2
$\begingroup$

A purported existence of a standard model of such theory as ZFC has been a cause of discomfort for a number of experts.

Taking a strong Platonist view of entities and their collections, etc., in a standard model of set theory such as ZFC as literally existing somewhere in the realm of the abstracta, one quickly reaches an inconsistency.

This is because, if sets, their power sets, etc., correspond to entities literally out there in the Platonist realm of the abstracta, and union (as governed by ZFC axioms) is merely the concatenation of such abstracta, then the entire arsenal is literally out there, which is of course inconsistent since "the set of all sets" cannot be a set in ZFC, as is well known.

The question is whether weaker Platonist and/or realist assumptions about sets in a putative standard model can be developed that would lead to an inconsistency in a subtler fashion?

More specifically: what kind of naive realist intuitions of sets do beginning students of set theory have that need to be rejected to avoid inconsistency?

$\endgroup$
  • 4
    $\begingroup$ Are you planning on disclosing your personal relationship with Jaykov Foukzon? $\endgroup$ – Asaf Karagila Jun 2 '18 at 18:27
  • 6
    $\begingroup$ I don't understand the first paragraph, it is way too informal. Can you try to elaborate more precisely what the purported issues are? $\endgroup$ – Andrés E. Caicedo Jun 2 '18 at 19:17
  • 6
    $\begingroup$ @Andrés: Just in case you missed it, Mikhail asked on MathOverflow recently a question where he asks if one of the recent "papers" by Jaykov Foukzon about the inconsistency of ZFC+"There exists a standard model" is a serious paper, since it was published in a "peer reviewed journal". Of course having received a tenured position, one can only imagine that Mikhail had the tools to make these assessments on his own, and chose to do this in some failed attempt to get more mainstream attention to the paper. (I say failed since it was closed and deleted, of course.) $\endgroup$ – Asaf Karagila Jun 2 '18 at 19:45
  • 5
    $\begingroup$ I'm afraid I still don't see what the problem is; I'll probably need a significantly more detailed explanation of the problem. I don't see why anybody would mention "the set of all sets", for instance, and conclude that there is a problem. At the moment, it feels like saying "imagine all natual numbers, with their relations. If we imagine all, then we also imagine their fractions. A contradiction, since $1/2$ is not a natural number." I suspect you mean something more serious. Is there a more nuanced argument you have in mind that are perhaps compressing too much in your presentation here? $\endgroup$ – Andrés E. Caicedo Jun 3 '18 at 12:16
  • 5
    $\begingroup$ I think your platonist's naivete is the hardest thing to believe here. A more nuanced argument by a less implausible platonist seems in order if you expect any serious answers. $\endgroup$ – Andrés E. Caicedo Jun 3 '18 at 14:20
3
$\begingroup$

A basic misunderstanding of sets at the intuitive level is to think of them as totalities of their elements, you see this sometimes phrased as: sets are nothing beyond their elements; a set is its elements, etc... philosophically re-phrasing this intuitive account is to say a set is the totality (or whole) of all of its elements; and formally speaking, since Mereology is the discipline devoted to understanding Part-Whole relation this is phrased as: a set is the mereological sum (or fusion) of its elements, or sometimes a set is the heap (conglomerate\aggregate) of its elements. That matter had been shown to be false as early as Bertrand Russell's work on mathematical logic (see: Introduction to Mathematical philosophy): let me quote that:

"We cannot take classes in the pure extensional way as simply heaps or conglomerations. If we were to attempt to do that, we should find it impossible to understand how there can be such a class as the null-class, which has no members at all and cannot be regarded as a “heap”; we should also find it very hard to understand how it comes about that a class which has only one member is not identical with that one member. I do not mean to assert, or to deny, that there are such entities as “heaps.” As a mathematical logician, I am not called upon to have an opinion on this point. All that I am maintaining is that, if there are such things as heaps, we cannot identify them with the classes composed of their constituents."

Bertrand Russell, Intorduction to Mathematical Philsosphy, p:146-147

Further work in Mereology and set theory reveals that "sets" [as of ZFC] begs more Ontology, in other words a set must have at least a part of it that is disjoint (do not share a common part) from the heap of all of its elements, and it is not composed just of the material of its elements, the more you define sets the more you are Ontologically committed to newer entities having new material in them, I think this was first attributed to Stanisław Leśniewski.

In David Lewis's Parts of Classes, a nice work on relationship between set theory and Mereology, one can see where that excess material of a set comes from. In nutshell he thinks of the existence of a singleton partial function $Lb$ [the notation is mine] that sends aggregates of atoms [objects having no proper parts] to atoms, so the atom that an aggregate is sent to under that singleton function would serve as a "label" for that aggregate, then he defines class as "aggregate of labels", and define epsilon membership "$\in$" as:

$x \in y \iff \exists l [l=Lb(x) \wedge l \ P \ y]$,

where $P$ signify "is a part of",

Now, under that definition, it is easy to see that a class do have a part of it that is disjoint of the aggregate of all of its elements, this simply would be the fusion of all atoms in the class that are not parts of what is labeled by a label that is a part of that class, more simply stated: what is remained from a class after taking out all elements of its UNION from it. And since we are speaking of well-founded models then there would always be an excess material into a set over the sum material of its elements. Now Lewis then goes to define "set" as a class that has a label under the singleton function, and of course a proper class would be a class for which no label is assigned by that singleton function.

Now the head post is speaking about some naive form intuition about sets plus some Platonism, whereby each set is an entity in the abstracta, i.e. the abstract Platonic realm, and thinks of "set Union" as merely the "concatenation of such abstracta", and here "concatenation" is just another word for mereological aggregate, and this claim is intuitively false, as seen from above. To re-phrase Lewis's views in your terms I'd say that the set union would be the concatenation of all labels of the concatenated abstracta. The problem is that we are not sure that the arsenal of all such abstracta has a Label! There is no axiom that states that every aggregate of labels must have a label, actually this axioms lead straightforwards to Russell's paradox. So you can see that the concatenation intuition leads to existence of a "class" of all the concatenated abstracta but not of a set of all of them.

So "sets" are not mere extensions [classes would be!] they encounter something else which in Lewis's view would be understood as labels, so set theory is about labeling of extensions, so an extension which is potentially a plurality would be labeled by a singular entity, and we take extensions of those singular entities and then label them by the singular, and so on, Lewis views set theory as the hierarchical inter-play of the plural and the singular.

I personally like to intuitively view "sets" simply as containers, and set member as an atom inside a container at some moment of time. We can rephrase the membership of ZF as "is contained in" and objects in the domain of ZFC to be some kind of containers [whether abstract or concrete]. Now if we extend ZFC with classes, then I take those classes to correspond to mereological aggregates of containers, and so classes are nearest to the idea of extensions (or concatenations in your terms), I also would define class membership in a separate manner from set membership (which I view as containment-ship really), a member of a class is a container that is a part of that class, and also mereologically I'd stipulate all containers as mereological atoms, since we are not meant with the proper parts of a container, we are meant with its containment action! So there is excess material involved with thinking of sets here, because they are clearly not the aggregate of what they contain, they are the container that contains all and only atoms of that aggregate.

So again your concatenation intuition would translate into saying that there would be an aggregate of containers but this need not necessarily have a container that contain all of its atoms, so again you won't have a SET union, you'll have a "class" union. And so there is no contradiction with ZF. The contradiction would raise if you think that every aggregate G must have a container C, i.e. C contains each atom that is part of G and only those. formally this is:

$\forall G \exists C \forall x (x \text { is conained in } C \leftrightarrow atom(x) \wedge x \ P \ G)$

This would be an example of a wrong intuition that would lead to paradoxes.

I personally like the container\aggregate distinction for set\class dichotomy because it provides a very sharp demarcating envisionment. Here a set won't be confused for a class except for singleton classes [classes having only one container as a part of them] and even then they'd have a separate membership relations unless we have a singleton set in itself, i.e. a container that contains itself [to allow for this we'll need to re-define set membership as being an atom in a container in a container, instead of just simply being an atom in a container], which is not raised with well founded sets.

Anyhow I like the container theory because I also think it is more trivial than sets, it doesn't beg Extensionality or well-foundedness or even Choice, and so it fulfills the fragment of ZFC that is axiomatized by Union, Power, Separation, Collection and Infinity, and so it can serve as a foundation for almost the whole of mathematics, and of set theory itself.

The errors spoken about above are the same intuitive errors behind Mirimanoff [re-spelled as: there exists a 'set' of all well-founded sets] and Burali-Forti.

$\endgroup$
  • 2
    $\begingroup$ Honestly, the issue with this answer is the same I have with the question, and it is that they are not about ZFC platonists (naive or otherwise), but rather about a (very?) naive approach to the "intuitive" notion of set theory that precedes any modern formal thinking about it. In this regard, any mention of ZFC seems misleading. A ZFC platonist does not think that "a set is the totality (or whole) of all of its elements", or that a standard model entails the existence of a "set of all sets". $\endgroup$ – Andrés E. Caicedo Jun 16 '18 at 21:04
  • 2
    $\begingroup$ I answered to this (very?) naive approach that I saw in the head post. As far as its relation to modern Platonist conception of ZFC, then this something else, the OP was asking about intuitive errors that can haunt beginner students of set theory, and these might not be related to the common modern Platonist thinking of it at all, but they do cross the mind of beginners and they've been addressed by specialist logicians as I referenced in the answer. $\endgroup$ – Zuhair Jun 16 '18 at 21:19
  • 1
    $\begingroup$ It's a long answer, I know. But is it possible not to have 12 revisions? Perhaps don't edit every correction in, but rather keep a copy on your computer that you edit, and once you think it's actually done post an edit? Or even better, once you think it's done post it for the first time. No one is chasing you to post faster. $\endgroup$ – Asaf Karagila Jun 17 '18 at 8:19
  • $\begingroup$ Hi @Andrés, I think you may have misunderstood my question. Notice that I never mentioned any entity called "naive ZFC platonist" that you mentioned. I don't quite understand what this could possibly mean, since by the time a fellow has absorbed the ZFC axioms at any level of detail, such a fellow is not "naive" anymore. What I referring to is a rather common phenomenon of a naive platonist who is aware of the existence of the traditional set theoretic foundations commonly labeled ZFC. $\endgroup$ – Mikhail Katz Jun 17 '18 at 16:47
  • $\begingroup$ As you know, these were developed in the first quarter of the 20th century. The naive platonist assumes that they formalize his views concerning sets. I never spoke of "naive ZFC platonists" but only of "naive platonists". Once such a naive platonist starts examining the limits of his views he is likely to run into paradoxes or inconsistencies. @Andrés $\endgroup$ – Mikhail Katz Jun 17 '18 at 16:47

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.