A boundedness theorem in ID1(W)

1986 ◽  
Vol 51 (4) ◽  
pp. 942-947
Author(s):  
Gerhard Jäger
Keyword(s):  
Fixed Point ◽  
Peano Arithmetic ◽  
Boundedness Theorem ◽  
Recursive Tree ◽  
Free Variables ◽  
Positive Formula ◽  
Image Position ◽  
Recursive Trees

In this paper we prove a boundedness theorem in the theory ID1(W). This answers a question asked by Feferman, for example in [3]. The background is the following.Let A[X, x] be an X-positive formula arithmetic in X. The theory ID1(PA) is an extension of Peano arithmetic PA by the following axioms:for arbitrary formulas F; PA is a constant for the least fixed point of A[X, x]. Set-theoretically, PA can be defined by recursion on the ordinals as follows:where is the first nonrecursive ordinal.Now let a ≺ b be the arithmetic relation which expresses that the recursive tree coded by a is a proper subtree of the tree coded by b, and defineThe least fixed point of Tree[X, x] is the set PTree of all well-founded recursive trees. We write W or Wα for PTree or , respectively. Since W is complete we have for all α < . If we define for each element a ∈ W its inductive norm ∣a∣ by ∣a∣≔ min{ξ: a ∈ Wξ}, then we have = {∣a∣: a ∈ W} and the elements of W can be used as codes for the ordinals less than .Assume that B[X, x] is an X-positive formula arithmetic in X with the only free variables X and x, and assume that QB is a relation that satisfiesIf we definethen we obviously have PB = IB.

MARGINALIA ON A THEOREM OF WOODIN

2017 ◽  
Vol 82 (1) ◽  
pp. 359-374
Author(s):  
RASMUS BLANCK ◽  
ALI ENAYAT
Keyword(s):  
Fixed Point ◽  
Peano Arithmetic ◽  
Countable Model ◽  
Recursively Enumerable ◽  
Recursively Enumerable Sets ◽  
Finite Set ◽  
Image Position

AbstractLet $\left\langle {{W_n}:n \in \omega } \right\rangle$ be a canonical enumeration of recursively enumerable sets, and suppose T is a recursively enumerable extension of PA (Peano Arithmetic) in the same language. Woodin (2011) showed that there exists an index $e \in \omega$ (that depends on T) with the property that if${\cal M}$ is a countable model of T and for some${\cal M}$-finite set s, ${\cal M}$ satisfies ${W_e} \subseteq s$, then${\cal M}$ has an end extension${\cal N}$ that satisfies T + We = s.Here we generalize Woodin’s theorem to all recursively enumerable extensions T of the fragment ${{\rm{I}\rm{\Sigma }}_1}$ of PA, and remove the countability restriction on ${\cal M}$ when T extends PA. We also derive model-theoretic consequences of a classic fixed-point construction of Kripke (1962) and compare them with Woodin’s theorem.

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

An extension of Ackermann's set theory

1972 ◽  
Vol 37 (4) ◽  
pp. 703-704
Author(s):  
Donald Perlis
Keyword(s):  
Set Theory ◽  
Free Variables ◽  
Image Position

Ackermann's set theory [1], called here A, involves a schemawhere φ is an ∈-formula with free variables among y1, …, yn and w does not appear in φ. Variables are thought of as ranging over classes and V is intended as the class of all sets.S is a kind of comprehension principle, perhaps most simply motivated by the following idea: The familiar paradoxes seem to arise when the class CP of all P-sets is claimed to be a set, while there exists some P-object x not in CP such that x would have to be a set if CP were. Clearly this cannot happen if all P-objects are sets.Now, Levy [2] and Reinhardt [3] showed that A* (A with regularity) is in some sense equivalent to ZF. But the strong replacement axiom of Gödel-Bernays set theory intuitively ought to be a theorem of A* although in fact it is not (Levy's work shows this). Strong replacement can be formulated asThis lack of A* can be remedied by replacing S above bywhere ψ and φ are ∈-formulas and x is not in ψ and w is not in φ. ψv is ψ with quantifiers relativized to V, and y and z stand for y1, …, yn and z1, …, zm.

A Generalization of a Fixed Point Theorem of Goebel, Kirk and Shimi

1976 ◽  
Vol 19 (1) ◽  
pp. 7-12 ◽  
Author(s):  
Joseph Bogin
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Compact Convex ◽  
Continuous Mapping ◽  
Normal Structure ◽  
Convex Banach Space ◽  
Uniformly Convex ◽  
Weakly Compact ◽  
Image Position

In [7], Goebel, Kirk and Shimi proved the following:Theorem. Let X be a uniformly convex Banach space, K a nonempty bounded closed and convex subset of X, and F:K→K a continuous mapping satisfying for each x, y∈K:(1)where ai≥0 and Then F has a fixed point in K.In this paper we shall prove that this theorem remains true in any Banach space X, provided that K is a nonempty, weakly compact convex subset of X and has normal structure (see Definition 1 below).

scholarly journals On the Zagreb Index of Random Recursive Trees

2011 ◽  
Vol 48 (04) ◽  
pp. 1189-1196 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Asymptotic Normality ◽  
Random Process ◽  
Central Limit ◽  
Topological Indices ◽  
Zagreb Index ◽  
Martingale Central Limit Theorem ◽  
Recursive Tree ◽  
Recursive Trees

We investigate the Zagreb index, one of the topological indices, of random recursive trees in this paper. Through a recurrence equation, the first two moments ofZn, the Zagreb index of a random recursive tree of sizen, are obtained. We also show that the random process {Zn− E[Zn],n≥ 1} is a martingale. Then the asymptotic normality of the Zagreb index of a random recursive tree is given by an application of the martingale central limit theorem. Finally, two other topological indices are also discussed in passing.

PROVABILITY LOGICS RELATIVE TO A FIXED EXTENSION OF PEANO ARITHMETIC

2018 ◽  
Vol 83 (3) ◽  
pp. 1229-1246
Author(s):  
TAISHI KURAHASHI
Keyword(s):  
Peano Arithmetic ◽  
Provability Logic ◽  
Consistent Extension ◽  
Image Position ◽  
Computably Enumerable

AbstractLet T and U be any consistent theories of arithmetic. If T is computably enumerable, then the provability predicate $P{r_\tau }\left( x \right)$ of T is naturally obtained from each ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of T. The provability logic $P{L_\tau }\left( U \right)$ of τ relative to U is the set of all modal formulas which are provable in U under all arithmetical interpretations where □ is interpreted by $P{r_\tau }\left( x \right)$. It was proved by Beklemishev based on the previous studies by Artemov, Visser, and Japaridze that every $P{L_\tau }\left( U \right)$ coincides with one of the logics $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α and β are subsets of ω and β is cofinite.We prove that if U is a computably enumerable consistent extension of Peano Arithmetic and L is one of $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α is computably enumerable and β is cofinite, then there exists a ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of some extension of $I{{\rm{\Sigma }}_1}$ such that $P{L_\tau }\left( U \right)$ is exactly L.

Positive Definite Sequence of Operators and a Fixed Point Theorem

1972 ◽  
Vol 15 (2) ◽  
pp. 295-295
Author(s):  
A. T. Dash
Keyword(s):  
Hilbert Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Positive Definite ◽  
Definite Sequence ◽  
Positive Definite Sequence ◽  
Image Position

The purpose of this note is to prove the following:Theorem. Let {An} be a positive definite sequence of operators on a Hilbert space H with A0=1. If A1f=f for some f in H, then Anf=f for all n.Note that a bilateral sequence of operators {An:n = 0, ±1, ±2,…} on H is positive definite iffor every finitely nonzero sequence {fn} of vectors in H [1].

Parabolic-like mappings

2014 ◽  
Vol 35 (7) ◽  
pp. 2171-2197 ◽  
Author(s):  
LUNA LOMONACO
Keyword(s):  
Fixed Point ◽  
Julia Set ◽  
The Family ◽  
Dynamical Properties ◽  
Image Position ◽  
Parabolic Fixed Point

In this paper we introduce the notion of parabolic-like mapping. Such an object is similar to a polynomial-like mapping, but it has a parabolic external class, i.e. an external map with a parabolic fixed point. We define the notion of parabolic-like mapping and we study the dynamical properties of parabolic-like mappings. We prove a straightening theorem for parabolic-like mappings which states that any parabolic-like mapping of degree two is hybrid conjugate to a member of the family $$\begin{eqnarray}\mathit{Per}_{1}(1)=\left\{[P_{A}]\,\bigg|\,P_{A}(z)=z+\frac{1}{z}+A,~A\in \mathbb{C}\right\}\!,\end{eqnarray}$$ a unique such member if the filled Julia set is connected.

scholarly journals Some fixed-point theorems on locally convex linear topological spaces

1975 ◽  
Vol 13 (2) ◽  
pp. 241-254 ◽  
Author(s):  
E. Tarafdar
Keyword(s):  
Fixed Point ◽  
Topological Space ◽  
Convex Subset ◽  
Fixed Point Theorems ◽  
Nonexpansive Mappings ◽  
Topological Spaces ◽  
The Family ◽  
Image Position ◽  
Commutative Family ◽  
Locally Convex

Let (E, τ) be a locally convex linear Hausdorff topological space. We have proved mainly the following results.(i) Let f be nonexpansive on a nonempty τ-sequentially complete, τ-bounded, and starshaped subset M of E and let (I-f) map τ-bounded and τ-sequentially closed subsets of M into τ-sequentially closed subsets of M. Then f has a fixed-point in M.(ii) Let f be nonexpansive on a nonempty, τ-sequentially compact, and starshaped subset M of E. Then f has a fixed-point in M.(iii) Let (E, τ) be τ-quasi-complete. Let X be a nonempty, τ-bounded, τ-closed, and convex subset of E and M be a τ-compact subset of X. Let F be a commutative family of nonexpansive mappings on X having the property that for some f1 ∈ F and for each x ∈ X, τ-closure of the setcontains a point of M. Then the family F has a common fixed-point in M.

Phase Changes in the Topological Indices of Scale-Free Trees

2013 ◽  
Vol 50 (02) ◽  
pp. 516-532 ◽  
Author(s):  
Qunqiang Feng ◽  
Zhishui Hu
Keyword(s):  
Nonnegative Integer ◽  
Topological Indices ◽  
Critical Values ◽  
Phase Changes ◽  
Zagreb Index ◽  
Asymptotic Behaviors ◽  
Scale Free ◽  
Recursive Tree ◽  
Free Tree ◽  
Recursive Trees

A scale-free tree with the parameter β is very close to a star if β is just a bit larger than −1, whereas it is close to a random recursive tree if β is very large. Through the Zagreb index, we consider the whole scene of the evolution of the scale-free trees model as β goes from −1 to + ∞. The critical values of β are shown to be the first several nonnegative integer numbers. We get the first two moments and the asymptotic behaviors of this index of a scale-free tree for all β. The generalized plane-oriented recursive trees model is also mentioned in passing, as well as the Gordon-Scantlebury and the Platt indices, which are closely related to the Zagreb index.

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