GENIUS FORUMS

Russell's paradox (set theory)

Let A be the set of all sets which do not contain themselves. Then does A contain itself? If it does, then by definition it does not; and if it does not, then by definition it does.

Russell’s paradox, like all paradoxes, comes about as a result of either shoddy reasoning or an erroneous premise, and as there is no fault to be found in Russell’s reasoning, the culprit here must be the premise: that there are some sets that are able to contain themselves; for if we deny that there is any set that can contain itself, then Russell’s ‘set of all sets that do not contain themselves’ becomes simply the ‘set of all sets’ (U), and as there can be no greater set than U, and as U cannot contain itself, it follows that U is neither an element of itself, nor of any other set. Consequently, the set of all sets that do not contain themselves (U) does not contain itself, and so there is no paradox to be found.

The notion that a set can contain itself is founded on a broader misconception: that a thing is nothing more that the sum of its parts; and that it is these parts alone that constitute the thing’s ‘self’ – there being nothing more. While it is true that a set is a collection of objects, it is more than that, for these objects must be in some way related; either physically, as in the case of the elements that comprise a bicycle, or logically, as the elements in a mathematical set. This logical relation, although not an element of the set per se, is an essential aspect of any set, for there can be no set without some relation between its elements. For example, a bicycle comprises a unique set of components that are essential to its being what it is, however, the mere presence of these bicycle components (elements) does not in itself constitute a bicycle, any more than does the mere presence of bricks and mortar constitute a wall. Only when its components are securely fastened together in the appropriate relation does a bicycle come into being; and the same is true of a set.

When Russell says “the set of all sets that do not contain themselves’’, not only is he expressing that characteristic whereby the individual elements are logically related, he is also identifying the set. That is to say, it is from the relation that a set takes its identity, and not from the elements wherein that relation is embodied. A clay pot, for example, is a ‘pot’ not because of the particles of clay that constitute it (it elements), but because the relation (mode of distribution) that holds between the individual particles of clay; for had the potter chosen to arrange those same particles differently, the result might well have been a plate, instead of a pot. A set, like every entity (distinct existent), must entail both an quality (elements) and a relation (form or identity), and if the elements are the content, then the relation is the container; for the two aspect are both interdependent and complementary.

Therefore, although it may be rightfully said that the elements of a set (its contents) are contained within the set (relation), the relation (set itself) cannot be said to be ‘self-contained’. Further, as the ‘set of all sets’ (U) is neither self-contained nor contained by another set, we may rightfully assert that U is finite and unbounded.

UA is underground for knowing without having to look ("main man").

A russell is name for "main main? (paradox)" (opens one to an x planet).

The inner processing of UA is that which you've described in Russell's paradox.

Jehu wrote: This logical relation, although not an element of the set per se, is an essential aspect of any set, for there can be no set without some relation between its elements.

There would be no elements at all if it wasn't for the relations between them, and as such to the containing set. So if we for one moment forget about the elements themselves, it's easy to see we've a collection of relations, the set itself being the defining relation, the elemental, and as such also an element in yet a different set as it necessarily has to relate to something.

the relation (set itself) cannot be said to be ‘self-contained’.

Only in the sense that it's defined by its relation to its context, in which it's contained. Its containment equals "itself".

as the ‘set of all sets’ (U) is neither self-contained nor contained by another set, we may rightfully assert that U is finite and unbounded.

The set of all sets would have no relation, no "counterpart" and as you wrote "it is from the relation that a set takes its identity", it follows the set of all sets does not exist, cannot even exist logically as an identity.

This logic I call the ghost logic. And a paradox is therefore a ghost. We get these paradoxes because we have been taught to use zero in our mathematical equations. Zero doesn't exist. What actually exists is +1 and -1.

+1 -1 is a ghost.

So the paradox creates a situation where something belongs to both groups at the same time. A sort of flickering image.

The best example of this is the Bose-Einstein condensate experiment. It a good example of not to use zero in math's. An group of atoms frozen to absolute ZERO all suddenly merged together just above zero. Basically they were in a state of paradox.. +1 -1, and could never be zero.

Diebert van Rhijn wrote:

There would be no elements at all if it wasn't for the relations between them, and as such to the containing set. So if we for one moment forget about the elements themselves, it's easy to see we've a collection of relations, the set itself being the defining relation, the elemental, and as such also an element in yet a different set as it necessarily has to relate to something.

Indeed!

Only in the sense that it's defined by its relation to its context, in which it's contained. Its containment equals "itself".

The set of all sets would have no relation, no "counterpart" and as you wrote "it is from the relation that a set takes its identity", it follows the set of all sets does not exist, cannot even exist logically as an identity.

When I say that a set is identical with its defining relation, the relation of which I speak is not that which holds between that set and other external sets, but that which holds between the elements of the set itself. What binds the ‘set of all things red’ together is that each element within the set partakes of the common quality (redness), and so they are all one with respect to this quality. Now, it is true that to say that something is ‘red’ would be completely pointless were it not for the fact that there are things that are ‘not red’, however, this is another matter entirely. It is true that the set of all sets (U) has no counterpart, for there is only that which is, and naught else. Nevertheless, it still has a relational component, for its elements (everything that is) are united by there partaking in that one universal quality called ‘being’ or ‘existence’.

While a relation (set) may be an element of some greater set, it cannot be an element of itself, for while it is true that a set necessarily contains all of the appropriate elements (constitutive causes), it cannot contain the appropriate relation (operative cause) – for the relation is the container, and a container cannot contain itself. That is to say, it cannot be both ‘container and content’, for all such dichotomous pairings are mutually exclusive.

Jehu wrote:When I say that a set is identical with its defining relation, the relation of which I speak is not that which holds between that set and other external sets, but that which holds between the elements of the set itself.

Since the contents of a set could be described as the defining relation between the elements and the elements are also defined by the same relation between each, the set contains itself: it's nothing but what it contains, defines - the elemental relation.

Nevertheless, it still has a relational component, for its elements (everything that is) are united by there partaking in that one universal quality called ‘being’ or ‘existence’.

So lets say this set U equals the defining relation of all its elements: 'existence'. We can call U itself also 'existence'. So the set called 'existence' contains, when we'd add all existing elements together, or even just one: 'existence'.

To me you just seem to be proving the idea that essentially the container contains always itself. The content is circumscribed by its container and the container is filled up by its [whole] content. For that reason they are the same in pure logical terms.

This line of thinking can help tremendously to understand the emptiness of "things" or why they are just "appearances", that is: containers limiting the boundless.

Diebert van Rhijn wrote: Since the contents of a set could be described as the defining relation between the elements and the elements are also defined by the same relation between each, the set contains itself: it's nothing but what it contains, defines - the elemental relation.

This is the point of contention, for if a relation holds between the elements of a set, which it must, then it cannot be an element itself. Now, I am not saying that a relation is something completely independent of the elements wherein it is embodied, but merely that there are two distinctly different aspects to any set: the elements and the relation which binds them together as a distinct entity (set). While it is the elements of the set that render it perceptible to the mind, it is the embodied relation which the mind recognizes and identifies (i.e., associates with a name).

So lets say this set U equals the defining relation of all its elements: 'existence'. We can call U itself also 'existence'. So the set called 'existence' contains, when we'd add all existing elements together, or even just one: 'existence'.

To me you just seem to be proving the idea that essentially the container contains always itself. The content is circumscribed by its container and the container is filled up by its [whole] content. For that reason they are the same in pure logical terms.

Indeed, the container (relation) partakes of an absolute, independent and immutable existence, while its contents (elements) partake of only a relative existence, are totally interdependent and subject to continuous alteration. Being (container and content) then, is not one, not many, not both and not neither; it simply is.

This line of thinking can help tremendously to understand the emptiness of "things" or why they are just "appearances", that is: containers limiting the boundless.

Yes, of course, for relative entities (things) are but shadows cast by cognizant awareness (light source) interacting with knowledge (opaque object); and are devoid of any intrinsic reality. They are truly the stuff of dreams.