Russell's paradox
If the mathematician Frege was a linguist in addition to being a mathematician, he was not discouraged by the paradox that Russell offered him and it might currently be a more solid theory sets.
It usually resolves the paradox of Russell by eliminating the possibility that a whole can be part of himself.
We thus depriving an unprecedented exploration of the theory of numbers. A set of things is an abstract thing. Nobody can "weigh" or "carry" a group because we are in the area of the symbol, not in that of the object.
Speaking of all of something implies that our thinking has reached the stage of conceptual language, that is to say that we are capable of mapping. Thus, any number is a set.
When Russell says that all coffee is not coffee itself, it might have been abused by design its own coffee. All coffee is a schematic, it is also a coffee, "coffee" is understood SYMBOL like (not as an object). We can say that all coffee is represented by the symbol COFFEE MAKER, which is both element (coffee) and set (the coffee in general).
All pots (or dogs) does not contain any concrete object (all dogs not contain any live dog ) .
Consider a comparison: all the women going to the locker room "woman" is represented by the symbol woman on the door, which means both a woman and all women. If we speak of all women concrete every woman is itself an indicator of the set, then a symbol, not a particular woman (which has no use for all women). Even if we decide that all women do not belong, that does not prevent any woman in all women remains a net .
Any sign, any word, any number serves as both a name (element) and as a symbolization (together).
Thus (in theory), the set of all sets is symbolized by the sign (or word) together, and it is only part of himself (if there is not).
The property of self-ownership (Continuum) is a guarantee of the existence of any item deemed to mean :
x = {x}
In sum, there is confusion between abstract thing and concrete thing (language and reality). In terms of concrete, the overall concept has no place. In a forest, you see that and tap "the trees". If you talk to "all trees", then this is the concept "tree" in question, not the object "tree". And every concept is set and element itself.
In Russell's paradox, there is an intruder language: all the sets that no longer belong. This set can not define EE (all together) because EA belongs to itself by necessity.
Either EA is set and everything belongs to himself. EA is not there any set does not belong.
Now the empty set belongs to itself (the empty set contains no element not empty, but contains himself ). So all and all sets are non-empty set belongs to itself virtually.
Reject the axiom of virtuality of a set is equivalent to reject not only all together, but also the empty set (since it is not real).
It therefore refers to any non-empty set Their belonging naturally (continuum) EXCEPT if and only if all its separate parts is arbitrary this set (matrix) .
Let E be a nonempty set. P (E) is included in E. virtually If it were not, we could construct P (E) real : {P} E distinct from E. So, if P (E) potential E, E naturally belongs to E.
other words, E "does not belong" in case it loses its singularity - which means that I always belongs to itself if is compared only to itself (hence the empty set and all sets). Property "does not belong" is only valid when applied to a plurality of sets (at least two), let it be that there may be a virtual pair of disjoint sets completely, which is impossible.
Indeed, if E is not empty, we have:
E = P0 (E), E not = P1 (E), E1 = P1 (E)
where P0 (E ) is the virtual totality of subsets of E, P1 (E) is the set of real parts of E, and E1 and P1 (E) form a pair which contains virtually E1 elements of P1 (E).
By adopting the convention that a set is a box and not a continuum (so in reality), we can do "as if" this set does not belong, but we must know that it is only a convention of writing unfit to approach the true meaning of the sets.
not x = {x} (where {} is a box or matrix).
We write: M = [x, {x}] where x is different from {x}. M is not a set in this case because the property "Does not belong" does not define a single set. We say that M is a non pair sets, or even a antipaire.
We show by contradiction that M can not be a single set.
Let E = {x} {x} where x is different from {x}. This set is impossible because {x} subset of E, does not belong to E, but P (E). {X} can not appear in E.
In fact, if {x} belongs to E still is an element of E and has been a part of E which is also part of E. So I belong to himself and x = {x}.
can also write: {x} {x} = {x} union {x} = {x}, which is the same.
Specifically:
x + {x} = {x, x} = {x}
x = {x} - {x} = {}, so {x, {x} = {} § {§ §}} and {, {§ §}} = {} + {} = {§ §}
§ The set {} is equal to the sum of its parts, so § § = {}, and M can not define a single set (x can never be non-empty M).
If E = M, any object can be set, since all of its parts is necessarily included (virtual) in this object.
The paradox arises simply not - except perhaps on a linguistic (*).
(*): reject the set of all sets the pretext that he belongs to itself negates the actual infinite. The great contradiction of Cantorian system based on the fact that he continues to regard the infinite as a "whole" if "all" calls into question his theorem. We can be both current and potential.
(*): a predicate does not define a set if contains its negation. Eg
- All sets that no longer belong.
- All non-sets.
- All colors are not colors.
- All the things that do not exist.
