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A SLIGHT REFORMATION OF SET THEORY
A SET
A collection of objects, known as the elements of the set,
specified in such a way that we can tell in principle whether
or not a given object belongs to it. E.g. the set of all prime
numbers, the set of zeros of the cosine function.
For each set there is a predicate (or property) which is
true for (posessed by) exectly those objects which are
elements of the set. The predicate may be defined by the set
or vice versa. Order and repetition of elements within the
set are irrelevant so, for example, 1, 2, 3 = 3, 2, 1 =
1, 3, 1, 2, 2.
Source: V.E.R.A. -- Virtual Entity of Relevant Acronyms December 2001
Notice how in the apparent definition shown above a set is explained to be 'a collection of objects'... (excetera)
Of course the term 'object' is meant to refer to object in the most generic and comprehensive sense of the word not limited merely to what are called "physical objects" but, instead, as referring to any object of study --any object of a thinking process (including physical objects, but not exclusively limited to them, but also including objects whose existence is purely in the abstract )
What the apparent definition does *not* ascribe to the mening of the word 'set' as being "a collection of a collection of objects along with the specified objects". Notice that the apparent definition for the word set does NOT read , "a collection of objects and also additionally the collection of objects as a whole itself" .
Yet, oddly enough, there have been those persons in set theory that have gone beyond the defintional parameters as to what the term 'set' refers to and thus made the bizarre claim that a set can go beyond having its specific elements as members, to proposing that a set can contain not only its elements but can allegedly also contain itself as a whole ! The set theorists who have proposed that notion call such a notion "self-containment", and allege that there are self-containing sets .
Unfortunately in the academic community there are often quite dodgy , murky non-accurate notions that are advocated by persons held in high renown , even professors of various fields of study that have considerable credentials and are venerated in various academic communities . Due to the mystification of credentials and renown in both mushc of the academic community at large and many areas of American and European culture even outside Academia, dodgy notions when adovated by persons of esteem and renown sometimes go largely unchallenged by many and even get manifested in textbooks as presuppositions that are scarcely given mcuh (if any argumentation) to support .
The philosopher Alfred North Whitehead once noted that the ultimate court of appeal is 'intrinsic reasonableness' and not whether someone of renown and prestige claims a belief to be true . Yet, unfortuantely, even in Academia mystifcation scarcely gets the chiding it deserves .
The notion that sets can contain themselves as a whole is, quite frankly , muddled thinking . The notion that sets can contain themselves as a whole is an example of what Gilbert Ryle called a 'category mistake' (a notion that involves the mistake of shifting what is contextually predicated of one category over into another category where it is *not* conceptually proximate) . Think about the issue deeply . It is indeed more accurate to speak of set like structures that contain other sets as members as meta-sets , not sets --since they involve a taxonomically different level of menaing .
A set is defined as containing several elements . A set is a collection of elements all sharing some specifying predicative feature (or limited grouping of features) which is inclusive of some meanings and ,hence , exclusionary of others .
A collection of elements is NOT the same as the specific elements . To speak of a set containing itself as an element makes about as much sense as claiming that air itself is one of the elements that makes up the compound air . Whcih is to say that it doesn't make sense at all . Air is a compound of elements (e.g. nitrogen , oxygen, carbon ect) yet to claim that air as a compund/ air as a whole is among the several elements that make up the compound air .
If one is going to make predications that shift the proposed relation of parts to the whole that comprises them in a way that makes no sense at least one could do it in a way that is funny and has more panache ---such as when poet Gertrude Stein apparently wrote in one of her poems ,
'Bees in a garden make a specialty of honey and so does honey .'
The collection of elements as a whole as considered under a different mode of presentation than the several specified elements is but what in philosophy would best be called a hypostasis or a *hypostatic* wholeness . Indeed such a whole is not somehow mysteriously greater than the sum of its parts . Such a whole does *not* at all have some mysterious faculty of being itself also some additional element along with the other elements that are its parts !
Thus, when some in the field of set theory claim that there are some *so-called* "paradoxes" in set theory (like the so-called "Russell Paradox" ) rest assured there are NO such authentic "paradoxes" and that the very use of such term is a misnomer ! For the claim of *so-called* "paradoxes" of *so-called* "self-containing sets" is based on a notion that is false ---it is a category mistake which involves a fallacy of scale to begin with ! The ontological status of set as a whole . of the collection itself --conceived momentarily apart from the several elements that make up the set --is a hypostasis and not an element !
It is rather odd that such an obvious fallacy goes apparently unchallenged in many printed discourses regarding set theory .
A SET
A collection of objects, known as the elements of the set,
specified in such a way that we can tell in principle whether
or not a given object belongs to it. E.g. the set of all prime
numbers, the set of zeros of the cosine function.
For each set there is a predicate (or property) which is
true for (posessed by) exectly those objects which are
elements of the set. The predicate may be defined by the set
or vice versa. Order and repetition of elements within the
set are irrelevant so, for example, 1, 2, 3 = 3, 2, 1 =
1, 3, 1, 2, 2.
Source: V.E.R.A. -- Virtual Entity of Relevant Acronyms December 2001
Notice how in the apparent definition shown above a set is explained to be 'a collection of objects'... (excetera)
Of course the term 'object' is meant to refer to object in the most generic and comprehensive sense of the word not limited merely to what are called "physical objects" but, instead, as referring to any object of study --any object of a thinking process (including physical objects, but not exclusively limited to them, but also including objects whose existence is purely in the abstract )
What the apparent definition does *not* ascribe to the mening of the word 'set' as being "a collection of a collection of objects along with the specified objects". Notice that the apparent definition for the word set does NOT read , "a collection of objects and also additionally the collection of objects as a whole itself" .
Yet, oddly enough, there have been those persons in set theory that have gone beyond the defintional parameters as to what the term 'set' refers to and thus made the bizarre claim that a set can go beyond having its specific elements as members, to proposing that a set can contain not only its elements but can allegedly also contain itself as a whole ! The set theorists who have proposed that notion call such a notion "self-containment", and allege that there are self-containing sets .
Unfortunately in the academic community there are often quite dodgy , murky non-accurate notions that are advocated by persons held in high renown , even professors of various fields of study that have considerable credentials and are venerated in various academic communities . Due to the mystification of credentials and renown in both mushc of the academic community at large and many areas of American and European culture even outside Academia, dodgy notions when adovated by persons of esteem and renown sometimes go largely unchallenged by many and even get manifested in textbooks as presuppositions that are scarcely given mcuh (if any argumentation) to support .
The philosopher Alfred North Whitehead once noted that the ultimate court of appeal is 'intrinsic reasonableness' and not whether someone of renown and prestige claims a belief to be true . Yet, unfortuantely, even in Academia mystifcation scarcely gets the chiding it deserves .
The notion that sets can contain themselves as a whole is, quite frankly , muddled thinking . The notion that sets can contain themselves as a whole is an example of what Gilbert Ryle called a 'category mistake' (a notion that involves the mistake of shifting what is contextually predicated of one category over into another category where it is *not* conceptually proximate) . Think about the issue deeply . It is indeed more accurate to speak of set like structures that contain other sets as members as meta-sets , not sets --since they involve a taxonomically different level of menaing .
A set is defined as containing several elements . A set is a collection of elements all sharing some specifying predicative feature (or limited grouping of features) which is inclusive of some meanings and ,hence , exclusionary of others .
A collection of elements is NOT the same as the specific elements . To speak of a set containing itself as an element makes about as much sense as claiming that air itself is one of the elements that makes up the compound air . Whcih is to say that it doesn't make sense at all . Air is a compound of elements (e.g. nitrogen , oxygen, carbon ect) yet to claim that air as a compund/ air as a whole is among the several elements that make up the compound air .
If one is going to make predications that shift the proposed relation of parts to the whole that comprises them in a way that makes no sense at least one could do it in a way that is funny and has more panache ---such as when poet Gertrude Stein apparently wrote in one of her poems ,
'Bees in a garden make a specialty of honey and so does honey .'
The collection of elements as a whole as considered under a different mode of presentation than the several specified elements is but what in philosophy would best be called a hypostasis or a *hypostatic* wholeness . Indeed such a whole is not somehow mysteriously greater than the sum of its parts . Such a whole does *not* at all have some mysterious faculty of being itself also some additional element along with the other elements that are its parts !
Thus, when some in the field of set theory claim that there are some *so-called* "paradoxes" in set theory (like the so-called "Russell Paradox" ) rest assured there are NO such authentic "paradoxes" and that the very use of such term is a misnomer ! For the claim of *so-called* "paradoxes" of *so-called* "self-containing sets" is based on a notion that is false ---it is a category mistake which involves a fallacy of scale to begin with ! The ontological status of set as a whole . of the collection itself --conceived momentarily apart from the several elements that make up the set --is a hypostasis and not an element !
It is rather odd that such an obvious fallacy goes apparently unchallenged in many printed discourses regarding set theory .
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