, worth looking at, but still in early stages

Zermelo’s Great... Oversight

SECTIONS

Zermelo and Set Theory

The Axiom of Abstraction and Russell’s Paradox

Zermelo’s Axiom of Separation and Its Paradox

Zermelo and Set Theory

(The reader will find this much more comprehensible if read after or in conjunction with Russell’s... Oversight.)

Ernst Zermelo is one of the most recognized names in mathematics in the 20th Century. His work, in particular his axiomatization of Set Theory (see ZF), is the most popular such (along with ZFC), and has deeply affected the development of the foundations of mathematics.

In his axiomatization of Set Theory, Zermelo attempted to avoid all constructions that were paradoxical, such as Russell’s Paradox (published in 1902, relating to the work of Frege) that derives from the Axiom of Abstraction. But Zermelo failed. Set Theory continued to be “paradoxical”. More, his axiomatization was so strict that mathematicians eventually insisted on adding the Axiom of Choice to allow constructions that were of interest, bringing paradox ever more strongly back into the fold.

In particular, in 1908 Zermelo put forward his axiomatization of set theory including his Axiom of Separation, a variant of the Axiom of Abstraction intended to avoid paradoxes such as Russell’s. But Zermelo (et al)... oversighted that it suffers from the same malady — or one similar — as the Axiom of Abstraction. Below will be detailed the “Axiom of Separation Paradox”, Zermelo’s Great... Oversight.

This author has had a few reservations as to whether this oversight deserves to be called “Great”, and may at some later date downgrade it to a “Topical Storm”.

SECTIONS

Zermelo and Set Theory

The Axiom of Abstraction and Russell’s Paradox

Zermelo’s Axiom of Separation and Its Paradox

The Axiom of Abstraction and Russell’s Paradox

The Axiom of Abstraction (which see for more formal detail) has to do with constructing sets from properties. The axiom (technically, an axiom schema) says that, given a property P (where P(x) indicates that P is true of x), there exists a set whose members are precisely those entities having property P. Formally it is written:
     ($y)("x)(xÎy « P(x))

Russell’s Paradox concerns the properties of self-membership and non-self-membership of sets: if we define a set of all sets that (have the property that they) are not members of themselves, the paradox lies in the question of whether this set is a member of itself; i.e. we have an “if it is, then it isn’t, and if it isn’t, then it is” type paradox. Russell attempted to avoid this paradox by developing his theory of types. Russell’s Paradox relates essentially to the Axiom of Abstraction, that many have found fault with and attempted to put right, but it has never been completely successfully resolved.

We can note that, like Cantor’s Paradox, Russell’s Paradox essentially has something to do with general considerations concerning the definition/construction of sets. This is examined in more detail in Russell’s... Oversight.

SECTIONS

Zermelo and Set Theory

The Axiom of Abstraction and Russell’s Paradox

Zermelo’s Axiom of Separation and Its Paradox

Zermelo’s Axiom of Separation and Its Paradox

In 1908 Zermelo introduced his Axiom of Separation to try to avoid paradoxes such as Russell’s Paradox, a paradox of self-membership and non-self-membership that is inherent in the Axiom of Abstraction. It is a variant of the Axiom of Abstraction (technically, they are both axiom schemas) which is given here as a reminder:

     ($y)("x)(xÎy « P(x))                       (Axiom of Abstraction)

Formally the Axiom of Separation is written:

     ($y)("x)(xÎy « xÎz & P(x))              (Axiom of Separation)

or, there exists a set y (that may be empty) that consists of all the members of some set z that have property P.

Zermelo overlooked, and it has been overlooked till now, that his Axiom of Separation suffers from the same type of illness as the Axiom of Abstraction.

There is a necessary consequence of the Axiom of Separation, one that follows from letting P be the property of non-self-membership, P(x) = xÏx, and letting x = y (forced by ("x)). Formally it can be written:

     ($y)(yÎy « yÎz & yÏy)

A usual surface analysis would be to point out how this does not force a contradiction because it allows both yÎy and yÎz & yÏy to be false, allowing a non-contradictory instance of y.

BUT, Zermelo (et al) overlooked the mess that ensues.

We get a situation where either:

1. if we take both sides of the « as being true:
   then y must be both and member of itself and not a member of itself simultaneously (i.e. yÎy is true and necessarily yÏy must be true since both yÎz  and/& yÏy must be true), as well as y necessarily being a member of z

2. if we take both sides of the « as being false:
   then y need not be both a member of itself and not a member of itself simultaneously, but since either way yÏy must be taken as true (at least it must if we allow the truth of the negation of yÎy to necessarily imply the truth of yÏy), then y cannot be a member of z

NOTE (avoiding non-standard possibilities): since y has the property of non-self-membership, i.e. yÏy, yÎy must be standardly false, and we get that:
     false « yÎy « (yÎz & yÏy) « (yÎz & true)
which last must be false to avoid a contradiction, thus yÎz must be false as well. So we pay a strange price for trying to avoid Russell’s Paradox by using Zermelo’s Axiom of Separation in that:

• a set y whose existence is posited by the Axiom of Separation specifically for the property of non-self-membership and specifically for membership in a set z must either:
  
  1) be both a member of itself and not a member of itself (simultaneously), and be a member of z, or
  
  2) not be a member of itself, but cannot be a member of z.
  
  I.e. if a set z has as a member the set y that is not a member of itself, then z cannot have y as a member. But, by standard set construction rules in Set Theory, we can easily make y a member of z, just as we do when we make up a set of all sets that are not members of themselves. (Zermelo was trying to avoid the paradoxes that resulted from this last.) And if we do... what happens to its existence?! At the very least its existence is no longer guaranteed by the Axiom of Separation. And if the Axiom of Separation is forced to give up even the “guarantee” of the existence of y, what kind of thing is this Axiom of Separation, and/or its “guarantee”?!
  
  The “universal” quantifier figures prominently in all this, too. Just what does “"x” — i.e. “for all x” — actually mean? Does it — or should it — already mean “for all x in some set z of ‘completely’ defined entities”? As it is standardly used, it brings with it “all” the “set of all things” or “set of all sets” paradoxes that were/still are considered fatal. By using “("x)”, Zermelo’s safety mechanism of “xÎz & P(x)” is only half-accessed. (For more details on all this, see Russell’s Great... Oversight.)
  
  All this constitutes the “Axiom of Separation Paradox”, overlooked till now.

This “Axiom of Separation Paradox” is Zermelo’s Great... Oversight. It merits status as a great oversight since it involves a variant of that same set self-membership/non-self-membership flaw that the Axiom of Abstraction did, the very flaw that Zermelo was trying to avoid.

The reason that the properties of self-membership and non-self-membership have these consequences in axiomatizations of Set Theory essentially has something to do with general considerations concerning the definition/construction of sets. For example, if we are constructing a “set of all sets that are not members of themselves”, there exists a serious question as to what stage of construction a set is in when we determine such properties of it such as “self-membership”. Can a set that is still under construction, i.e. has not yet been completely constructed- finished-“completed”, be added as a member to any set, let alone itself, also still under construction?!

E.g., if we have the set {1,2,3...n+1}, the nth successor set of {1}, and we then make that set a member of another set, we should either expect this new member to remain fixed at {1,2,3...n+1} as we continue (and not continue to grow as more successor sets of {1,2,3...n+1} are constructed), and finally finish, the construction of the new set, or we should not be surprised when we get dynamic entities whose properties are changing with “time”, e.g. giving us a set that alternates being a member of itself and not a member of itself.

The “universal” quantifier figures prominently in all this. Just what does “"x” — i.e. “for all x” — actually mean? Does it — or should it — already mean “for all x in some set z of completely defined entities”? Zermelo should never have overlooked this question since it is the same question he was attempting to answer with his Axiom of Separation. Neither should Russell, nor any other 20th or 21st Century logician, mathematician, nor a good selection of philosophers. It may be that Zermelo’s Axiom of Separation is trying to hint to us that there is this problem of making elements of sets from incompletely constructed entities and thinking that everything will somehow magically be all right when the incompletely defined entities are defined more and more completely, until they are “completely”, if ever, whatever any of that means, and what it means is very much in paradoxical question.

Another essential problem that is perennially overlooked with the Axiom of Abstraction and the Axiom of Separation is that the property function P(x) is so general that it goes way beyond what is mathematically well-defined, even well beyond what is mathematical. At the very least, this opens the door to partial recursion, even for “mathematical” properties. A partial recursive function PRF(x) is not necessarily well-defined for “"x” in the function’s domain; in fact, it is not only not well-defined, it can go off into La-La Land for a given x, never to return (like Charlie ’neath the streets of Boston). This La-La Land phenomenon, studied somewhat carefully in the Theory of Partial Recursive Functions, is studiously ignored in Logic, Set Theory (where it is relevant to e.g. set construction paradoxes), and analysis (where it is relevant to e.g. the convergence of infinite sequences/series). Also studiously ignored are things also ignored by all logics extant as of 2006 CE, among which are: time(s), space(s), point(s) of view(s), each and all of which can be taken as qualifiers/qualifications on properties and propositions.

This is all examined in more detail in Russell’s Great... Oversight where it is suggested that if we insist on fixing the state of an entity made a member of a set under construction that we can avoid the unacceptable paradoxes that come with the Axiom of Abstraction and Axiom of Separation.