Search

For $$n\in \omega $$ n ∈ ω , the weak choice principle $$\textrm{RC}_n$$ RC n is defined as follows: For every infinite setXthere is an infinite subset$$Y\subseteq X$$ Y ⊆ X with a choice function on$$[Y]^n:=\{z\subseteq Y:|z|=n\}$$ [ Y ] n : = { z ⊆ Y : | z | = n } . The choice principle $$\textrm{C}_n^-$$ C n - states the following: For every infinite family ofn-element sets, there is an infinite subfamily$${\mathcal {G}}\subseteq {\mathcal {F}}$$ G ⊆ F with a choice function. The choice principles $$\textrm{LOC}_n^-$$ LOC n - and $$\textrm{WOC}_n^-$$ WOC n - are the same as $$\textrm{C}_n^-$$ C n - , but we assume that the family $${\mathcal {F}}$$ F is linearly orderable (for $$\textrm{LOC}_n^-$$ LOC n - ) or well-orderable (for $$\textrm{WOC}_n^-$$ WOC n - ). In the first part of this paper, for $$m,n\in \omega $$ m , n ∈ ω we will give a full characterization of when the implication $$\textrm{RC}_m\Rightarrow \textrm{WOC}_n^-$$ RC m ⇒ WOC n - holds in $${\textsf {ZF}}$$ ZF . We will prove the independence results by using suitable Fraenkel-Mostowski permutation models. In the second part, we will show some generalizations. In particular, we will show that $$\textrm{RC}_5\Rightarrow \textrm{LOC}_5^-$$ RC 5 ⇒ LOC 5 - and that $$\textrm{RC}_6\Rightarrow \textrm{C}_3^-$$ RC 6 ⇒ C 3 - , answering two open questions from Halbeisen and Tachtsis (Arch Math Logik 59(5):583–606, 2020). Furthermore, we will show that $$\textrm{RC}_6\Rightarrow \textrm{C}_9^-$$ RC 6 ⇒ C 9 - and that $$\textrm{RC}_7\Rightarrow \textrm{LOC}_7^-$$ RC 7 ⇒ LOC 7 - .

We give necessary and sufficient conditions, both algebraic and geometric, for a quadrilateral to be the level set of the sum of the distances to m ≥ 2 different lines.

Linear and Multilinear Algebra, 70 (20), ISSN:1026-7573, ISSN:0308-1087, ISSN:1563-5139

In geometry, a Heron triangle is a triangle with rational side lengths and integral area. We investigate Heron triangles and their elliptic curves. In particular, we provide some new results concerning Heron triangles and give elementary proofs for some results concerning Heronian elliptic curves.

Given a cubic K. Then for each point P there is a conic $$C_P$$ associated to P. The conic $$C_P$$ is called the polar conic of K with respect to the pole P. We investigate the situation when two conics $$C_0$$ and $$C_1$$ are polar conics of K with respect to some poles $$P_0$$ and $$P_1$$ , respectively. First we show that for any point Q on the line $$P_0P_1$$ , the polar conic $$C_Q$$ of K with respect to Q belongs to the linear pencil of $$C_0$$ and $$C_1$$ , and vice versa. Then we show that two given conics $$C_0$$ and $$C_1$$ can always be considered as polar conics of some cubic K with respect to some poles $$P_0$$ and $$P_1$$ . Moreover, we show that $$P_1$$ is determined by $$P_0$$ , but neither the cubic nor the point $$P_0$$ is determined by the conics $$C_0$$ and $$C_1$$ .

For a set M, fin(M) denotes the set of all finite subsets of M, M² denotes the Cartesian product M × M, [M]² denotes the set of all 2-element subsets of M, and seq¹⁻¹(M) denotes the set of all finite sequences without repetition which can be formed with elements of M. Furthermore, for a set S, let |S| denote the cardinality of S. Under the assumption that the four cardinalities , |[M]², |M|², |fin(M)|, |seq¹⁻¹(M)| are pairwise distinct and pairwise comparable in ZF, there are six possible linear orderings between these four cardinalities. We show that at least five of the six possible linear orderings are consistent with ZF., Annals of Pure and Applied Logic, 174 (2), ISSN:0168-0072, ISSN:0003-4843

More than a century ago, Norman Anning conjectured that it is possible to arrange 48 points on a circle, such that all distances between the points are integer numbers and are all among the solutions of the diophantine equationx2 + xy + y2 = 72 132 192 312:We shall obtain Anning's conjecture as a consequence of a far more general geometrical result.Key words: Quadratic diophantine equations, quadratic forms, plane integral point sets.

Linear Algebra and its Applications, 609, ISSN:0024-3795, ISSN:1873-1856

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles' Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter's construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter's construction we provide a new parametrisation of elliptic curves with torsion group Z/2ZxZ/8Z and give some configurations with all their points on a cubic curve., Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry, 63 (4), ISSN:0138-4821, ISSN:2191-0383

A pair (a,b) of positive integers is a pythagorean pair if a2+b2=□ (i.e., a2+b2 is a square). A pythagorean pair (a,b) is called a double-pythapotent pair if there is another pythagorean pair (k,l) such that (ak,bl) is a pythagorean pair, and it is called a quadratic pythapotent pair if there is another pythagorean pair (k,l) which is not a multiple of (a,b), such that (a2k,b2l) is a pythagorean pair. To each pythagorean pair (a,b) we assign an elliptic curve Γa,b with torsion group Z/2Z×Z/4Z, such that Γa,b has positive rank over Q if and only if (a,b) is a double-pythapotent pair. Similarly, to each pythagorean pair (a,b) we assign an elliptic curve Γa with torsion group Z/2Z×Z/8Z, such that Γa has positive rank over Q if and only if (a,b) is a quadratic pythapotent pair. Moreover, in the later case we obtain that every elliptic curve Γ with torsion group Z/2Z×Z/8Z is isomorphic to a curve of the form Γa, where (a,b) is a pythagorean pair. As a side-result we get that if (a,b) is a double-pythapotent pair, then there are infinitely many pythagorean pairs (k,l), not multiples of each other, such that (ak,bl) is a pythagorean pair; the analogous result holds for quadratic pythapotent pairs., Journal of Number Theory, 233, ISSN:0022-314X, ISSN:1096-1658

We give a projective proof of the butterfly porism for cyclic quadrilaterals and present a general reversion porism for polygons with an arbitrary number of vertices on a conic. We also investigate projective properties of the porisms., Comment: 12 figures

In diesem Kapitel wollen wir die bisher hergeleitete Theorie anwenden, umein Apollonisches Beruhrungsproblem zu losen. Bei dem ausgesuchten Problem geht es darum, zu drei gegebenen Kreisen einen vierten Kreis zu konstruieren, welcher jeden der drei gegebenen Kreise beruhrt.

In diesem Kapitel betrachten wir eine Abbildung der Ebene auf sich selbst, die sogenannte Inversion am Kreis, und untersuchen geometrische Eigenschaften dieser Abbildung. Insbesondere werden wir sehen, dass diese Abbildung Kreise und Geraden auf Kreise und Geraden abbildet, aber nicht unbedingt Kreise in Kreise und Geraden in Geraden.

Sometimes it is convenient to extend a given signature L by adding new non-logical symbols which have to be properly deffned within the language L or with respect to a given L-theory T.

In this section, we will show that ZF is suffciently strong to prove that PA is consistent.

The goal of this chapter is to develop the formal language of First-Order Logic from scratch. At the same time, we introduce some terminology of the so-called metalanguage, which is the language we use when we speak about the formal language (e.g., when we want to express that two strings of symbols are equal).

The key ingredient for Gӧdel’s Incompleteness Theorems is the so-called Gӧdelisation process which allows us to code terms, formulae and even proofs within PA.

By Gӧdel’s Completeness Theorem 5.5 we know that every consistent theory T has a model, and if T has an infinite model, then it also has arbitrarily large models.

Linear Algebra and its Applications, 589, ISSN:0024-3795, ISSN:1873-1856

There are two different views on a given set of formulae ϕ, namely the syntactical view and the semantical view.

In this chapter, we take a closer look at Peano Arithmetic (PA) which we have defined in Chapter 1. In particular, we prove within PA some basic arithmetical results, starting with the commutativity and associativity of addition and multiplication, culminating in some results about coprimality.

In this chapter, we shall present and discuss the axioms of Zermelo-Fraenkel Set Theory including the Axiom of Choice, denoted ZFC. It will turn out that within this axiom system, we can develop all of first-order mathematics, and therefore, the axiom system ZFC serves as a foundation of mathematics.

It follows from Godel's Completeness Theorem that a theory is consistent if and only if it has a model. In particular, the consistency of Peano Arithmetic follows from \(\mathbb{N} \models \mathbf{PA}\).

In Example 1.2 we gave a proof of 1 + 1 = 2 in 17 (!) proof steps. At that point you may have asked yourself: If it takes that much effort to prove such a simple statement, how can one ever prove any non-trivial mathematical result using formal proofs? This objection is of course justified; however, we will show in this chapter how one can simplify formal proofs using some methods of proof such as proofs by cases or by contradiction.

In 1931, Gӧdel proved his FIRST INCOMPLETENESS THEOREM which states that if PA is consistent, then it is incomplete, i.e.

In this chapter, we will first construct a model of the real numbers using Cauchy sequences of rational numbers. We also present a second model of the real numbers according to A'Campo [1].

In Chapter 10, we have seen that Peano Arithmetic PA is incomplete. Moreover, if we omit the Induction Schema and replace it by the axiom \(\forall x(x=0\vee\exists y(x=sy))\), stating that every number is either 0 or has a predecessor, then the resulting theory, called Robinson Arithmetic, is also incomplete.

The goal of this chapter is to show that every consistent L-theory has a model, no matter whether the signature L is countable or uncountable.

Zermelo writes in [59, p. 262] that he was not able to show that the seven axioms for Set Theory given in that article are consistent. Even though it is essential whether a theory is consistent or not, we know that whenever a theory is strong enough to prove the axioms of PA, then there is no way to prove its consistency within this theory (see Chapter 11).

The exponential pencil Gλ:=G1(G−10G1)λ−1, generated by two conics G0,G1, carries a rich geometric structure: It is closed under conjugation, it is compatible with duality and projective mappings, it is convergent for λ→±∞ or periodic, and it is connected in various ways with the linear pencil gλ=λG1+(1−λ)G0. The structure of the exponential pencil can be used to characterize the position of G0 and G1 relative to each other., Beiträge zur Algebra und Geometrie = Contributions to algebra and geometry, 59 (3), ISSN:0138-4821, ISSN:2191-0383

For a set M, let $${\text {seq}}(M)$$ denote the set of all finite sequences which can be formed with elements of M, and let $$[M]^2$$ denote the set of all 2-element subsets of M. Furthermore, for a set A, let denote the cardinality of A. It will be shown that the following statement is consistent with Zermelo–Fraenkel Set Theory $$\textsf {ZF}$$ : There exists a set M such that and no function is finite-to-one.

A positive integer $A$ is called a congruent number if $A$ is the area of a right-angled triangle with three rational sides. Equivalently, $A$ is a congruent number if and only if the congruent number curve $y^2=x^3-A^2x$ has a rational point $(x,y)\in\mathbb Q^2$ with $y\neq 0$. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order., Comment: 8 pages, one figure

We investigate the relationship between various choice principles and nth-root functions in rings. For example, we show that the Axiom of Choice is equivalent to the statement that every ring has a square-root function. Furthermore, we introduce a choice principle which implies that every integral domain has an nth-root function (for odd integers n), and introduce another choice principle which is equivalent to the Prime Ideal Theorem restricted to certain ideals. Finally, we investigate the dependencies between the two new choice principles and a choice principle for families of n-element sets., Results in Mathematics, 74 (1)

The Journal of Symbolic Logic, 59 (1), ISSN:1520-4812, ISSN:1943-5886

In the middle and late nineteenth century, members of the then small mathematical community began to look for a rigorous foundation of Mathematics. In accordance with the Euclidean model for reason, the ideal foundation consists of a few simple, clear principles, so-called axioms, on which the rest of knowledge can be built via firm and reliable thoughts free of contradictions. However, at the time it was not clear what assumptions should be made and what operations should be allowed in mathematical reasoning.

We have already seen that there may exist \(2^{\mathfrak{c}}\) pairwise non-isomorphic Ramsey ultrafilters (see Proposition 14.10), that there may exist \(\mathfrak{c}\) Ramsey ultrafilters (see Proposition 23.6), and in the previous chapter we constructed a model in which there are no Ramsey ultrafilters (see Proposition 26.23). In this chapter, we conclude this issue by showing that for each cardinal κ with 0 ≤ κ ≤ ω2, there is a model of ZFC in which there are exactly κ pairwise non-isomorphic Ramsey ultrafilters.

We shall introduce and discuss in this chapter the axioms of Zermelo–Fraenkel Set Theory with the Axiom of Choice.

In 1904, Zermelo published his first proof that every set can be well-ordered. The proof is based on the so-called Axiom of Choice, denoted AC, which, in Zermelo’s words, states that the product of an infinite totality of sets, each containing at least one element, itself differs from zero (i.e., the empty set). The full theory ZF + AC, denoted ZFC, is called Set Theory.

In this chapter we investigate 13 cardinal characteristics and their relations to one another. A cardinal characteristic of the continuum is an uncountable cardinal number which is less than or equal to \(\mathfrak{c}\) that describes a combinatorial or analytical property of the continuum. Like the power of the continuum itself, the size of a cardinal characteristic is often independent from ZFC. However, some restrictions on possible sizes follow from ZFC, and we shall give a complete list of what is known to be provable in ZFC about their relation. Later in Part III, but mainly in Part IV, we shall see how one can diminish or augment some of these 13 cardinals without changing certain other cardinals. In fact, these cardinal characteristics are also used to investigate combinatorial properties of the various forcing notions introduced in Part IV.

In this chapter we investigate Suslin’s Problem, which is the question whether every linearly ordered set which is dense, unbounded, complete, and satisfies the countable chain condition is also separable. It will be shown that the answer to Suslin’s Problem is independent of ZFC.

In diesem Kapitel untersuchen wir harmonische Punkte in Bezug auf Kreise. Dabei werden die Apollonischen Kreise von zentraler Bedeutung sein.