Cont: Deeper than primes - Continuation 2

[Hevneren]

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, ...}]).

So the correct term is multiplicity, then.

 

[doronshadmi]

So the correct term is multiplicity, then.

No, the current term is copy, since copies as defined in [1, 1, 2, 3, 4, 4, 4, 5], are not multiplicities as defined in {(1,2),(2,1),(3,1),(4,3),(5,1)}.

 

[doronshadmi]

They are not copies.

They are copies of the same mutlset, exactly because, the cardinality of, for example [A, A, A, A, A] is 5.

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

In is irrelevant since the number of copies of 2 under set {1, 2, 3} is exactly one copy.

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?

A = (for example) {1, 2, 3, ...}

The point that you have failed to understand is the irreducibly of the cardinality of multiset [A, A, A, A, A] into the cardinality of set {A}.

Because of this irreducibly, there can be more than one mapping among the As under multiset [A, A, A, A, A] where only one of the possible mappings is bijection.

 

[jsfisher]

No, the current term is copy, since copies as defined in [1, 1, 2, 3, 4, 4, 4, 5], are not multiplicities as defined in {(1,2),(2,1),(3,1),(4,3),(5,1)}.

You are not talking about Mathematics, then. You are focused on something else where you have substituted your own fantasies and misunderstandings in place of established terms and concepts. Whenever you'd like to return to actual Mathematics, let us know.

 

[jsfisher]

...<snip of most of the chaotic concepts and terminology>...

there can be more than one mapping among the As under multiset [A, A, A, A, A] where only one of the possible mappings is bijection.

So much wrong in so little space.

Mappings do not require multisets. Mappings require only a domain, a co-domain, and the relationship of every element of the domain to an element of the co-domain.

The domain and co-domain may be the same thing, there is nothing restricting that, and whether the mapping is a bijection is purely dependent on the mapping. At no point does the membership of the domain and co-domain in some multiset ever become relevant.

Moreover, there can be more than one mapping that is a bijection for a give domain/co-domain pair.

 

[Little 10 Toes]

So the correct term is multiplicity, then.

No, the current term is copy, since copies as defined in [1, 1, 2, 3, 4, 4, 4, 5], are not multiplicities as defined in {(1,2),(2,1),(3,1),(4,3),(5,1)}.

No, the correct term is multiplicity. Lets look at the whole quote from wikipedia for multiset^WP that you dishonestly cut short.

Your original post:

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

For example, {a, a, b} and {a, b} are different multisets although they are the same set

https://en.wikipedia.org/wiki/Multiset

Since multiset is a generalization of the concept of set

In mathematics, a multiset (aka bag or mset) is a generalization of the concept of a set

https://en.wikipedia.org/wiki/Multiset

then {a, a, b} is an example of a multiset with redundancy degree > 1, where {a, b} is a multiset with redundancy degree = 1, known by the name "set".

So, you have no argument.

Why can't you use the full Wikipedia quote? Is it because it uses words that already have definitions?

Here is the first few lines from Wikipedia on the definition of multiset^WP

In mathematics, a multiset (aka bag or mset) is a generalization of the concept of a set that, unlike a set, allows multiple instances of the multiset's elements. For example, {a, a, b} and {a, b} are different multisets although they are the same set. However, order does not matter, so {a, a, b} and {a, b, a} are the same multiset.

The multiplicity of an element is the number of instances of the element in a specific multiset. For example, an infinite number of multisets exist which contain only elements a and b, varying only by multiplicity:

• The unique set {a, b} contains only elements a and b, each having multiplicity 1
• In multiset {a, a, b}, a has multiplicity 2 and b has multiplicity 1
• In multiset {a, a, a, b, b, b}, a and b both have multiplicity 3

Once again, you have established terms and ideas. Then you say that they are wrong. And you dishonestly quote things that show you are wrong.

 

[doronshadmi]

Mappings do not require multisets.

Since multiset is a generalization of the concept of a set (set is simply a multiset with one copy for each member), you have no case.

Mappings require only a domain, a co-domain, and the relationship of every element of the domain to an element of the co-domain.

In case of the relationship between two copies of infinite sets under a given multiset, there can be also non-bijective mapping, as addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864.

Moreover, there can be more than one mapping that is a bijection for a give domain/co-domain pair.

It is irrelevat to the relationship between two copies of infinite sets under a given multiset.

 

[doronshadmi]

You are not talking about Mathematics, then. You are focused on something else where you have substituted your own fantasies and misunderstandings in place of established terms and concepts. Whenever you'd like to return to actual Mathematics, let us know.

In mathematics, a multiset is a generalization of the concept of a set, such that a set is the particular case of a multiset with one copy for each member.

It means that (for example) multiset [A,A] is irreducible to multiset [A], and if A = (for example) {1,2,3,...} (known also as the set of all natural numbers), then under [A,A] (which is irreducible to [A]) the mapping between As copies is not necessarily bijection (as adderessed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864).

Generally, you arbitrarily reject http://www.internationalskeptics.com/forums/showpost.php?p=12160476&postcount=2884.

So jsfisher and Little 10 Toes, you have no argument, exactly because by your notion of multiplicity, for example, [A,A] is reducible into [A] (which is not the case by the notion of copy (where by this notion, for example, [A,A] is irreducible into [A]).

Mathematics is not limited to any particular agreement among any group of people, and this is exactly its strength (to those who are interested, http://www.internationalskeptics.com/forums/showpost.php?p=12130235&postcount=2837 is another example of the artificial restrictions of jsfisher (which is a member of a particular group of people that have a particular agreement of Mathematics) on Mathematics).

 

[jsfisher]

In mathematics, a multiset is a generalization of the concept of a set

You continue to respond to strawmen completely disconnected from the very statements you quote.

...such that a set is the particular case of a multiset with one copy for each member.

And you continue to misinterpret or completely ignore established concepts and terminology.

It is as if your efforts to confuse and obfuscate were deliberate.

 

[doronshadmi]

You continue to respond to strawmen completely disconnected from the very statements you quote.



And you continue to misinterpret or completely ignore established concepts and terminology.

It is as if your efforts to confuse and obfuscate were deliberate.

Jsfisher, [A,A] is irreducible into [A] no matter what "established concepts and terminology" of your particular group of people, is used.

And in case that A = {1,2,3,...} bijection is only one alternative between the two copies of A under mutiset [A,A].

Your, so called, "established concepts and terminology" is no more like a door in the middle of a given yard, which one does not necessarily have to use, in order to cross the yard form one side to the other side.

Actually what you call "established concepts and terminology" is exactly like a dogma of some religion, and I am not going to argue with anyone here about his\her religious dogma.

Generally, Mathematics is not limited to any particular agreement among any group of people, and this is exactly its strength, which is something that you still can't comprehend.

 

[jsfisher]

Jsfisher, [A,A] is irreducible into [A]

No one has said [A, A] and [A] are the same. Why to you keep bringing it up?

 

[Hevneren]

And in case that A = {1,2,3,...} bijection is only one alternative between the two copies of A under mutiset [A,A]

Could we all agree on using the term instances instead of copies, just to reduce the confusion a bit? Wikipedia uses instances in the quote you supplied. I don't necessarily have an opinion of which of the two is better, but instances seems to be more widely used.

 

[phiwum]

Could we all agree on using the term instances instead of copies, just to reduce the confusion a bit? Wikipedia uses instances in the quote you supplied. I don't necessarily have an opinion of which of the two is better, but instances seems to be more widely used.

I think you're on the right path here.

The fundamental issue, which may be beyond certain participants, is an issue between syntax and semantics. In the multiset represented by "[2,2]", there are two distinct symbols representing the same thing, namely the number 2. One of those symbols occurs to the left of the other, but it is nonsense to think that there are two different numbers 2.

Of course, most folks in this thread are participating in a fool's errand to convince others that they are cranks. I stopped doing that back in the Usenet days. Eventually.

 

[caveman1917]

The fundamental issue, which may be beyond certain participants, is an issue between syntax and semantics. In the multiset represented by "[2,2]", there are two distinct symbols representing the same thing, namely the number 2. One of those symbols occurs to the left of the other, but it is nonsense to think that there are two different numbers 2.

A = A. It really doesn't get more fundamental than that.

 

[phiwum]

A = A. It really doesn't get more fundamental than that.

So long as one remembers that there's a difference between syntax and semantics and that Rand was a simpleton.

ETA: not that I take you for an objectivist, from what I recall.

 

[caveman1917]

So long as one remembers that there's a difference between syntax and semantics and that Rand was a simpleton.

Not sure I'm getting your point here with the reference?

ETA: not that I take you for an objectivist, from what I recall.

No, I'm a proper egoist. She wasn't just a simpleton, she was a bourgeois simpleton!

 

[phiwum]

Not sure I'm getting your point here with the reference?



No, I'm a proper egoist. She wasn't just a simpleton, she was a bourgeois simpleton!

[qimg]http://www.internationalskeptics.com/forums/picture.php?albumid=1305&pictureid=11699[/qimg]

A=A is a standard trope of Rand's objectivism, that's all. I didn't think that's where you were going with that equation, but just thought I'd point it out.

 

[caveman1917]

A=A is a standard trope of Rand's objectivism, that's all. I didn't think that's where you were going with that equation, but just thought I'd point it out.

Ah no, apologies, I meant "A = A" in the sense of the law of identity. As in "it doesn't get more fundamental than that" given that the error is a violation of the law of identity, effectively saying 2 ≠ 2.

And the law of identity is pretty fundamental to doing any sort of logic or math. I mean, you could, like, do away with the law of the excluded middle and construct multi-valued logics if you want. But the law of identity? That one's probably about as fundamental as it goes.

That's what I meant.

 

[Little 10 Toes]

In mathematics, a multiset is a generalization of the concept of a set, such that a set is the particular case of a multiset with one copy for each member.

Circular definition.

Coins are part of money, such that money is specifically coins.

You've been called out by me again with dishonest definitions.

From Wikipedia's definition of multiset^WP

In mathematics, a multiset (aka bag or mset) is a generalization of the concept of a set that, unlike a set, allows multiple instances of the multiset's elements.

It means that (for example) multiset [A,A] is irreducible to multiset [A], and if A = (for example) {1,2,3,...} (known also as the set of all natural numbers), then under [A,A] (which is irreducible to [A]) the mapping between As copies is not necessarily bijection (as adderessed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864).

There are parts of your post that I do agree on. Multisets allow multiple instances of the same element. Most people will use the letter N for the set of natural numbers. And using the definition of multiset, it can't be reduced.

Generally, you arbitrarily reject http://www.internationalskeptics.com/forums/showpost.php?p=12160476&postcount=2884.

So jsfisher and Little 10 Toes, you have no argument, exactly because by your notion of multiplicity, for example, [A,A] is reducible into [A] (which is not the case by the notion of copy (where by this notion, for example, [A,A] is irreducible into [A]).

Mathematics is not limited to any particular agreement among any group of people, and this is exactly its strength (to those who are interested, http://www.internationalskeptics.com/forums/showpost.php?p=12130235&postcount=2837 is another example of the artificial restrictions of jsfisher (which is a member of a particular group of people that have a particular agreement of Mathematics) on Mathematics).

And you're wrong again. You can't reduce a multiset. Why are you trying to do? Why can't you understand the definition?

 

[jsfisher]

I find it very amusing that one of the commonly cited examples for the utility of multisets is for prime factorizations. 120 is 2^{[SIZE=-3]3[/SIZE]}3^{[SIZE=-3]1[/SIZE]}5^{[SIZE=-3]1[/SIZE]}, and the latter sure looks like a multiset to me.

I would not say that were deeper than primes, though.