Higher order reflection principles

1989 ◽  
Vol 54 (2) ◽  
pp. 474-489 ◽  
Author(s):  
M. Victoria Marshall R.
Keyword(s):  
Measurable Cardinal ◽  
Reflection Principle ◽  
Higher Order ◽  
Weakly Compact ◽  
Class Theory ◽  
First Order ◽  
Transitive Set ◽  
Reflection Principles ◽  
Image Position ◽  
Order Reflection

In [1] and [2] there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See [3]. These axioms are formulated in first order logic with ∈:(A1)Extensionality.(A2)Class specification. Ifϕis a formula andAis not free inϕ, thenNote that “xis a set“ can be written as “∃u(x∈u)”.(A3)Subsets.Note also that “B⊆A” can be written as “∀x(x∈B→x∈A)”.(A4)Reflection principle. Ifϕ(x)is a formula, thenwhere “uis a transitive set” is the formula “∃v(u∈v) ∧ ∀x∀y(x∈y∧y∈u→x∈u)” andϕPuis the formulaϕrelativized to subsets ofu.(A5)Foundation.(A6)Choice for sets.We denote byB1the theory with axioms (A1) to (A6).The existence of weakly compact and-indescribable cardinals for everynis established inB1by the method of defining all metamathematical concepts forB1in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see [1]. There is a proof of the consistency ofB1assuming the existence of a measurable cardinal; see [4] and [5]. In [6] several set and class theories with reflection principles are developed. In them, the existence of inaccessible cardinals and some kinds of indescribable cardinals can be proved; and also there is a generalization of indescribability for higher-order languages using only class parameters.The purpose of this work is to develop higher order reflection principles, including higher order parameters, in order to obtain other large cardinals.

The completeness of the first-order functional calculus

1949 ◽  
Vol 14 (3) ◽  
pp. 159-166 ◽  
Author(s):  
Leon Henkin
Keyword(s):  
Functional Calculus ◽  
Propositional Calculus ◽  
Higher Order ◽  
New Method ◽  
Completeness Proof ◽  
New Approach ◽  
First Order ◽  
Formal Systems ◽  
The Subject ◽  
Image Position

Although several proofs have been published showing the completeness of the propositional calculus (cf. Quine (1)), for the first-order functional calculus only the original completeness proof of Gödel (2) and a variant due to Hilbert and Bernays have appeared. Aside from novelty and the fact that it requires less formal development of the system from the axioms, the new method of proof which is the subject of this paper possesses two advantages. In the first place an important property of formal systems which is associated with completeness can now be generalized to systems containing a non-denumerable infinity of primitive symbols. While this is not of especial interest when formal systems are considered as logics—i.e., as means for analyzing the structure of languages—it leads to interesting applications in the field of abstract algebra. In the second place the proof suggests a new approach to the problem of completeness for functional calculi of higher order. Both of these matters will be taken up in future papers.The system with which we shall deal here will contain as primitive symbolsand certain sets of symbols as follows:(i) propositional symbols (some of which may be classed as variables, others as constants), and among which the symbol “f” above is to be included as a constant;(ii) for each number n = 1, 2, … a set of functional symbols of degree n (which again may be separated into variables and constants); and(iii) individual symbols among which variables must be distinguished from constants. The set of variables must be infinite.

Iterated reflection principles and the ω-rule

1982 ◽  
Vol 47 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Ulf R. Schmerl
Keyword(s):  
Reflection Principle ◽  
Peano Arithmetic ◽  
Equivalent System ◽  
Formal Similarity ◽  
Sound System ◽  
Formal Systems ◽  
Ordinal Notations ◽  
Axiomatizable Theory ◽  
Reflection Principles ◽  
Image Position

The ω-rule,with the meaning “if the formula A(n) is provable for all n, then the formula ∀xA(x) is provable”, has a certain formal similarity with a uniform reflection principle saying “if A(n) is provable for all n, then ∀xA(x) is true”. There are indeed some hints in the literature that uniform reflection has sometimes been understood as a “formalized ω-rule” (cf. for example S. Feferman [1], G. Kreisel [3], G. H. Müller [7]). This similarity has even another aspect: replacing the induction rule or scheme in Peano arithmetic PA by the ω-rule leads to a complete and sound system PA∞, where each true arithmetical statement is provable. In [2] Feferman showed that an equivalent system can be obtained by erecting on PA a transfinite progression of formal systems PAα based on iterations of the uniform reflection principle according to the following scheme:Then T = (∪dЄ, PAd, being Kleene's system of ordinal notations, is equivalent to PA∞. Of course, T cannot be an axiomatizable theory.

On an Ackermann-type set theory

1973 ◽  
Vol 38 (3) ◽  
pp. 410-412
Author(s):  
John Lake
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Large Cardinal ◽  
Predicate Calculus ◽  
First Order ◽  
Free Variables ◽  
Individual Constant ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Universal Closure

Ackermann's set theory A* is usually formulated in the first order predicate calculus with identity, ∈ for membership and V, an individual constant, for the class of all sets. We use small Greek letters to represent formulae which do not contain V and large Greek letters to represent any formulae. The axioms of A* are the universal closures ofwhere all free variables are shown in A4 and z does not occur in the Θ of A2.A+ is a generalisation of A* which Reinhardt introduced in [3] as an attempt to provide an elaboration of Ackermann's idea of “sharply delimited” collections. The language of A+ is that of A*'s augmented by a new constant V′, and its axioms are A1–A3, A5, V ⊆ V′ and the universal closure ofwhere all free variables are shown.Using a schema of indescribability, Reinhardt states in [3] that if ZF + ‘there exists a measurable cardinal’ is consistent then so is A+, and using [4] this result can be improved to a weaker large cardinal axiom. It seemed plausible that A+ was stronger than ZF, but our main result, which is contained in Theorem 5, shows that if ZF is consistent then so is A+, giving an improvement on the above results.

scholarly journals 02. Social Pathologies, Reflexive Pathologies, and the Idea of Higher-Order Disorders

2015 ◽  
Author(s):  
Arto Laitinen
Keyword(s):  
Social Reality ◽  
Social Criticism ◽  
Higher Order ◽  
Second Order ◽  
Third Order ◽  
First Order ◽  
The Third ◽  
The Social ◽  
Order Reflection

This paper critically examines Christopher Zurn’s suggestion mentioned above that various social pathologies (pathologies of ideological recognition, maldistribution, invisibilization, rationality distortions, reification and institutionally forced self-realization) share the structure of being ‘second-order disorders’: that is, that they each entail ‘constitutive disconnects between first-order contents and secondorder reflexive comprehension of those contents, where those disconnects are pervasive and socially caused’ (Zurn, 2011, 345-346). The paper argues that the cases even as discussed by Zurn do not actually match that characterization, but that it would be premature to conclude that they are not thereby social pathologies, or that they do not have a structure in common. It is just that the structure is more complex than originally described, covering pervasive socially caused evils (i) in the social reality, (ii) in the first order experiences and understandings, (iii) in the second order reflection as discussed by Zurn, and also (iv) in the ‘third order’ phenomenon concerning the pre-emptive silencing or nullification of social criticism even before it takes place 

scholarly journals MODEL-THEORETIC PROPERTIES OF ULTRAFILTERS BUILT BY INDEPENDENT FAMILIES OF FUNCTIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 103-134 ◽  
Author(s):  
M. MALLIARIS ◽  
S. SHELAH
Keyword(s):  
Random Graph ◽  
Measurable Cardinal ◽  
Degree Of Saturation ◽  
Stable Theory ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Graph Type ◽  
Inductive Construction ◽  
Image Position ◽  
Precise Degree

Abstract Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on λ (i.e., a point after which all antichains of ${\cal P}\left( \lambda \right)/{\cal D}$ have cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter ${\cal D}$ on λ so that any ultrafilter extending ${\cal D}$ fails to ${\lambda ^ + }$ -saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable cardinal κ, we construct a regular ultrafilter on $\lambda > \kappa$ which is λ-flexible but not ${\kappa ^{ + + }}$ -good, improving our previous answer to a question raised in Dow (1985). Third, assuming a weakly compact cardinal κ, we construct an ultrafilter to show that ${\rm{lcf}}\left( {{\aleph _0}} \right)$ may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.

scholarly journals A STRONG REFLECTION PRINCIPLE

2017 ◽  
Vol 10 (4) ◽  
pp. 651-662 ◽  
Author(s):  
SAM ROBERTS
Keyword(s):  
Set Theory ◽  
Reflection Principle ◽  
Higher Order ◽  
Second Order ◽  
Proper Class ◽  
Strong Reflection ◽  
Reflection Principles ◽  
Extendible Cardinals ◽  
Order Set

AbstractThis article introduces a new reflection principle. It is based on the idea that whatever is true in all entities of some kind is also true in a set-sized collection of them. Unlike standard reflection principles, it does not re-interpret parameters or predicates. This allows it to be both consistent in all higher-order languages and remarkably strong. For example, I show that in the language of second-order set theory with predicates for a satisfaction relation, it is consistent relative to the existence of a 2-extendible cardinal (Theorem 7.12) and implies the existence of a proper class of 1-extendible cardinals (Theorem 7.9).

Reflection principles and iterated consistency assertions

1979 ◽  
Vol 44 (1) ◽  
pp. 33-35 ◽  
Author(s):  
George Boolos
Keyword(s):  
Standard Model ◽  
Reflection Principle ◽  
Incompleteness Theorem ◽  
First Order ◽  
Starting Point ◽  
The Standard Model ◽  
Reflection Principles ◽  
Order Arithmetic

This paper compares the strength of two sorts of sentences of PA (classical first-order arithmetic with induction): reflection principles and sentences that may be called iterated consistency assertions.Let Bew(x) be the standard provability predicate for PA, and for any sentence S of PA, let ⌈S⌉ be the numeral for the Gödel number of S. The reflection principle for S is the sentence Bew(⌈S⌉) → S, and a reflection principle is simply the reflection principle for some sentence. Nothing false (in the standard model for PA) is provable in PA, and therefore every reflection principle is true. Löb's theorem asserts that S is provable (in PA) if the reflection principle for S is provable.We shall suppose that the 0-ary propositional connectives ⊤ and ⊥ are taken as primitives in the formulation of PA. We define the iterated consistency assertions Conm by: Con0 = ⊤; Conm−1 = − Bew(⌈ − Conm⌉). Con1 may be taken to be the sentence of PA that expresses the consistency of PA; Conn−1, the sentence that expresses the consistency of PA ⋃ {Conn}.Our starting point is the observation that Con1 is equivalent (in PA) to the reflection principle for ⊥. (The second incompleteness theorem thus follows in a well-known way from Löb's theorem: if PA is consistent, then ⊥ is not provable, the reflection principle for ⊥ is not provable, and the consistency of PA is not provable either.)

Leibniz on Phenomenal Consciousness

Vivarium ◽  
2014 ◽  
Vol 52 (3-4) ◽  
pp. 333-357 ◽  
Author(s):  
Christian Barth
Keyword(s):  
Phenomenal Character ◽  
Representational Content ◽  
Phenomenal Consciousness ◽  
Higher Order ◽  
Middle Ground ◽  
First Order ◽  
Cognitive Operations ◽  
Contemporary Standard ◽  
Order Reflection

The main aim of this paper is to show that we can extract an elaborate account of phenomenal consciousness from Leibniz’s (1646-1716) writings. Against a prevalent view, which attributes a higher-order reflection account of phenomenal consciousness to Leibniz, it is argued that we should understand Leibniz as holding a first-order conception of it. In this conception, the consciousness aspect of phenomenal consciousness is explained in terms of a specific type of attention. This type of attention, in turn, is accounted for in terms of cognitive appetites aiming at knowledge about a represented object by means of initiating cognitive operations on representational content. Furthermore, against the view that Leibniz holds a reifying account, it is argued that Leibniz accepts an epistemic account of phenomenal character. According to this view, the phenomenal character of phenomenally conscious states rests on the confusing effect of imperfect acts of attention directed towards representational contents. Holding this view, Leibniz finds fruitful middle ground between contemporary standard positions like higher-order theories, representationalist conceptions, and qualia accounts of phenomenal consciousness. His position possesses resources to meet several objections these standard accounts are confronted with.

Bernays and Set Theory

2009 ◽  
Vol 15 (1) ◽  
pp. 43-69 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Higher Order ◽  
Reflection Principles ◽  
Order Reflection

AbstractWe discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles.

ON ${\omega _1}$-STRONGLY COMPACT CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 266-278 ◽  
Author(s):  
JOAN BAGARIA ◽  
MENACHEM MAGIDOR
Keyword(s):  
Measurable Cardinal ◽  
Second Order ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Relational Structures ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Strongly Compact Cardinals ◽  
Order Reflection

Abstract An uncountable cardinal κ is called ${\omega _1}$ -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$ -complete ultrafilter on I. We show that the first ${\omega _1}$ -strongly compact cardinal, ${\kappa _0}$ , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$ . We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$ . Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$ .

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