A remark on method in transfinite algebra

Zorn, Max

doi:10.1090/s0002-9904-1935-06166-x

Cited by 166 publications

(43 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Transversal Functional Analysis we obtain that the first equality in (11) holds. This with (12) give a good reason for the preceding equalities in (11).…”

Section: Proposition 3 Let X Be a Linear Space And Let (X ρ) Be An mentioning

confidence: 70%

See 1 more Smart Citation

Transversal functional analysis

Taskovic¹

2014

Math Morav

View full text Add to dashboard Cite

Short survey. This paper provides an introduction to the ideas and methods of transversal functional analysis based on the transversal sets theory. A unifying concept that lies at the heart of transversal functional analysis is that of a transversal normed linear space. Transversal upper normed spacesLet X be a linear space over K (:= R or C). The mapping x → ||x|| : X → [a, b] for some 0 ≤ a < b < +∞ or x → x : X → [a, b) for some 0 ≤ a < b ≤ +∞ is called an upper transversal seminorm (or upper seminorm) iff: ||x|| ≥ a for every x ∈ X, ||λx|| = |λ| ||x|| for all λ ∈ K and x ∈ X, and if there is a function g :for all x, y ∈ X.Further, x → ||x|| is called an upper transversal norm (or upper norm) iff in addition: ||x|| = a if and only if x = 0.An upper transversal normed space (X, || · ||) over K consists of a linear space X over K together with an upper transversal norm x → ||x||.2010 Mathematics Subject Classification. Primary: 04A25, 05A15, 47H10, 54E15; Secondary: 54H25, 54E24.

show abstract

“…Transversal Functional Analysis we obtain that the first equality in (11) holds. This with (12) give a good reason for the preceding equalities in (11).…”

Section: Proposition 3 Let X Be a Linear Space And Let (X ρ) Be An mentioning

confidence: 70%

“…hence, because of the fact that for every x ∈ X\{0} the upper norm of the vector x/ x is equaly 1, it follows fact that A = sup x =1 Ax , which is one of the equalities in (11). Since…”

Section: Proposition 3 Let X Be a Linear Space And Let (X ρ) Be An mentioning

confidence: 99%

Transversal functional analysis

Taskovic¹

2014

Math Morav

View full text Add to dashboard Cite

show abstract

“…In fact, Version 5 (the proof of which uses Zorn's Lemma [21]) wasn't established yet in this form when Noether wrote her paper. Birkhoff's first aim was to generalize Noether's Irreducible Decomposition Theorem to general algebras whose congruence lattices satisfy the ascending chain condition.…”

Section: Let Us Recall Now That Version 5 Produces For a Commutative mentioning

confidence: 99%

Nullstellen and Subdirect Representation

Tholen

2013

Appl Categor Struct

View full text Add to dashboard Cite

David Hilbert's solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hilbert-NoetherBirkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere. The aim of this article is to analyze this linkage in the most elementary terms and then, based on our work of the 1980s, to present a general categorical framework for Birkhoff's theorem. IntroductionThe first purpose of this article is to exhibit in elementary algebraic terms the linkage between Hilbert's celebrated Nullstellensatz [6] for systems of polynomial equations and Birkhoff's Subdirect Representation Theorem [2] for general algebras, as facilitated by Noether's work [15] on the decomposition of ideals in rings. Hence, in Section 1 we give a brief tour of the development of the Nullstellensatz through Hilbert, Noether and Birkhoff. We do so not from a strictly historical perspective; rather, in today's language we present six versions of the theorem as marked by these three great mathematicians and show how their proofs are interrelated, referring to them as the HNB Theorems.A seventh and an eighth version are given in Section 3, after a discussion of the categorical notion of subdirect irreducibility in Section 2. We illuminate the notion by examples, both traditional and unconventional, in particular in comma categories. We touch upon the dual notion only briefly, but refer the reader to substantial recent work by Matias Menni [14] on Lawvere's concept of cohesion in this context. With a suitable notion of finitariness we formulate the all-encompassing seventh version of the HNB Theorem without recourse to any limits or colimits in the ambient category. The morphism version of it leads to atypical factorizations of morphisms, in the sense that even in standard categories like that of sets one obtains factorizations of maps in a constructive manner (without recourse to choice) which, however, may not be obtained in a functorial way.Acknowledgement I am indebted to Bill Lawvere for his suggestion to study the topic of this paper and his subsequent interest in and comment on a preliminary (but extended) version of this work, presented in part at The European Category Theory Meeting in Haute Bodeux (Belgium) in 2003. I am also grateful to László Márki for his careful reading of an earlier version of this paper and for detecting an inaccuracy that may be traced back to Birkhoff's original paper (see 2.3(4) below).

show abstract

“…2 A simple statement of this axiom due to Rosser ([52], p. 88) is "If λ is a set of nonempty, nonoverlapping sets, then there is a set γ which has exactly one member in common with each member of λ." Given that λ can vary, which is usually a transfinite cardinal, there are many versions of the axiom, with Zorn's Lemma [79] usually being assumed to assert it holds for most levels that most mathematicians deal with, although Specker [64] showed that it does not hold at the ultimate level of the universe as a whole, using the law of the excluded middle, or reductio ad absurdum, thus rendering absurd a constructivist assertion of this theorem as a general disproof of the axiom. Specker's result uses the Quine's New Foundations approach.…”

Section: P 4 ) [Italics In Original]mentioning

confidence: 99%

On the Foundations of Mathematical Economics

Rosser

2012

New Math. and Nat. Computation

View full text Add to dashboard Cite

Abstract:Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the Bolzano-Weierstrass, Hahn-Banach, and intermediate value theorems, and then the implications of these arguments for such "crown jewels" of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel's incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him.

show abstract

A remark on method in transfinite algebra

Cited by 166 publications

References 0 publications

Transversal functional analysis

Transversal functional analysis

Nullstellen and Subdirect Representation

On the Foundations of Mathematical Economics

Contact Info

Product

Resources

About