the most fundamental properties is being empty or non-empty

By what metric did you determine that this was the most fundamental property?

Keep in mind, too, that equal sets are indistinguishable. The consequence of this is that there is only one empty set.

 

[jsfisher]

It would be useless to stipulate the existence of a set without identifying it (as you seem to now be using the word, identify). That would be where that "such that" part of the proposition comes into play.

There is a limiting case (a pre-set theory assumption) wherein a non-specific set appears, but the subtlety of what and why that is is beyond scope of this thread, and it does not damage the general statement, above.

 

[doronshadmi]

It would be useless to stipulate the existence of a set without identifying it (as you seem to now be using the word, identify). That would be where that "such that" part of the proposition comes into play.

So, no. Set Theory is not founded on set existence as separate from set identity.

You have missed this:

More generally, Mathematics (abstract or non-abstract) is useful exactly because it is discovered and invented, where the invented depends on the discovered, but not vice versa.

By what metric did you determine that this was the most fundamental property?

By the invented (subjective) level of members, that are variant (they may exist or not exist) w.r.t the invariant level of set, which is discovered (objective), or in other words, this level is always exists, independently of the invented (subjective) variant level of members.

It has to be stressed that level of members is an invention that is defined in some moment in time, but it does not mean that, for example, "whenever a set y is a member of X, then S is also a member of X" is a process in time, it simply demonstrates the inaccessibility of the invented (subjective) level of members of a given set, to the discovered (objective) level of set.

Keep in mind, too, that equal sets are indistinguishable.

Exactly, at the discovered (objective) invariant level of set, all is known is that there exists something, independently the moment of discovery. At this level there is no plurality like sets, but this level enables the plurality at the invented (subjective) variant level of members.

The consequence of this is that there is only one empty set.

There are no consequences at the discovered (objective) variant level of set, only the awareness of existence.

Based on this discovered (objective) invariant level of set, the invented (subjective) variant level of members is given, which enables to conclude whether a given set is, at least, empty or non-empty.

 

[jsfisher]

You have missed this:

You continue to confuse discount with miss, ignore, and didn't understand.

Now, would you care to actually respond to my post, or will you continue to back reference posts where you didn't?

 

[doronshadmi]

EDIT:

You continue to confuse discount with miss, ignore, and didn't understand.

You continue to confuse objective existence that does not need any identity in order to exist, with subjective existence that defines identities to the objective existence.

The subjective existence depends on the objective existence, but not vice versa, and this dependency is shown exactly because objective and subjective levels of existence are both used in some given framework (can be shown in http://www.internationalskeptics.com/forums/showpost.php?p=10023964&postcount=3759).

Now, would you care to actually respond to my post, or will you continue to back reference posts where you didn't?

Now would you care to actually read my posts (until their end), in order to follow the notion of the hierarchy of dependency, or will you continue your one level notion and your partial reading and response about my posts (as clearly can be seen, for example, in your http://www.internationalskeptics.com/forums/showpost.php?p=10023892&postcount=3757 post)?

 

[doronshadmi]

EDIT:

There is a limiting case (a pre-set theory assumption) wherein a non-specific set appears, but the subtlety of what and why that is is beyond scope of this thread, and it does not damage the general statement, above.

I explicitly use a notion of hierarchy of dependency of the subjective on the objective (and not vice versa) and this notion is used also by ZFC, for example:

"There exists a set X (the objective level) such that (the subjective level that depends on the objective level, but not vice versa) 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"


So your argument about a limiting case is irrelevant, exactly because my notion is not only about the objective level of existence (where no hierarchy of dependency between the objective and the subjective is involved).

 

[jsfisher]

I explicitly use a notion of hierarchy of dependency of the subjective on the objective (and not vice versa) and this notion is used also by ZFC, for example:

"There exists a set X (the objective level) such that (the subjective level that depends on the objective level, but not vice versa) 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"

The parenthetical comments in italics are your fantasy, not anything in ZFC. You simply are making stuff up, as you so frequently do.

You post also relates in no way to the post of mine you quoted. So there is no evidence you understood my post at all. This lends further support to the conclusion your reading comprehension issues are at the root of why you fabricate instead of understand.

Also, I note your continued parroting of "S" in your posts as if you believed it has meaning stripped from its original context. Even more evidence of failures in understanding.

ETA:
By the way, in what way is "the empty set {}" different from "the empty set" or from "{}"? Or do you just feel compelled to restate things redundantly by repeating things?