The membership relation

The MEMBERSHIP RELATION and its underlying definitions are presented in this white paper (knowledge base).

Undirecting membership in models of Anti-Foundation

It is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\in y$$ x ∈ y or $$y\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if $$x\in x$$ x ∈ x for some x) or multiple edges (if $$x\in y$$ x ∈ y and $$y\in x$$ y ∈ x for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.

AN EXTENSION OF A THEOREM OF ZERMELO

We show that if $(M,{ \in _1},{ \in _2})$ satisfies the first-order Zermelo–Fraenkel axioms of set theory when the membership relation is ${ \in _1}$ and also when the membership relation is ${ \in _2}$, and in both cases the formulas are allowed to contain both ${ \in _1}$ and ${ \in _2}$, then $\left( {M, \in _1 } \right) \cong \left( {M, \in _2 } \right)$, and the isomorphism is definable in $(M,{ \in _1},{ \in _2})$. This extends Zermelo’s 1930 theorem in [6].

Mereology

Mereology is the theory of the part–whole relation and of derived operations such as the mereological sum. (The sum of several things is the smallest thing of which they are all parts.) It was introduced by Leśniewski to avoid Russell’s paradox. Unlike the set-membership relation, the part–whole relation is transitive. This makes mereology much weaker than set theory, but gives the advantage of ontological parsimony. For example, mereology does not posit the proliferation of entities found in set theory, such as ∅⁣,{∅⁣},{{∅⁣}},…. Mereology has occasioned controversy: over whether many things really have a mereological sum if they are either scattered or, even worse, of different categories; over the uniqueness of sums; and over Lewis’ claim that the non-empty subsets of a set are literally parts of it.

The Apparent Autonomy of Singular Group Agents

Chapter 5 shows how to extend the multiple agents account of plural agency to the case of grammatically singular group action sentences in a way that explains some of the features of singular group action sentences that were identified in Chapter 3 as suggesting that a reductive account was implausible. First, it shows how to integrate the time indexed membership relation into the account. Second, it explains how this enables us to understand singular group action sentences in which it appears that such groups do things through changes in their membership in a way that only appeals to the agents who are members of it at any given time. Third, it shows that the fact that it appears that singular group agents could have had different members than they do is just a matter of their being picked out via descriptions which could have had different denotations.

Group Membership

Chapter 11 is concerned with the nature of the membership relation relevant to sorting agents under the concept of a mob or organization and the nature of groups organized under such relations. First, it develops a taxonomy of different sorts of groups, distinguishing natural groups, institutions or organizations, and mobs and crowds. Second, it provides an account of membership in institutions, the key idea of which is that membership is a matter of having a determinable status role characterized by the nature of the relevant institution. Third, it takes up a counting puzzle connected with the reductive view of organizations. If an organization at a time is nothing but its members, then two clubs with the same members are the same, but this seems counterintuitive. Finally, it gives an analysis of membership specifically in mobs and crowds.

Chaitin's halting probability and the compression of strings using oracles

If a computer is given access to an oracle—the characteristic function of a set whose membership relation may or may not be algorithmically calculable—this may dramatically affect its ability to compress information and to determine structure in strings, which might otherwise appear random. This leads to the basic question, ‘given an oracle A , how many oracles can compress information at most as well as A ?’ This question can be formalized using Kolmogorov complexity. We say that B ≤ LK A if there exists a constant c such that K A ( σ )< K B ( σ )+ c for all strings σ , where K X denotes the prefix-free Kolmogorov complexity relative to oracle X . The formal counterpart to the previous question now is, ‘what is the cardinality of the set of ≤ LK -predecessors of A ?’ We completely determine the number of oracles that compress at most as well as any given oracle A , by answering a question of Miller ( Notre Dame J. Formal Logic , 2010, 50 , 381–391), which also appears in Nies ( Computability and randomness . Oxford, UK: Oxford University Press, 2009. Problem 8.1.13); the class of ≤ LK -predecessors of a set A is countable if and only if Chaitin's halting probability Ω is Martin-Löf random relative to A .

Extensional quotients for type theory and the consistency problem for NF

Quine's “New Foundations” (NF) was first presented in Quine [10] and later on in Quine [11]. Ernst Specker [15, 13], building upon a previous result of Ehrenfeucht and Mostowski [5], showed that NF is consistent if and only if there is a model of the Theory of Negative (and positive) Types (TNT) with full extensionality that admits of a “shifting automorphism,” but the existence of such a model remains an open problem.In his [8], Ronald Jensen gave a partial solution to the problem of the consistency of Quine's NF. Jensen considered a version of NF—referred to as NFU—in which the axiom of extensionality is weakened to allow for Urelemente or “atoms.” He showed, modifying Specker's theorem, that the existence of a model of TNT with atoms admitting of a “shifting automorphism” implies the consistency of NFU, proceeding then to exhibit such a model.This paper presents a reduction of the consistency problem for NF to the existence of a model of TNT with atoms containing certain “large” (unstratified) sets and admitting a shifting automorphism. In particular we show that such a model can be “collapsed” to a model of pure TNT in such a way as to preserve the shifting automorphism. By the above-mentioned result of Specker's, this implies the consistency of NF.Let us take the time to explain the main ideas behind the construction. Suppose we have a certain universe U of sets, built up from certain individuals or “atoms.” In such a universe we have only a weak version of the axiom of extensionality: two objects are the same if and only if they are both sets having the same members. We would like to obtain a universe U′ that is as close to U as possible, but in which there are no atoms (i.e., the only memberless object is the empty set). One way of doing this is to assign to each atom ξ, a set a (perhaps the empty set), inductively identifying sets that have members that we are already committed to considering “the same.” In doing this we obtain an equivalence relation ≃ over U that interacts nicely with the membership relation (provided we have accounted for multiplicity of members, i.e., we have allowed sets to contain “multiple copies” of the same object). Then we can take U′ = U/≃, the quotient of U with respect to ≃. It is then possible to define a “membership” relation over U′ in such a way as to have full extensionality. Relations such as ≃ are referred to as “contractions” by Hinnion and “bisimulations” by Aczel.

Replacement and collection in intuitionistic set theory

Intuitionistic Zermelo-Fraenkel set theory, which we call ZFI, was introduced by Friedman and Myhill in [3] in 1970. The idea was to have a theory with the same axioms as ordinary classical ZF, but with Heyting's predicate calculus HPC as the underlying logic. Since some classically equivalent statements are intuitionistically inequivalent, however, it was not always obvious which form of a classical axiom to take. For example, the usual formulation of the axiom of foundation had to be replaced with a principle of transfinite induction on the membership relation in order to avoid having excluded middle provable and thus making the logic classical. In [3] the replacement axiom is taken in its most common classical form:In the presence of the separation axiom,this is equivalent to the axiomIt is this schema Rep that we shall call the replacement axiom.Friedman and Myhill were able to show in [3] that ZFI has a number of desirable “constructive” properties, including the existence property for both numbers and sets. They were not able to determine, however, whether ZFI is proof-theoretically as strong as ZF. This is still open.Three years later, in [2], Friedman introduced a theory ZF− which is like ZFI except that the replacement axiom is changed to the classically equivalent collection axiom:He showed that ZF− is proof-theoretically of the same strength as ZF, and he asked whether ZF− is equivalent to ZFI.