Prime Ideal Theorems and systems of finite character

summary:\font\jeden=rsfs7 \font\dva=rsfs10 We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if \text{\jeden S} is a system of finite character then so is the system of all collections of finite subsets of \bigcup \text{\jeden S} meeting a common member of \text{\jeden S}), the Finite Cutset Lemma (a finitary version of the Teichm"uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth's Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma

The Axiom of Choice

We will discuss the 9th axiom of Zermelo-Fraenkel set theory with choice, which is often abbreviated ZFC, since it includes the axiom of choice (AC). AC is a controversial axiom that is mathematically equivalent to many well known theorems and has an interesting history in set theory. This thesis is a combination of discussion of the history of the axiom and the reasoning behind why the axiom is controversial. This entails several proofs of theorems that establish the fact that AC is equivalent to such theorems and notions as Tychonoff\u27s Theorem, Zorn\u27s Lemma, the Well-Ordering Theorem, and many more

Canonical Forms for Matrices over Polynomial Rings

One of the important concepts in matrix algebra is rank of matrices. If the entries of such matrices are from fields or principal ideal domains, then this concept of rank is well-defined. However, when such matrices are defined over the ring of polynomials F[x_1, . . . , x_k ], k ≥ 2 (polynomial matrices in more than one indeterminate), the concept of rank has different but inequivalent definitions. Despite this flaw, some theories, in relation to ranks, can still be applied to polynomial matrices in more than one indeterminate. One of the outcomes of these theories is that lower and upper bounds for ranks of such polynomial matrices in more than one indeterminate can be obtained. Just like matrices over fields or principal ideal domains can be reduced to some simpler or canonical forms, there are algorithms that can be used to reduce matrices over polynomial rings in more than one indeterminate to some simpler forms, though these reduced forms do not always tell the ranks of such polynomial matrices in more than one indeterminate. In this thesis, these algorithms will be presented with examples

Paul Lorenzen -- Mathematician and Logician

This open access book examines the many contributions of Paul Lorenzen, an outstanding philosopher from the latter half of the 20th century. It features papers focused on integrating Lorenzen's original approach into the history of logic and mathematics. The papers also explore how practitioners can implement Lorenzen’s systematical ideas in today’s debates on proof-theoretic semantics, databank management, and stochastics. Coverage details key contributions of Lorenzen to constructive mathematics, Lorenzen’s work on lattice-groups and divisibility theory, and modern set theory and Lorenzen’s critique of actual infinity. The contributors also look at the main problem of Grundlagenforschung and Lorenzen’s consistency proof and Hilbert’s larger program. In addition, the papers offer a constructive examination of a Russell-style Ramified Type Theory and a way out of the circularity puzzle within the operative justification of logic and mathematics. Paul Lorenzen's name is associated with the Erlangen School of Methodical Constructivism, of which the approach in linguistic philosophy and philosophy of science determined philosophical discussions especially in Germany in the 1960s and 1970s. This volume features 10 papers from a meeting that took place at the University of Konstanz