The free one-generated left distributive algebra: basics and a simplified proof of the division algorithm

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Richard Laver ◽  
Sheila Miller
Keyword(s):  
Normal Form ◽  
Set Theory ◽  
Large Cardinal ◽  
Braid Groups ◽  
Division Algorithm ◽  
Free Left ◽  
Distributive Law ◽  
The Law

AbstractThe left distributive law is the law a· (b· c) = (a·b) · (a· c). Left distributive algebras have been classically used in the study of knots and braids, and more recently free left distributive algebras have been studied in connection with large cardinal axioms in set theory. We provide a survey of results on the free left distributive algebra on one generator, A, and a new, simplified proof of the existence of a normal form for terms in A. Topics included are: the confluence of A, the linearity of the iterated left division ordering <L of A, the connections of A to the braid groups, and an extension P of A obtained by freely adding a composition operation. This is followed by a simplified proof of the division algorithm for P, which produces a normal form for terms in A and is a powerful tool in the study of A.

Core Logic and the Paradoxes

2017 ◽  
Author(s):  
Neil Tennant
Keyword(s):  
Normal Form ◽  
Set Theory ◽  
Free Logic ◽  
Semantic Paradoxes ◽  
Law Of Excluded Middle ◽  
The Law ◽  
The Liar ◽  
Excluded Middle

The Law of Excluded Middle is not to be blamed for any of the logico-semantic paradoxes. We explain and defend our proof-theoretic criterion of paradoxicality, according to which the ‘proofs’ of inconsistency associated with the paradoxes are in principle distinct from those that establish genuine inconsistencies, in that they cannot be brought into normal form. Instead, the reduction sequences initiated by paradox-posing proofs ‘of ⊥’ do not terminate. This criterion is defended against some recent would-be counterexamples by stressing the need to use Core Logic’s parallelized forms of the elimination rules. We show how Russell’s famous paradox in set theory is not a genuine paradox; for it can be construed as a disproof, in the free logic of sets, of the assumption that the set of all non-self-membered sets exists. The Liar (by contrast) is still paradoxical, according to the proof-theoretic criterion of paradoxicality.

A NORMAL FORM FOR THE FREE LEFT DISTRIBUTIVE LAW

1994 ◽  
Vol 04 (04) ◽  
pp. 499-528 ◽  
Author(s):  
PATRICK DEHORNOY
Keyword(s):  
Normal Form ◽  
New Normal ◽  
Free Left ◽  
Distributive Law ◽  
Primitive Recursive ◽  
Made In

We construct a new normal form for one variable terms up to left distributivity. The proof that this normal form exists for every term is considerably simpler than the corresponding proof for the forms previously introduced by Richard Laver. In particular the determination of the present normal form can be made in a primitive recursive way.

Richard Laver. The left distributive law and the freeness of an algebra of elementary embeddings. Advances in mathematics, vol. 91 (1992), pp. 209–231. - Richard Laver. A division algorithm for the free left distributive algebra. Logic Colloquium '90, ASL summer meeting in Helsinki, edited by J. Oikkonen and J. Väänänen, Lecture notes in logic, no. 2, Springer-Verlag, Berlin, Heidelberg, New York, etc., 1993, pp. 155–162. - Richard Laver. On the algebra of elementary embeddings of a rank into itself. Advances in mathematics, vol. 110 (1995), pp. 334–346. - Richard Laver. Braid group actions on left distributive structures, and well orderings in the braid groups. Journal of pure and applied algebra, vol. 108 (1996), pp. 81–98. - Patrick Dehornoy. An alternative proof of Laver's results on the algebra generated by an elementary embedding. Set theory of the continuum, edited by H. Judah, W. Just, and H. Woodin, Mathematics Sciences Research Institute publications, vol. 26, Springer-Verlag, New York, Berlin, Heidelberg, etc., 1992, pp. 27–33. - Patrick Dehornoy. Braid groups and left distributive operations. Transactions of the American Mathematical Society, vol. 345 (1994), pp. 115–150. - Patrick Dehornoy. A normal form for the free left distributive law. International journal of algebra and computation, vol. 4 (1994), pp. 499–528. - Patrick Dehornoy. From large cardinals to braids via distributive algebra. Journal of knot theory and its ramifications, vol. 4 (1995), pp. 33–79. - J. R. Steel. The well-foundedness of the Mitchell order. The journal of symbolic logic, vol. 58 (1993), pp. 931–940. - Randall Dougherty. Critical points in an algebra of elementary embeddings. Annals of pure and applied logic, vol. 65 (1993), pp. 211–241. - Randall Dougherty. Critical points in an algebra of elementary embeddings, II. Logic: from foundations to applications, European logic colloquium, edited by Wilfrid Hodges, Martin Hyland, Charles Steinhorn, and John Truss, Clarendon Press, Oxford University Press, Oxford, New York, etc., 1996, pp. 103–136. - Randall Dougherty and Thomas Jech. Finite left-distributive algebras and embedding algebras. Advances in mathematics, vol. 130 (1997), pp. 201–241.

2002 ◽  
Vol 8 (4) ◽  
pp. 555-560
Author(s):  
Aleš Drápal
Keyword(s):  
New York ◽  
Critical Points ◽  
American Mathematical Society ◽  
Braid Groups ◽  
Alternative Proof ◽  
Elementary Embedding ◽  
Oxford University ◽  
Elementary Embeddings ◽  
Free Left ◽  
Distributive Law

The consistency problem for positive comprehension principles

1989 ◽  
Vol 54 (4) ◽  
pp. 1401-1418 ◽  
Author(s):  
M. Forti ◽  
R. Hinnion
Keyword(s):  
Set Theory ◽  
Difficult Problem ◽  
Large Cardinal ◽  
Positive Theory ◽  
Construction Principle ◽  
Short Summary ◽  
Free Construction ◽  
Unpublished Thesis ◽  
Consistency Problem ◽  
Original Construction

Since Gilmore showed that some theory with a positive comprehension scheme is consistent when the axiom of extensionality is dropped and inconsistent with it (see [1] and [2]), the problem of the consistency of various positive comprehension schemes has been investigated. We give here a short classification, which shows clearly the importance of the axiom of extensionality and of the abstraction operator in these consistency problems. The most difficult problem was to show the consistency of the comprehension scheme for positive formulas, with extensionality but without abstraction operator. In his unpublished thesis, Set theory in which the axiom of foundation fails [3], Malitz solved partially this problem but he needed to assume the existence of some unusual kind of large cardinal; as his original construction is very interesting and his thesis is unpublished, we give a short summary of it. M. Forti solved the problem completely by working in ZF with a free-construction principle (sometimes called an anti-foundation axiom), instead of ZF with the axiom of foundation, as Malitz did.This permits one to obtain the consistency of this positive theory, relative to ZF. In his general investigations about “topological set theories” (to be published), E. Weydert has independently proved the same result. The authors are grateful to the Mathematisches Forshungsinstitut Oberwolfach for giving them the opportunity of discussing these subjects and meeting E. Weydert during the meeting “New Foundations”, March 1–7, 1987.

PRIMITIVE INDEPENDENCE RESULTS

2003 ◽  
Vol 03 (01) ◽  
pp. 67-83
Author(s):  
HARVEY M. FRIEDMAN
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Power Set ◽  
Independence Results

We present some new set and class theoretic independence results from ZFC and NBGC that are particularly simple and close to the primitives of membership and equality (see Secs. 4 and 5). They are shown to be equivalent to familiar small large cardinal hypotheses. We modify these independendent statements in order to give an example of a sentence in set theory with 5 quantifiers which is independent of ZFC (see Sec. 6). It is known that all 3 quantifier sentences are decided in a weak fragment of ZF without power set (see [4]).

scholarly journals MARKOV AND ARTIN NORMAL FORM THEOREM FOR BRAID GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 951-961 ◽  
Author(s):  
L. A. BOKUT ◽  
V. V. CHAYNIKOV ◽  
K. P. SHUM
Keyword(s):  
Normal Form ◽  
Braid Groups ◽  
Form Theorem ◽  
Normal Form Theorem

In this paper we will present the results of Artin–Markov on braid groups by using the Gröbner–Shirshov basis. As a consequence we can reobtain the normal form of Artin–Markov–Ivanovsky as an easy corollary.

Frege's theorem in a constructive setting

1999 ◽  
Vol 64 (2) ◽  
pp. 486-488 ◽  
Author(s):  
John L. Bell
Keyword(s):  
Set Theory ◽  
Natural Numbers ◽  
The Family ◽  
Power Set ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Image Position ◽  
Excluded Middle ◽  
Intuitionistic Set Theory

By Frege's Theorem is meant the result, implicit in Frege's Grundlagen, that, for any set E, if there exists a map υ from the power set of E to E satisfying the conditionthen E has a subset which is the domain of a model of Peano's axioms for the natural numbers. (This result is proved explicitly, using classical reasoning, in Section 3 of [1].) My purpose in this note is to strengthen this result in two directions: first, the premise will be weakened so as to require only that the map υ be defined on the family of (Kuratowski) finite subsets of the set E, and secondly, the argument will be constructive, i.e., will involve no use of the law of excluded middle. To be precise, we will prove, in constructive (or intuitionistic) set theory, the followingTheorem. Let υ be a map with domain a family of subsets of a set E to E satisfying the following conditions:(i) ø ϵdom(υ)(ii)∀U ϵdom(υ)∀x ϵ E − UU ∪ x ϵdom(υ)(iii)∀UV ϵdom(5) υ(U) = υ(V) ⇔ U ≈ V.Then we can define a subset N of E which is the domain of a model of Peano's axioms.

A NORMAL FORM FOR BRAIDS

2008 ◽  
Vol 17 (06) ◽  
pp. 697-732 ◽  
Author(s):  
XAVIER BRESSAUD
Keyword(s):  
Normal Form ◽  
Regular Language ◽  
Braid Groups ◽  
New Normal ◽  
Geometric Action

We present a seemingly new normal form for braids, where every braid is expressed using a word in a regular language on some simple alphabet of elementary braids. This normal form stems from analysing the geometric action of braid groups on curves in a punctured disk.

scholarly journals Strong Logics of First and Second Order

2010 ◽  
Vol 16 (1) ◽  
pp. 1-36 ◽  
Author(s):  
Peter Koellner
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
Close Correspondence ◽  
First Order ◽  
Second Order Logic

AbstractIn this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing.

Regressive partitions and Borel diagonalization

1989 ◽  
Vol 54 (2) ◽  
pp. 540-552 ◽  
Author(s):  
Akihiro Kanamori
Keyword(s):  
Set Theory ◽  
Natural Generalization ◽  
Large Cardinal ◽  
Unit Interval ◽  
Combinatorial Proof ◽  
Consistency Strength ◽  
Several Variables ◽  
Borel Measurable ◽  
Mahlo Cardinals ◽  
Level Analysis

Several rather concrete propositions about Borel measurable functions of several variables on the Hilbert cube (countable sequences of reals in the unit interval) were formulated by Harvey Friedman [F1] and correlated with strong set-theoretic hypotheses. Most notably, he established that a “Borel diagonalization” proposition P is equivalent to: for any a ⊆ co and n ⊆ ω there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. In later work (see the expository Stanley [St] and Friedman [F2]), Friedman was to carry his investigations further into propositions about spaces of groups and the like, and finite propositions. He discovered and analyzed mathematical propositions which turned out to have remarkably strong consistency strength in terms of large cardinal hypotheses in set theory.In this paper, we refine and extend Friedman's work on the Borel diagonalization proposition P. First, we provide more combinatorics about regressive partitions and n-Mahlo cardinals and extend the approach to the context of the Erdös cardinals In passing, a combinatorial proof of a well-known result of Silver about these cardinals is given. Incorporating this work and sharpening Friedman's proof, we then show that there is a level-by-level analysis of P which provides for each n ⊆ ω a proposition almost equivalent to: for any a ⊆ co there is an ω-model of ZFC + ∃κ(κ is n-Mahlo) containing a. Finally, we use the combinatorics to bracket a natural generalization Sω of P between two large cardinal hypotheses.

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