Class Of Finite Sets Research Articles

Finiteness Classes and Small Violations of Choice

We study properties of certain subclasses of the Dedekind finite sets (addressed as finiteness classes) in set theory without the axiom of choice (AC) with respect to the comparability of their elements and to the boundedness of such classes, and we answer related open problems from Herrlich’s “The Finite and the Infinite.” The main results are as follows: 1. It is relatively consistent with ZF that the class of all finite sets is not the only finiteness class such that any two of its elements are comparable. 2. The principle “Small Violations of Choice” (SVC)—introduced by A. Blass—implies that the class of all Dedekind finite sets is bounded above. 3. “The class of all Dedekind finite sets is bounded above” is true in every permutation model of ZFA in which the class of atoms is a set, and in every symmetric model of ZF. 4. There exists a model of ZFA set theory in which the class of all atoms is a proper class and in which the class of all infinite Dedekind finite sets is not bounded above. 5. There exists a model of ZF in which the class of all infinite Dedekind finite sets is not bounded above.

The Finite and the Infinite

In ZF set theory finiteness classes are introduced and their stability under basic set theoretical constructions are being investigated. Typical results are: 1. The class of finite sets is the smallest finiteness class. 2. The class of Dedekind-finite sets is the largest finiteness class. 3. The class of almost finite sets is the largest summable finiteness class. 4. Equivalent are: a. There is only one finiteness class. b. The union of each family of 1-element sets, indexed by a Dedekind-finite set, is Dedekind-finite. c. The axiom of choice, for countable families of Dedekind-finite sets. d. The shrinking principle for families (Xi)i ∈ I of sets, indexed by a Dedekind-finite set I (i.e., there exists a family (Yi)i ∈ I of pairwise disjoint subsets Yi of Xi with \(\underset{i\in I}{\bigcup} Y_i = \underset{i\in I}{\bigcup} X_i\)). 5. In suitable ZF-models there exist families \(({\mathfrak{A}}_r)_{r\in{\mathbb{R}}}\) of finiteness classes such that $$r < s \Longrightarrow {\mathfrak{A}}_r \subsetneqq {\mathfrak{A}}_s.$$

Ehrenfeucht–Fraïssé games in finite set theory

In this article, with Ehrenfeucht–Fraïssé games we prove that Δ 1 ≠ Δ 0 on BFR , which implies Δ ≠ Δ 0 on BFR , and thus solve an open problem raised by Albert Atserias in his dissertation ( Δ 0 , Δ 1 , Δ are fragments of first order logic and BFR is a class of finite sets which in essence is equivalent to a class of finite pure arithmetic structures with built-in BIT predicate).

Learning commutative deterministic finite state automata in polynomial time

We consider the problem of learning the commutative subclass of regular languages in the on-line model of predicting {0,1∼-valued functions from examples and reinforcements due to Littlestone [7,4]. We show that the entire class of commutative deterministic finite state automata (CDFAs) of an arbitrary alphabet sizek is predictable inO(sk) time with the worst case number of mistakes bounded above byO(skk logs), wheres is the number of states in the target DFA. As a corollary, this result implies that the class of CDFAs is also PAC-learnable from random labeled examples in timeO(sk) with sample complexity\(O\left( {\tfrac{1}{ \in }\left( {\log \tfrac{1}{\delta } + s^k k\log s} \right)} \right)\), using a different class of representations. The mistake bound of our algorithm is within a polynomial, for a fixed alphabet size, of the lower boundO(s+k) we obtain by calculating the VC-dimension of the class. Our result also implies the predictability of the class of finite sets of commutative DFAs representing the finite unions of the languages accepted by the respective DFAs.

Polynomials and finite sets connected with an identity

It is shown that the polynomials satisfying the identityf(x) f(x + 1) = f(x 2 +x − a), wherea either belongs to a field of characteristic zero or is transcendental over a prime field of characteristic exceeding 2, are precisely those of the form(x 2 −a) n ; thus extending a result proved by Nathanson in the complex case. The result is not, in general, true in characteristic 2. Additionally, a class of finite sets, considered by Nathanson in connection with the identity, is completely determined.

Rice Theorems for ∑n-1 Sets

In [3] Hay proves generalizations of Rice's Theorem and the Rice-Shapiro Theorem for differences of recursively enumerable sets (d.r.e. sets). The original Rice Theorem [5, p. 3G4, Corollary B] says that the index set of a class C of r.e. sets is recursive if and only if C is empty or C contains all r.e. sets. The Rice-Shapiro Theorem conjectured by Rice [5] and proved independently by McNaughton, Shapiro, and Myhill [4] says that the index set of a class C of r.e. sets is r.e. if and only if C is empty or C consists of all r.e. sets which extend some element of a canonically enumerable class of finite sets. Since a d.r.e. set is a difference of r.e. sets, a d.r.e. set has an index associated with it, namely, the pair of indices of the r.e. sets of which it is the difference.

A noninitial segment of index sets

Let {Wk}k ≥ 0 be a standard enumeration of all recursively enumerable (r.e.) sets. If A is any class of r.e. sets, let θA denote the index set of A, i.e., θA = {k ∣ Wk ∈ A}. The one-one degrees of index sets form a partial order ℐ which is a proper subordering of the partial order of all one-one degrees. Denote by ⌀ the one-one degree of the empty set, and, if b is the one-one degree of θB, denote by the one-one degree of . Let . Let {Ym}m≥0 be the sequence of index sets of nonempty finite classes of finite sets (classified in [5] and independently, in [2]) and denote by am the one-one degree of Ym. As shown in [2], these degrees are complete at each level of the difference hierarchy generated by the r.e. sets. It was proved in [3] that, for each m ≥ 0,(a) am+1 and ām+1 are incomparable immediate successors of am and ām, and(b) .For m = 0, since Y0 = θ{⌀}, it follows from (a) that(c) .Hence it follows that(d) {⌀, , ao, ā0, a1, ā1 is an initial segment of ℐ.

A discrete chain of degrees of index sets

Let {Wi} be a standard enumeration of all recursively enumerable (r.e.) sets, and for any class A of r.e. sets, let θA denote the index set of A = {n ∣ Wn ∈ A}. (Clearly, .) In [1], the index sets of nonempty finite classes of finite sets were classified under one-one reducibility into an increasing sequence {Ym}, 0 ≤ m &lt; ∞. In this paper we examine further properties of this sequence within the partial ordering of one-one degrees of index sets. The main results are as follows: (1) For each m, Ym &lt; Ym + 1 and &lt; Ym + 1; (2) Ym is incomparable to ; (3) Ym + 1 and ; are immediate successors (among index sets) of Ym and m; (4) the pair (Ym + 1, ) is a “least upper bound” for the pair (Ym, ) in the sense that any successor of both Ym and is ≥ Ym + 1or; (5) the pair (Ym, ) is a “greatest lower bound” for the pair (Ym + 1, ) in the sense that any predecessor of both Ym + 1 and is ≤ Ym or . Since and all Ym are in the bounded truth-table degree of K, this yields some local information about the one-one degrees of index sets which are “at the bottom” in the one-one ordering of index sets.