Deeper than primes - Continuation

Status
Not open for further replies.
Little 10 Toes said:
let's start with what you got right.
Click to expand...
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.
 
Last edited:
doronshadmi said:
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)).
Click to expand...

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
...
Click to expand...

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.
 
jsfisher said:
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.
Click to expand...

You wrote:
jsfisher said:
At the set-theoretic level, cardinality is a relative measure.
Click to expand...
Please explain in more details what exactly do you mean by "cardinality is a relative measure" ?
 
 
jsfisher said:
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.
Click to expand...

jsfisher said:
Again, set theory provides no way to determine how many of an element are in a set, only whether it is in the set.
Click to expand...
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.
 
Last edited:
doronshadmi said:
What is written above does not change my argument
Click to expand...

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 "}".
Click to expand...

You made that up. Unless you can establish this personal invention of yours in actual mathematics, your argument dies on the vine right here.
 
jsfisher said:
Really? What about the part where you referred to numbers?
Click to expand...
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(y) is also a member of X.
 
doronshadmi said:
Let's start with what you still do not get.
Click to expand...
Ok then.

Any given set exists as an abstract mathematical object, whether it does or does not have members.
Click to expand...
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 "}".
Click to expand...
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.
 
doronshadmi said:
The part that is referred to numbers, comes after the part that is referred to the set theory level.
Click to expand...

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.
Click to expand...

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.
Click to expand...

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:
Click to expand...

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(y) is also a member of X.
Click to expand...

Yes, and...?

A slightly hacked restatement of the Axiom doesn't get you to any conclusion. Where did you want to go from here?
 
Little 10 Toes said:
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.
Click to expand...
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.
 
jsfisher said:
A slightly hacked restatement of the Axiom doesn't get you to any conclusion. Where did you want to go from here?
Click to expand...

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.
 
Last edited:
EDIT:

Let's understand the following expressions:

3380195a8703c35a0552323381e606ef.png


c313514351bfc97bf7ff8da1f74018ae.png


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).
 
Last edited:
doronshadmi said:
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).
Click to expand...

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?
 
jsfisher said:
So much wrong in so little space. "Already exists"? Are you now imposing some time relationship for sets?
Click to expand...
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.

jsfisher said:
And finally, where oh where does notation in ZFC have the significance you claim?
Click to expand...
jsfisher said:
Are you making up things again/still? If not, express what you mean within the constructs of ZFC.
Click to expand...
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(y) is also a member of X.

jsfisher said:
X cannot be a member of A if A does not exist.
Click to expand...
"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.
 
Last edited:
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(y) 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).
 
Last edited:
EDIT:

Here is Russell's paradox formal symbolic expression:

be0bd5b59ed09618db939d03dbd6b22f.png


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.
 
Last edited:
EDIT:

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

jsfisher said:
"Objective level"? You continue to make stuff up. What does this mean and how does it fit into ZFC?
Click to expand...
It is answered in the post, but you ignored it:
doronshadmi said:
"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.
Click to expand...

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.
 
Last edited:
doronshadmi said:
It is answered in the post
Click to expand...

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.
 
jsfisher said:
Already, adj., before this time : before now : before that time : so soon : so early.
Click to expand...
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.
 
Last edited:
jsfisher said:
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.
Click to expand...
As long as you simply ignore my detailed answers to to you, there is no communication, what so ever, between us.

jsfisher said:
but as long as you continue to attack Mathematics itself
Click to expand...
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.
 
Last edited:
doronshadmi said:
No jsfisher, "Already", as I use it....
Click to expand...

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.
 
doronshadmi said:
That is not a set in any theory that does not distinguish between the objective and subjective levels of a given set, so?
Click to expand...

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

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.
 
jsfisher said:
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.
Click to expand...
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.
 
doronshadmi said:
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.
Click to expand...


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.
 
jsfisher said:
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.
Click to expand...
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)?
 
Last edited:
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(y) 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.
 
Last edited:
doronshadmi said:
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).
Click to expand...

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
Click to expand...

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 said:
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.
Click to expand...

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.
 
jsfisher said:
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.
Click to expand...

You have missed this:
doronshadmi said:
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.
Click to expand...



jsfisher said:
By what metric did you determine that this was the most fundamental property?
Click to expand...
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(y) 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.

jsfisher said:
Keep in mind, too, that equal sets are indistinguishable.
Click to expand...
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.

jsfisher said:
The consequence of this is that there is only one empty set.
Click to expand...
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.
 
Last edited:
EDIT:

jsfisher said:
You continue to confuse discount with miss, ignore, and didn't understand.
Click to expand...
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).

jsfisher said:
Now, would you care to actually respond to my post, or will you continue to back reference posts where you didn't?
Click to expand...
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)?
 
Last edited:
EDIT:

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


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(y)" 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?
 
Last edited:
Status
Not open for further replies.

Back
Top Bottom