Cont: Deeper than primes - Continuation 2

[Hevneren]

Phiwum and I understand multisets in terms of how the concept is defined. You, Doronshadmi, don't bother with such formalities.

By the way, will you as some point tell us all what you had in mind with this (A,1) and (A,2) notation or will that be just another mystery of doronetics?

I think (A,1) represents the first A in the (apparently ordered) (multi)set, while (A,2) represents the second one. But I don't know which one of them is the original A and which one is the copy, or if the original A is stored in a safe place somewhere else, so nobody can put it in a set where it doesn't belong.

 

[doronshadmi]

I think (A,1) represents the first A in the (apparently ordered) (multi)set, while (A,2) represents the second one.

No, the second parameter, for example, in (A,1), (A,2), (A,3) etc. ... , defines the number of copies of a given member (the first parameter, which in this case is A) in a given multiset.

 

[jsfisher]

If (x,y) is meant as an ordered pair where x is an element and y is whole number representing multiplicity, then, sure, that is a way to embed the multiset concept into set theory. (It isn't perfect, but it is workable.)

E.g. { (2,3), (3,1), (5,1) } would be the multiset of the prime factors of 120 under this model.

Doronshadmi, since no one has argued otherwise, why do you continue to argue as if we had? You have been doing that a lot of late, arguing against positions nobody has taken.

 

[doronshadmi]

If (x,y) is meant as an ordered pair where x is an element and y is whole number representing multiplicity, then, sure, that is a way to embed the multiset concept into set theory.

If (x,y) is meant as an ordered pair where x is an element and y is any cardinal number > 0 representing redundancy among x, , then, sure, that is a way to embed elements in to multisets, where a multiset is a generalization of the concept of set (a set is simply a multiset with redundancy = 1).

If (x,y) is meant as an ordered pair where x is an element and y is whole number representing multiplicity, then, sure, that is a way to embed the multiset concept into set theory. (It isn't perfect, but it is workable.)

E.g. { (2,3), (3,1), (5,1) } would be the multiset of the prime factors of 120 under this model.

Doronshadmi, since no one has argued otherwise, why do you continue to argue as if we had? You have been doing that a lot of late, arguing against positions nobody has taken.

jsfisher, you are still missing the notion of redundancy as defined, for example, in {(2,3)}, because your notion is restricted only to redundancy = 1 (defined in this case as {(2,1),(2,1),(2,1)}).

Your restriction is in terms of notions and not in terms notations, exactly because whole your mathematical understanding is reduced into redundancy = 1.

Here are your own words:

No matter how it is expressed, a single 2 is sufficient for all its needs.

and this is exactly the notion of redundancy = 1 (defined in this case as {(2,1),(2,1),(2,1)}).

So jsfisher, until this vary moment you have no clue what is my argument in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864.

Yet you continue your off topic replies even if you have no clue about my argument, which is not restricted to redundancy = 1.

 

[phiwum]

To those who follow the last discussion, jsfisher and phiwum understand, for example [A,A] in terms of {(A,1),(A,1)} and not in terms of {(A,2)} as I do.

In other words, their notion is restricted only to (A,1).

Nonsense. The multiset [A,A] is obviously represented in set theory as {(A,2)}. Duh.

But A had no referent here, unless you mean that for all sets A this is the case. Which is, of course, true.

Thus, in [A,A], the single set A is represented twice, but it is nonetheless the same A.

 

[jsfisher]

jsfisher, you are still missing the notion of redundancy as defined, for example, in {(2,3)}, because your notion is restricted only to redundancy = 1 (defined in this case as {(2,1),(2,1),(2,1)}).

Why do you say that? Other than your insistence on using the wrong term (multiplicity is the correct one), [2, 2, 2] and { (2,3) } are identical constructs (in terms of the context of this thread).

It is not clear why you brought up { (2,1), (2,1), (2,1) }.

Just two more instances of you arguing against things no one else claimed.

 

[doronshadmi]

More of the same is not just the same.

A = {1,2,3,...} (the set of all natural numbers)

[A,A] is more of the same, such that bijection is only one case among the As under [A,A], as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864.

 

[jsfisher]

More of the same is not just the same.

A = {1,2,3,...} (the set of all natural numbers)

[A,A] is more of the same

That is an odd way to express it, but no one has argued to the contrary. Will all of your posts be striking out against strawmen?

...such that bijection is only one case among the As under [A,A]

The multiset [A, A] does not include nor imply any sort of mapping from A to A, bijection or otherwise. It is just a multiset with A as a member with multiplicity of 2.

Mappings, on the other hand, require three things: a domain, a co-domain, and a functional relationship from domain to co-domain.

Please stop conflating the two.

 

[doronshadmi]

The multiset [A, A] does not include nor imply any sort of mapping from A to A, bijection or otherwise.

I agree with you.

It is just a multiset with A as a member with multiplicity of 2.

[A, A] is an example of more of the same, such that redundancy > 1.

Since by convectional mathematics multiplicity is not understood in terms of redundancy, one wrongly understands, for example, {(A,2)} in terms of {(A,1),(A,1)} without being aware of his\her failure.

Here is a concrete example:

Since jsfisher is unaware of the redundancy in [2,2,2] he wrongly understands {(2,3)} (which is actually an expression that does not ignore the redundancy), in terms of {(2,1),(2,1),(2,1)} which ignores the redundancy in [2,2,2], where this reply

It is not clear why you brought up { (2,1), (2,1), (2,1) }.

very simply demonstrates jsfisher's non-awareness of the considered subject.

Mappings, on the other hand, require three things: a domain, a co-domain, and a functional relationship from domain to co-domain.

If mappings is considered among the As under [A, A] such that A is an infinite set, then since [A, A] is more of the same (where redundancy is not ignored) bijection is only one case among the As under [A,A], as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864.

In other words, jsfisher, you continue your off topic replies, exactly because you understand {(A,2)} (redundancy is not ignored) in terms of {(A,1),(A,1)} (redundancy is ignored).

 

[doronshadmi]

As for phiwum, he wrongly thinks that I claim that [2,2] has two different members:

but it is nonsense to think that there are two different numbers 2.

But this is not my argument.

My argument simply does not ignore the redundancy > 1 in a given multiset, and in case that the redundancy > 1 is not ignored among the mappings between redundant infinite members, bijection is not necessarily the one and only one mapping, as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864.

 

[jsfisher]

The multiset [A, A] does not include nor imply any sort of mapping from A to A, bijection or otherwise.

I agree with you.

It is just a multiset with A as a member with multiplicity of 2.

[A, A] is an example of more of the same, such that redundancy > 1.

No, you do not get to redefine Mathematics in your desperate quest to disprove it.

 

[doronshadmi]

No, you do not get to redefine Mathematics in your desperate quest to disprove it.

Mathematics is not a property of anyone or any group of people not in the past, not now and not in the future.

Your desperate quest to close it under dogmas actually transforms it into a religion.

A more fruitful way is to reply in details to http://www.internationalskeptics.com/forums/showpost.php?p=12165528& , but currently it seems that your best reply is "you do not get to redefine Mathematics".

 

[jsfisher]

Mathematics is not a property of anyone or any group of people not in the past, not now and not in the future.

Property rights have nothing to do with your misrepresentation of Mathematics.

 

[Little 10 Toes]

Mathematics is not a property of anyone or any group of people not in the past, not now and not in the future.

Your desperate quest to close it under dogmas actually transforms it into a religion.

A more fruitful way is to reply in details to http://www.internationalskeptics.com/forums/showpost.php?p=12165528& , but currently it seems that your best reply is "you do not get to redefine Mathematics".

Doronshadmi, can I get my orange juice metric please? Extra 50 weight.

 

[doronshadmi]

Property rights have nothing to do with your misrepresentation of Mathematics.

You get Mathematics as if it is the non-redefined property of a group of people that you are one of its members, and this is exactly your misrepresentation of Mathematics.

 

[Little 10 Toes]

How about instead of making up words with your own private definitions, you work on using the correct terms?

 

[doronshadmi]

Let's generalize the mathematical notion of strict membership (which is a particular case of Fuzzy logic).

Let E be a placeholder of any entity.

Let # be a placeholder of any strict membership.

Let (E,#) be a generalization of any strict membership of any entity.

Multiset is a generalization of the concept of set.

Any entity with strict membership 0 ( notated as (E,0) ), can't be defined but as E (it is strictly not a member of any multiset).

Let [] (empty multiset) be a multiset of any entity with strict membership 0 ( notated as (E,0) ).

Any entity with strict membership 1 ( notated as (E,1), can be defined as E = E (it is a strict and unique member of any multiset).

Convectional mathematics is the mathematical framework of strict membership of the form (E,1), and it needs the axiomatic method in order to define [] in terms of (E,1), instead of simply deduce it in terms of strict membership 0 ( notated as (E,0) ).

Let A = [1,2,3,… etc.]

Let S = [A,A]

S is a (E,2) mathematical framework, which enables mapping among infinite multisets that is not restricted to (E,1) mathematical framework, such that bijection is only a particular case among As under S, exactly because A is a strict but non-unique member of S (as very simply addressed in http://www.internationalskeptics.com/forums/showpost.php?p=12158706&postcount=2864).

Exactly as one can't claim that strict membership is the one and only one possible membership (since it is simply a particular case of fuzzy logic), one also can't claim that (E,1) is the one and only possible strict membership.

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Edited by Agatha: 

Edited to remove rule 12 breach

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Mathematics is indeed deeper than primes.

 

[doronshadmi]

Here is some quote taken from Russell's article about the axiom of choice :

The boots-and-socks metaphor was given in 1919 by Russell 1993, pp. 125–127. He suggested that a millionaire might have ℵ₀ pairs of boots and ℵ₀ pairs of socks.

Among boots we can distinguish right and left, and therefore we can make a selection of one out of each pair, namely, we can choose all the right boots or all the left boots; but with socks no such principle of selection suggests itself, and we cannot be sure, unless we assume the multiplicative axiom, that there is any class consisting of one sock out of each pair.

Russell generally used the term "multiplicative axiom" for the axiom of choice. Referring to the ordering of a countably infinite set of pairs of objects, he wrote:

There is no difficulty in doing this with the boots. The pairs are given as forming an ℵ₀, and therefore as the field of a progression. Within each pair, take the left boot first and the right second, keeping the order of the pair unchanged; in this way we obtain a progression of all the boots. But with the socks we shall have to choose arbitrarily, with each pair, which to put first; and an infinite number of arbitrary choices is an impossibility. Unless we can find a rule for selecting, i.e. a relation which is a selector, we do not know that a selection is even theoretically possible.

Russell then suggests using the location of the centre of mass of each sock as a selector.

The notion that defines a set as a collection of distinct members, is actually based on an act of selection.

In order to be aware of it, all one needs is to use the notion of membership 0, such that exactly one element (out of infinitely many totally isolated and identical elements (for example: infinitely many identical socks that are not members of any set)) is chosen as a member of a given set (in this case the used notion is membership 1, such that, for example, {2} , {3,2} , {1,2,3,...} etc. are all cases of membership 1).

{2,2} , {2,3,2} , {1,2,2,2,2,2,2,2,3,...} etc. are all examples of membership > 1, such that more than one element (out of infinitely many totally isolated and identical elements (for example: infinitely many identical socks that are not members of any set)) are chosen as members of a given multiset.

If multiset is defined as a generalization of the concept of set, then:

[] is the case that no element (out of infinitely many totally isolated and identical elements (for example: infinitely many identical socks that are not members of any set)) is chosen as a member of a given multiset.

[2] is the case that one element (out of infinitely many totally isolated and identical elements (for example: infinitely many identical socks that are not members of any set)) is chosen as a member of a given multiset.

[2,2] is the case that two elements (out of infinitely many totally isolated and identical elements (for example: infinitely many identical socks that are not members of any set)) are chosen as members of a given multiset.

etc. ... ad infinitum.

Convectional mathematics is simply the particular case of choosing exactly one element (out of infinitely many totally isolated and identical elements (for example: infinitely many yellow identical socks that are not members of any multiset, infinitely many rad identical socks that are not members of any multiset etc. ad infinitum)) such that infinitely many membership 1 multisets have no more than one yellow sock and one rad sock.

 

[doronshadmi]

By understanding the notion of membership in terms of membership = or > 0, one understands that, for example, X is not a member of [X], since X is a membership 0 case and [X] is a membership 1 case.

Those who understand membership only in terms of membership 1 case, simply missing it.

The real problem is to claim that only membership 1 case can be considered as real mathematics.

 

[doronshadmi]

Also, there is only one N; it just happens to be a member of your unnecessary multiset twice.

jsfisher's notion about mathematical objects simply ignores the possible mappings between multiple copies of them under a given multiset.

Multiset is a generalization of the concept of set.

Under multiset S=[N] the possible mapping can't be but bijection.

But under multiset M=[N, N] the possible mapping can't be only bijection.

Since by jsfisher's notion of mutisets M is reducible into S, M is an unnecessary mathematical object, and as a result the possible mapping between multiple Ns, can't be but bijection.

"it just happens to be a member of your unnecessary multiset twice." is a concrete example of hands waving approach of the considered subject, which simply can't deal with http://www.internationalskeptics.com/forums/showpost.php?p=12173799&postcount=2958 or http://www.internationalskeptics.com/forums/showpost.php?p=12194360&postcount=2959.