A Non-Representation Theorem for Gödel-Bernays Set Theory

AbstractHere we investigate the classes of representable directed cylindric algebras of dimension α introduced by Németi [12]. can be seen in two different ways: first, as an algebraic counterpart of higher order logics and second, as a cylindric algebraic analogue of Quasi-Projective Relation Algebras. We will give a new, “purely cylindric algebraic” proof for the following theorems of Németi: (i) is a finitely axiomatizable variety whenever α ≥ 3 is finite and (ii) one can obtain a strong representation theorem for if one chooses an appropriate (non-well-founded) set theory as foundation of mathematics. These results provide a purely cylindric algebraic solution for the Finitization Problem (in the sense of [11]) in some non-well-founded set theories.

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The Hahn representation theorem for ℓ-groups in ZFA

Journal of Symbolic Logic ◽

10.2307/2586553 ◽

2000 ◽

Vol 65 (2) ◽

pp. 519-524

Author(s):

D. Gluschankof

Keyword(s):

Set Theory ◽

Representation Theorem ◽

Group Structure ◽

Ordered Group ◽

Representation Theorems ◽

Ordered Groups ◽

Lattice Ordered Groups ◽

Totally Ordered Group ◽

Basic Concepts ◽

The Axiom Of Choice

In [7] the author discussed the relative force —in the set theory ZF— of some representation theorems for ℓ-groups (lattice-ordered groups). One of the theorems not discussed in that paper is the Hahn representation theorem for abelian ℓ-groups. This result, originally proved by Hahn (see [8]) for totally ordered groups and half a century later by Conrad, Harvey and Holland for the general case (see [4]), states that any abelian ℓ-group can be embedded in a Hahn product of copies of R (the real line with its natural totally-ordered group structure). Both proofs rely heavily on Zorn's Lemma which is equivalent to AC (the axiom of choice).The aim of this work is to point out the use of non-constructible axioms (i.e., AC and weaker forms of it) in the proofs. Working in the frame of ZFA, that is, the Zermelo-Fraenkel set theory where a non-empty set of atoms is allowed, we present alternative proofs which, in the totally ordered case, do not require the use of AC. For basic concepts and notation on ℓ-groups the reader can refer to [1] and [2]. For set theory, to [11].

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Graphic Deduction Based on Set

JUCS - Journal of Universal Computer Science ◽

10.3897/jucs.2020.069 ◽

2020 ◽

Vol 26 (10) ◽

pp. 1331-1342

Author(s):

Xia He ◽

Guoping Du ◽

Long Hong

Keyword(s):

Artificial Intelligence ◽

Set Theory ◽

Representation Theorem ◽

Basic Concept ◽

Formal Language ◽

Deductive Reasoning ◽

Graphical Representation ◽

Formal Logic ◽

New Method ◽

Symbolic Logic

Based on basic concept of symbolic logic and set theory, this paper focuses on judgments and attempts to provide a new method for the study of logic. It establishes the formal language of the extension of judgment J*, and formally describes a, e, i, o judgment, and thus gives set theory representation and graphical representation that can distinguish between universal judgments and particular judgments. According to the content of non-modal deductive reasoning in formal logic, it gives weakening theorem, strengthening theorem and a number of typical graphical representation theorem (graphic theorem), where graphic deduction is carried out. Graphic deduction will be beneficial to the research of artificial intelligence, which is closely related to judgment and deduction in logic.

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A measure representation theorem for strong partition cardinals

Journal of Symbolic Logic ◽

10.2307/2273389 ◽

1982 ◽

Vol 47 (1) ◽

pp. 161-168 ◽

Cited By ~ 1

Author(s):

E. M. Kleinberg

Keyword(s):

Set Theory ◽

Representation Theorem ◽

Axiom Of Choice ◽

Partition Relation ◽

Subsequent Work ◽

Partition Relations ◽

Measure Representation ◽

The Axiom Of Choice ◽

Measurable Cardinals ◽

Strong Partition Cardinals

There are two main axiomatic extensions of Zermelo-Fraenkel set theory without the axiom of choice, that associated with the axiom of determinateness, and that associated with infinite exponent partition relations. Initially, the axiom of determinateness, henceforth AD, was the sole tool available. Using it, set theorists in the late 1960s produced many remarkable results in pure set theory (e.g. the measurability of ℵ1) as well as in projective set theory (e.g. reduction principles for ). Infinite exponent partition relations were first studied successfully soon after these early consequences of AD. They too produced measurable cardinals and not only were the constructions here easier than those from AD—the results gave a far clearer picture of the measures involved than had been offered by AD. In general, the techniques offered by infinite exponent partition relations became so attractive that a great deal of the subsequent work from AD involved an initial derivation from AD of the appropriate infinite exponent partition relation and then the derivation from the partition relation of the desired result.Since the early 1970s work on choiceless extensions of ZF + DC has split mainly between AD and its applications to projective set theory, and infinite exponent partition relations and their applications to pure set theory. There has certainly been a fair amount of interplay between the two, but for the most part the theories have been pursued independently.Unlike AD, infinite exponent partition relations have shown themselves amenable to nontrivial forcing arguments. For example, Spector has constructed models for interesting partition relations, consequences of AD, in which AD is false. Thus AD is a strictly stronger assumption than are various infinite exponent partition relations. Furthermore, Woodin has recently proved the consistency of infinite exponent partition relations relative to assumptions consistent with the axiom of choice, in particular, relative to the existence of a supercompact cardinal. The notion of doing this for AD is not even considered.

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