Cont: Deeper than primes - Continuation 2

[jsfisher]

The set of all natural numbers is one set. You can, however, include it multiple times in a multiset, like you did here.

Did you know that Doronshadmi has claimed, quite strongly, too, that 2 is not necessarily a member of the set containing 2 as a member? (Normal sets, too, not multisets.)

 

[Hevneren]

Did you know that Doronshadmi has claimed, quite strongly, too, that 2 is not necessarily a member of the set containing 2 as a member? (Normal sets, too, not multisets.)

No, I must have missed that one. That's WAY deeper than primes. I suspect that some of this twisted thinking comes from object-oriented computer programming. In Java, for instance, you can create two identical lists of ascending Integer objects. The objects themselves will occupy unique memory locations, so in some sense the number 2 from one list won't be a member of the other list.

 

[Hevneren]

Not under a multiset with redundancy degree > 1 (where multiset is a generalization of the concept of set, such that a set is actually a multiset with redundancy degree = 1, known by the name "set").

Please define redundancy degree. Is it the greatest multiplicity found for any member in a multiset? And why do you think it's useful to make up yet another undefined term?

 

[doronshadmi]

Please define redundancy degree.

It is the number of copies of each given member in a given multiset.

And why do you think it's useful to make up yet another undefined term?

The independency of the copies (>1) with respect to each other, enable understanding of the concept of infinite cardinality, which is not restricted only to multisets with redundancy degree = 1, as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864.

 

[jsfisher]

It is the number of copies of each given member in a given multiset.

You don't need to coin new terms when perfectly usable words are already assigned. Multiplicity will do nicely for the case at hand.

In the multiset^*, [A, B, C, B, C, C], the multiplicity of A, B, and C are 1, 2, and 3, respectively.


^*Note that I have adopted the sometimes-used convention of delineating multisets with brackets, reserving braces for normal sets.

 

[doronshadmi]

Did you know that Doronshadmi has claimed, quite strongly, too, that 2 is not necessarily a member of the set containing 2 as a member? (Normal sets, too, not multisets.)

More precisely, 2 is not the same as (for example) {2}.

 

[doronshadmi]

You don't need to coin new terms when perfectly usable words are already assigned. Multiplicity will do nicely for the case at hand.

In the multiset^*, [A, B, C, B, C, C], the multiplicity of A, B, and C are 1, 2, and 3, respectively.


^*Note that I have adopted the sometimes-used convention of delineating multisets with brackets, reserving braces for normal sets.

This reply demonstrates why one misses the possible independency of some given mutiset's members (with more than one copy) with respect to each other (addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864).

 

[jsfisher]

More precisely, 2 is not the same as (for example) {2}.

Of course those two are not the same. However, you have said in the past that the former was not a member of the latter.

 

[jsfisher]

This reply demonstrates why one misses the possible independency of some given mutiset's members (with more than one copy) with respect to each other (addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864).

Independency? A bit of an archaic word. Independence is in the common parlance.

Be that as it may, identical members of a multiset have no independence of each other. All the A's in the multiset [A, A, A, A, A] are the same A.

 

[doronshadmi]

Of course those two are not the same. However, you have said in the past that the former was not a member of the latter.

This is my last reply to your rad herring.

2 is not the same as (for example) {2} exactly because it is not necessarily defined in terms of sets.

 

[doronshadmi]

All the A's in the multiset [A, A, A, A, A] are the same A.

Once again you demonstrate your restriction to [A, A, A, A, A] = {A}, such that |[A, A, A, A, A]| = |{A}| = 1

 

[jsfisher]

2 is not the same as (for example) {2} exactly because it is not necessarily defined in terms of "pure" sets.

Why do you keep responding with this strawman?

And again, of course 2 and {2} are different. Nobody has said otherwise. You, however, have claimed that 2 was not a member of {2}.

 

[jsfisher]

Once again you demonstrate your restriction to [A, A, A, A, A] = {A}, such that |[A, A, A, A, A]| = 1

I claimed none of that.

[A, A, A, A, A] is a multiset of 5 elements (and therefore its cardinality is 5).

Gee, it is almost as if you think repeated elements in a multiset must somehow be different from each other. Of course, that would be silly and make the whole concept of multisets meaningless, so you couldn't possibly think that, right?

 

[doronshadmi]

Gee, it is almost as if you think repeated elements in a multiset must somehow be different from each other.

It is clear that you think that (for example) two copies of members of a given multiset with the same elements (same members, if you wish), can't be independent of each other.

But these are your conceptual problems, no less no more.

 

[phiwum]

Sorry. I meant the terms that Doronshadmi was using. Specifically, "redundancy degree".

Apologies for missing your point.

 

[jsfisher]

It is clear that you think that (for example) two copies of members of a given multiset with the same elements (same members, if you wish), can't be independent of each other.

They are not copies, just the same element appearing more that once.

In {1, 2, 3} and {2, 10}, 2 appears in both sets. It is the same 2. They have no independence from each other because they are the same 2.

Were they independent, then they could behave differently. There would be something that distinguished one from the other. But there isn't anything like that.

 

[Hevneren]

It is the number of copies of each given member in a given multiset.

OK, so what's the redundancy degree of [1, 1, 2, 3, 4, 4, 4, 5]?

 

[doronshadmi]

EDIT:

They are not copies, just the same element appearing more that once.

They are more than one copies of members under the same multiset.

In {1, 2, 3} and {2, 10}, 2 appears in both sets.

Be aware that my argument about cardinality holds only between more than one copies of members (which are infinite sets) under the same multiset (addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864).

In case of (for example) [{1, 2, 3, ...}, {1, 2, 3, ...}] there are more than one copies of members (which are infinite sets) under the same multiset.

In case of (for example) {{1, 3, 4, ...}, {1, 2, 3, ...}} there are no more than one copies of members (which are infinite sets) under the same set.

There would be something that distinguished one from the other.

This something is the difference of cardinality between (for example) |[{1, 2, 3, ...}, {1, 2, 3, ...}]|=2 and |{{1, 2, 3, ...}}|=1


Generally, you still try to deduce multiset (which has redundancy degrees > 1) in terms of set (which has redundancy degree = 1).

 

[doronshadmi]

OK, so what's the redundancy degree of [1, 1, 2, 3, 4, 4, 4, 5]?

There are several redundancy degrees: 2 for 1,1 , 1 for 2,3 and 5 , and 3 for 4,4,4.

But again, my argument holds only in case of more than one copy of a given infinite set under a given multiset
(for example: [{1, 2, 3, ..}, {1, 2, 3, ...}]).

 

[jsfisher]

They are more than one copies of members under the same multiset.

They are not copies. That 2 in the set {1, 2, 3} isn't a copy, either.

...
This something is the difference of cardinality between (for example) |[{1, 2, 3, ...}, {1, 2, 3, ...}]|=2 and |{{1, 2, 3, ...}}|=1

You are responding to a question that was not asked. The point you failed to address is that you have claimed independency for the members of [A, A, A, A, A]. If those A's of that multiset are independent of each other, there must be some property that could distinguish them. What is it? What makes them different?