, worth looking at, but still in early stages

Russell’s Great... Oversights

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Russell and Set Theory

Automata, Recursive Functions, and Set Theory

Defining Versus Constructing

Analyzing Russell’s Paradox

Russell and Set Theory

Bertrand Russell is one of the most famous names in mathematics, logic and philosophy in the 20th Century. His work has deeply affected the development of the foundations of mathematics. A paradigmatically crucial paradox in Set Theory bears his name: Russell’s Paradox. It is a paradox of self-membership and non-self-membership based on the Axiom of Abstraction (of Frege). It was to avoid such paradoxes that Zermelo put forward his Axiom of Separation, but... oversighted by Zermelo (et al), it too allows almost the same paradox.

In his statement of his paradox, and in later attempts at resolution of it, Russell overlooked some essentials. These essentials have to do with defining/constructing sets. Although these oversights have become more obvious with the advent of the Computer Age, they still should not have been overlooked even in the 19th Century, let alone since, especially not by mathematicians and philosophers concentrating on the logic of semantic constructions. They will be examined in more detail below.

SECTIONS

Russell and Set Theory

Automata, Recursive Functions, and Set Theory

Defining Versus Constructing

Analyzing Russell’s Paradox

Automata, Recursive Functions, and Set Theory

When Set Theory was being conceived at the end of the 19th Century, Automata Theory and Recursive Function Theory were not yet visible over the horizon. Set Theory has yet to notice, let alone come to terms with, some of the implications of these newer theories.

One characteristic difference between them and Set Theory is in their treatment of infinity. Cantor had wanted his infinite sets to be “completed infinities”, by which he meant that one did not have to keep adding elements to them to make them “infinite” (the root meaning of which in Latin is “unfinished”). (We can note that Cantor failed to explicitly notice and take into account that there is an essential relationship between “completed” and “finitary”; he also failed to take this into account implicitly. But all variants of his Set Theory so far have many more problems than that comment suggests.) The entire infinity (or “transfinity”, a term that has not yet become popular) of the elements of the set was there, “completed”. This is an essential concept in Set Theory, and the treatment of transfinite cardinals depends on it. E.g. how can one construct the power set of a set until that set is “completed”?

The set of all natural numbers is held to have cardinality a₀, the first transfinite cardinal and a completed infinity. But if we set up a Turing machine to write 1s on its tape, 1 for each natural number, Automata Theory does not allow the halting of the Turing machine when it “completes” writing all the 1s. In fact, it does not even have a concept of “completing” such an activity, and thus the emphasis on the concept of “algorithm”, which by definition always halts after a “finite” number of atomic (and also “finite”, especially in time) operations.

This all may seem to be somewhat frivolous to even mention, but referring to these theories and the computers that they helped spawn will help when attempting to gain insight into Set Theory’s “completed infinities”, and the paradoxes that arise when defining/constructing “completed” sets.

SECTIONS

Russell and Set Theory

Automata, Recursive Functions, and Set Theory

Defining Versus Constructing

Analyzing Russell’s Paradox

Defining Versus Constructing

Set Theory was conceived in a world that was teetering on a brink. That brink can be given a buzzword: “time”. Einstein was just about to bring the attention of the whole world back to a concept that was ignored by being taken for granted: “time”.

The world of mathematics still has not come to terms with it, even though so-called applied mathematics have “time” as a variable for various functions, as a parameter, i.e. at least mentioning it by label. All branches of mathematics,  geometry being a classic example, are considered timeless truths, at least implicitly. E.g. if a theorem is provable, then in a sense it pre-existed the human discovery of its proof. And when logic examines syllogisms such as “if P É Q, then if P then Q”, it gives the variables P and Q different values at different “times”, but logic does not really have an internal concept of “time”, which lack is consistent with the “timeless truth value” of logic that to this day goes without saying.

The early set theorists were also more than a little naive when they thought sets as being collections of anything that could be conceived. They also thought of all these sets as somehow pre-existing their human conception, which is how they could speak of a set that included itself as a member. They didn’t really consider the possibility that any computer programmer would consider, that when one is constructing a set, the set being constructed is not yet defined and in a sense does not yet exist when it comes to making it a member of itself. A computer programmer could work around that, making one of its elements point to its set’s own main table entry and then changing the status field of the main table entry for the set from “under construction” to “successfully completed”, but this is done with great care for later programming involving the software set. When done carefully (to e.g. lock out access till successful construction has been completed), this simultaneously changes the set and itself as an element (infinitely recursively, but an anti-Cantorian “uncompleteable infinity”). Set Theorists still do Set Theory variants of constructing sets that are members of themselves, but without the careful consideration for what they are doing.

But there is rather more to consider in this situation where we have been mistaking our sense of the timelessness of mathematical truths for pre-existence of mathematical entities in the definition-construction sense.

• We tend to logically verify a truth for one set of values of what turns out to be set of variables, then use that truth later like a check, but with a different amount filled in above the signature.

The mishmash of errors is much more nebulously complex than this quasi-intro to an introduction suggests, and we will only scratch the surface here, but this is no excuse for logicians of Russell’s competence and stature to mistake one thing for another in these regards.

The whole group of interrelated concepts of “define”, “property”, “rules of construction” have not been fully appreciated with regard to “time” and the questions of “existence”, especially “pre-existence”. We will use Automata Theory and Recursive Function Theory as offering competing paradigms for these things, and pursue a “comparative religion” approach when peering into the nebula.

It is easy to see how we would construct a set that does not have itself as a member, but how would we construct a set that does have itself as a member? (We do not want to forget the computer paradigm given above, but it is a question that needs to be explored much more fully in Set Theory.) When we start to construct a set, it is not yet “completed”, it is not yet “defined”. How can we add an entity, e.g. a set, as a member of our being constructed set if that entity has not yet been  “defined”, “constructed” or “completed”? If it has not yet been defined, is the member that we just added forever the entity in its “undefined” state? Or does it become whatever the entity is later defined as? This latter has been what has been accepted implicitly, essentially without question (because mathematical and logical entities are “timeless”?), but should have been explicitly challenged if only to explore it well. We start seeing the problem of “time” in Set Theory that has been overlooked by mathematicians for over a century.

• When we define/construct sets, we forget the state of the elements when we make them members. When the construction of the set is “completed”, and the elements that are its members somehow change after that “completion” (as opposed to “at the instant of/by the act of completion”), we are talking about sets being non-self-equal. This is unacceptable until we resolve the “timelessness”/“time” issue satisfactorily.

SECTIONS

Russell and Set Theory

Automata, Recursive Functions, and Set Theory

Defining Versus Constructing

Analyzing Russell’s Paradox

Analyzing Russell’s Paradox

When an entity, e.g. a set, has been “constructed” in the sense of being finished or “completed”, it somehow seems... questionable to go changing that entity/set, to add members, delete members, or, even more invisibly, to modify members. (At least if we do, we forgo “timelessness” in a sense. And at the very least we will need paradigms from e.g. the world of computers.) But this is what is done when in e.g. Russell’s Paradox.

Russell “constructed” the “set of all sets that are not members of themselves (and no others)”, and then asked the question “is this set a member of itself?” If it is, then it cannot be, and if it isn’t then it must be.

At first it’s hard to tell whether Russell’s question/paradox is just horribly naive (e.g. naively “constructed” when we consider the insights quasi-introduced above), or whether it is important. It is both, and that makes it paradigmatic in an extremely significant sense.

The set Russell “defined”, the “set of all sets that are not members of themselves (and no others)”, had not yet been defined when it was in the process of being defined. (Notice how quickly the question/paradox starts seeming naive.) The sets other than itself may all have been constructed by the time Lord Russell starts his construction... unless they were defined as... well, it gets sticky.

•  The meaning of “all x” “("x)” is absolutely in question. The meaning of “all x in z” ("xÎz) makes rather more sense, but still... Do we know for sure that the set z has been “defined”-“constructed”-“completed”, and, most importantly, “fixed” in the sense of “no further change to the set or its elements” (no new elements, no elements deleted, and no modifications of any elements).

This brings up issues that have remained unexplored in Set Theory:

•  In Set Theory, a set is ostensibly defined solely by its members. But we often consider sets to have properties such as membership in other sets. Since a set y is “defined”-“constructed”-“completed”, and its elements are ostensibly “fixed”, that it has x as a member can reasonably be considered a property of y (and a “fixed” property as well). But that a set x is a member of another set y is questionable as a property of x, if, in fact, a set is defined solely by its members. Self-membership as a property, though, seems to dodge this question.

•  If we have e.g. 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} (and not continue to grow as more successor sets of {1,2,3...n+1} are constructed) as we continue, 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. This latter makes sense in computer theory, makes rather less sense in strict automata theory where the accent is on algorithms that halt by definition, and makes no sense in classical mathematics where such dynamism has never had a place in the “timeless” logic its foundations.

•  The “universal” quantifier figures prominently in all this since it has an essential “set of all sets/things” quality to it. 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 sets” or “set of all things” paradoxes that were considered fatal. Russell should never have overlooked this question since it is the same question he was attempting to point out with his Paradox. Neither should Zermelo, nor any other 20th or 21st Century logician or mathematician.

I.e. we have to find a good answer to further questions, such as: “how can we add the undefined or incompletely defined set to itself as it is being defined?” (A computer programmer, as we looked at above, could probably think of several ways, but we haven’t explored enough to know if any of these would satisfy the needs of a competent Set Theory.) “How can we do so in such a way that, when we finish constructing the set, the unfinished version of itself that we added itself becomes finished?” “What does that mean and do we even want to do that?” “Do we want to allow e.g. the option of leaving the member set unfinished?”

The next point is noticeable to computer programmers immediately. If the set (of all etc.) is a member of itself, then it isn’t, and if it isn’t, then it is. Automata Theory and (Partial) Recursive Function Theory, both standard theories in Computer Science, have already explored the anti-Cantorian “uncompleteable infinity” resolutions to this. And if a computer programmer had programmed the software to make the set a member of itself, then it would be so, despite the fact that it no longer seems to us to be eligible for self-membership. The programmer could then have the software “unfinish” the set, revoke the self-membership, and “finish” the set again, despite the fact that it then seems to etc. The programmer could even program the software to alternately “finish” the set, “unfinish” it, change it, “finish” it again, etc. an infinity of times that would “never” yield to Cantor’s desire for a “completed infinity”, i.e. that would not halt after completing the infinity in question (a₀, a₁,...). And there are indefinitely many more possibilities.

A computer programmer would find the original desire for a set of all sets that are not members of themselves to be... naive, at least if that desire was for a “timeless” mathematical truth, an unchanging mathematical entity. The software version of self-membership rarely if ever makes computer sense.

Cantor’s Paradox has more or less the same flaw as Russell’s Paradox, one of construction. Until the issues put forth above are resolved competently, we cannot legitimately define-construct a set of “all sets” until we have defined-constructed “all sets”, and we cannot legitimately define-construct the power set of a set until we have defined-constructed the set.

Neither Russell’s Theory of Types nor Zermelo’s Axiom of Separation competently address these issues. In fact, Zermelo’s Axiom of Separation has the Axiom of Separation Paradox inherent in it, just as the Axiom of Abstraction (of Frege) has Russell’s Paradox inherent in it. (The Axiom of Separation Paradox can be successfully resolved by paying attention to the lack of completion of the definition-construction of entities.) These paradoxes are inescapable if one ignores the inherent difficulties (ignoring them tends to makes them fatal flaws) in defining-constructing entities in terms of as yet undefined-unconstructed entities.

Until Set Theory gets a less “timeless” conception, it seems necessary to forgo the now obviously questionable practices of playing fast and loose with definitions and constructions. One should not be able to add an entity that is not yet defined-constructed to a set being defined-constructed. Once an entity has been defined-constructed, it should not be allowed to be modified, directly, either more visibly e.g. by changing the membership of a set, or more invisibly e.g. by changing a member of a set.

It bears repetition:

•  Set Theory needs to start being far more careful about the definition-construction of sets, and that means not only of the collection of the elements, but of the elements themselves. If the construction of the set is “completed”, and the elements that are its members somehow change after that “completion” (as opposed to “at the instant of/by the act of completion”, which still has its own problems), we are distinctly talking about sets being non-self-equal. This is unacceptable until we resolve the “timelessness”/“time” issue satisfactorily. Until then we need to accent the “fixed” aspect of sets and elements when defining/constructing sets or those entities that are/will be their elements/members.

•  We have tried to perfect the timeless logic we inherited from the Greeks. But we have failed to note that it starts to fail badly in situations where time is involved, such as when defining-constructing sets of all sets that are not members of themselves. The development of multi-modal and paraconsistent logics attest to this and other... shortcomings.

And it bears repetition to explicitly question how Lord Russell could have... oversighted the definition-construction issues raised above in his famous paradox. These issues do not really need paradigms from computer programming to recognize them nor to find the simple-minded resolution (yet quite adequate until we want to include something like the paradigms of computer science in Set Theory) of not allowing incompletely defined-constructed to be added as members to sets under construction, even if our languages allow the well-flawed-formulas to be formed seemingly flawlessly.

Relatedly, another essential problem that is perennially overlooked, e.g. with the Axiom of Abstraction and the Axiom of Separation, is that the property function P(x) is so general that it opens the door to partial recursion. A partial recursive function PRF(x) is not necessarily well-defined for “"x”; 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 “Charley ’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, and only occasionally noticed in Set Theory, and then noted as “paradox” and dismissed.

Although the much newer Automata/Computer Theory and Recursive Function Theory, which includes the theory of partial recursive functions that are “not defined” for all values in their domains (they recurse to an infinity that both practically and formally within those theories cannot be “completed”), offer many examples of how there can be serious problems with definition and construction, such examples should never have been needed. Russell et al had the many examples of classical mathematics, e.g. of polynomials that with some combinations of coefficients do not have (“real”) roots. A set of all sets that are not subsets of themselves is like a polynomial that does not have a “real” solution. It would have been incredible if Russell had even hesitated, but that he never noticed the parallels even though there exists a somewhat chaotic complex of seemingly minor thises-and-thats... truly Russell’ Great... Oversights.

•  We need to start developing logics, mathematics and philosophies that can keep up with the world of time and its change that we find ourselves in. Our Greek logic of absolute timeless propositions and truth values, of absolute timeless (and unambiguous, non-fuzzily defined, yet incompletely defined since abstract) entities (e.g. the objects timelessly referred to by timeless propositions), and of absolute timeless (and again unambiguously defined) logical operators (when does x really “imply” y? and what does that mean? “smoke implies fire”? “x ‘causes’ fire”? we usually want to get “causes” from “implies” even though it is only everything together that “really” causes... uh, everything together) needs to give way to a logic that can at least keep up with the video game realities of computer programming.