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Cardinal and Ordinal Numbers
... The aim of this section is to briefly recall the notion of ordinals, which play an important role in our theory. An excellent introduction to this topic may be found in (Hrbacek and Jech, 1999, Chapter 6), while the classical reference is (Sierpiński, 1965). ...
... The specific construction of ordinals is not important for our purposes, and we therefore discuss ordinals somewhat informally. We refer to (Hrbacek and Jech, 1999, Chapter 6) or (Sierpiński, 1965) for a careful treatment. ...
How quickly can a given class of concepts be learned from examples? It is common to measure the performance of a supervised machine learning algorithm by plotting its "learning curve", that is, the decay of the error rate as a function of the number of training examples. However, the classical theoretical framework for understanding learnability, the PAC model of Vapnik-Chervonenkis and Valiant, does not explain the behavior of learning curves: the distribution-free PAC model of learning can only bound the upper envelope of the learning curves over all possible data distributions. This does not match the practice of machine learning, where the data source is typically fixed in any given scenario, while the learner may choose the number of training examples on the basis of factors such as computational resources and desired accuracy. In this paper, we study an alternative learning model that better captures such practical aspects of machine learning, but still gives rise to a complete theory of the learnable in the spirit of the PAC model. More precisely, we consider the problem of universal learning, which aims to understand the performance of learning algorithms on every data distribution, but without requiring uniformity over the distribution. The main result of this paper is a remarkable trichotomy: there are only three possible rates of universal learning. More precisely, we show that the learning curves of any given concept class decay either at an exponential, linear, or arbitrarily slow rates. Moreover, each of these cases is completely characterized by appropriate combinatorial parameters, and we exhibit optimal learning algorithms that achieve the best possible rate in each case. For concreteness, we consider in this paper only the realizable case, though analogous results are expected to extend to more general learning scenarios.
... Clearly, one can find many references for chains being frequently regarded as special lattices, but here we rather need distinguished properties of chains instead of the standard ones. For this reason we first recall the basic properties of chains together with some modifications (compare [1] and [6]). Next, we introduce the more general concept of a chain fibration which can be regarded as a chain analogue of the well-known concept of a topological fibration. ...
... Moreover, one can see that if o G 5" (ft 6 S + ), then for any x € S (y € S) such that x < a (y > b) we have x € S~ and y € S + . This implies that S~ and S + are convex subchains of 5 such that for any x G S~ and y 6 we have x < y. Hence we conclude that S~ 0 S + = S. Similarly, as in the previous case, we see that a defines anti oisomorphisms er+~ : S + -> S~ and cr~+ : S~ -• S + . ...
... The discussion of ordinals is borrowed from [BHM + 21]. For a thorough treatment of the subject, the interested reader is referred to [HJ99,Sie58]. We consider some set S. A well ordering of S is defined to be any linear ordering < so that every non-empty subset of S contains a least element. ...
In this paper we study the problem of multiclass classification with a bounded number of different labels $k$, in the realizable setting. We extend the traditional PAC model to a) distribution-dependent learning rates, and b) learning rates under data-dependent assumptions. First, we consider the universal learning setting (Bousquet, Hanneke, Moran, van Handel and Yehudayoff, STOC '21), for which we provide a complete characterization of the achievable learning rates that holds for every fixed distribution. In particular, we show the following trichotomy: for any concept class, the optimal learning rate is either exponential, linear or arbitrarily slow. Additionally, we provide complexity measures of the underlying hypothesis class that characterize when these rates occur. Second, we consider the problem of multiclass classification with structured data (such as data lying on a low dimensional manifold or satisfying margin conditions), a setting which is captured by partial concept classes (Alon, Hanneke, Holzman and Moran, FOCS '21). Partial concepts are functions that can be undefined in certain parts of the input space. We extend the traditional PAC learnability of total concept classes to partial concept classes in the multiclass setting and investigate differences between partial and total concepts.
... We can now easily adapt the proof of l.l(ii), as given in [3], to prove the following result. Definition. ...
This paper is a continuation of [ 1; 2 ]. In [ 2 ], I stated that I had been unable to construct examples of planes satisfying various conditions. Some of the examples that I have since constructed are given below. A discussion of one-dimensional absolute geometries, with examples, will be given in a separate paper. The relevant parts of [ 1 ] and [ 2 ] are [ 1 , § 1, § 2 up to 2.4; 2 , § 2]. We shall use the notation and terminology of [ 1; 2 ]; the axioms Cl*-C4* and C4** (referred to below) can all be found in [ 1 ].
We shall show here that spaces of dimension greater than 1 exist, both Archimedean and non-Archimedean, satisfying Cl*-C4*, in which not all points are isometric, and that C4** does not follow from Cl*-C4* in non- Archimedean geometries of dimension greater than 1.
... The negation of a statement with a definite description affects not only the predicate but also the characterization of the conclusion the description influences. 6. Regarding the notion of infinity and the distinctions between the different types of infinities discovered by Cantor (Cantor 1915;Sierpinski 1965), we should point out that Spinoza, 1 3 200 years earlier, had already incorporated this concept that is so controversial in contemporary mathematics. We can quote as evidence the Letter XII addressed to Meyer where three types of infinity are distinguished. ...
If the words in Spinoza’s Ethics are considered as symbols, then certain words in the definitions of the Ethics can be replaced with symbols from set theory and we can reexamine Spinoza’s first definitions within a logical–mathematical frame. The authors believe that, some aspects of Spinoza’s work can be explained and illustrated through mathematics. A semantic relation between the definitions of the philosopher and set theory is presented. It is explained each chosen symbol. The ontological argument is developed through modal logic. And finally, we present some conclusions drawn from this work.
... Given f 1 , f 2 ∈ Fld( C ) say that f 1 C f 2 if either Dom(f 1 ) ≤ Dom(f 2 ) and f 1 (i) ∼ f 2 (i) for all i < Dom(f 1 ); or if j is the "first difference", f 1 (j) = b 1 , a 1 , and f 2 (j) = b 2 , a 2 , then either b 1 ≺ B b 2 or b 1 ≡ B b 2 and a 1 ≺ A a 2 . It is readily seen that C is a WPS, and using well-known facts about ordinal arithmetic (see section 16 of [25] for example) Rnk(C) = α β . That W ρ (ξ) is an ǫ-number follows by induction on ξ. ...
Schemes may be used to define systems of ordinal functions. Such systems may be used to obtain lower bounds on the smallest repeat point. Using a scheme Σ CTT , it is shown that the smallest ordinal θ T such that there is no T-separating set at θ T is at least an ordinal which is larger than the analog of the Bachmann-Howard ordinal for the Bachmann hierarchy starting at the critical point enumerator C. This holds for any coherent sequence satisfying o(U(κ)(β)) = β. The notion of an O-scheme in L[A] is defined. The notion in L[U] is used to give sufficient conditions for non-existence of repeat points. A function W is defined in L[U], and it is shown using C and W that in L[U], if θ < θ T then Pow(κ) ⊆ L θ [U].
... This is an implication of the following mathematical result, stated and proved in Sierpinski (1965), p. 70-71. ...
This paper presents easily verifiable sufficient conditions on sequence spaces that guarantee representation of preference orders. Our approach involves identifying a suitable subset of the set of alternatives, such that (a) the preference order is representable on this subset, and (b) the subset has the property that for each alternative, there is some element in this subset which is indifferent to it. We follow Wold in choosing this subset to be the diagonal. Our first result uses a weak monotonicity condition (on the diagonal), and a substitution condition, and may be identified as the essence of Wold's contribution. In the second result, we show that one can obtain a Wold-type representation result when weak monotonicity is replaced by a weak continuity condition. We use the countable order-dense characterization of representability in the proofs of both results, thereby integrating the contributions of Wold (1943) and Debreu (1954). Through a series of examples we show that our representation results are robust; they cannot be improved upon by dropping any of our conditions. An example is also presented to show that existence of degenerate indifference classes is compatible with the representation of monotone preferences. Our study thereby indicates that while the presence of substitution possibilities can be useful in representing preferences, they are not necessary for such results to hold.
... We denote by Ord the class of ordinals. For a thorough exposition of ordinals we refer to the classical handbooks such as [9] and [8]. ...
Diekert, Matiyasevich and Muscholl proved that the existential first-order theory of a trace monoid over a finite alphabet is decidable. We extend this result to a natural class of trace monoids with infinitely many generators. As an application, we prove that for every ordinal $\lambda$ less than $\varepsilon_0$, the existential theory of the set of successor ordinals less than $\lambda$ equipped with multiplication is decidable.
... Proof. There are uncountably many countable ordinals [16]. Thus there are uncountably many countable ordered sets. ...
We show that locally solvable subgroups of PLo(I) are countable. Then for each countable ordered set, we construct a locally solvable subgroup of Thompson's Group F. We develop machinery for understanding embeddings from solvable subgroups into solvable subgroups. Finally, we apply this machinery to show the ordered sets used in our construction are invariant under isomorphisms between the groups constructed. Therefore, we effectively distinguish the groups and provide uncountably many non-isomorphic locally solvable, hence elementary amenable, subgroups of Thompson's Group F.
... Set of negative integers has a max but no min. ORDER TYPE [Sierpinski (1965)] A strictly ordered set Y(<) is of order type ω if Y(<) is similar to N(<). σ if Y(<) is similar to I(<) where I is the set of integers, and µ if Y contains a non-empty subset Y ′ such that the strictly ordered set Y ′ (<) is of order type σ. ...
Princeton Workshop on Infinite Value
... We assume the standard conventions on ordinal number arithmetic[Sie65]. The successor of an ordinal α is α + 1. Addition is non-commutative and left-cancellative, that is, let n be a finite ordinal, then ...
In the lambda calculus a term is solvable iff it is operationally relevant.
Solvable terms are a superset of the terms that convert to a final result
called normal form. Unsolvable terms are operationally irrelevant and can be
equated without loss of consistency. There is a definition of solvability for
the lambda-value calculus, called v-solvability, but it is not synonymous with
operational relevance, some lambda-value normal forms are unsolvable, and
unsolvables cannot be consistently equated. We provide a definition of
solvability for the lambda-value calculus that does capture operational
relevance and such that a consistent proof-theory can be constructed where
unsolvables are equated attending to the number of arguments they take (their
"order" in the jargon). The intuition is that in lambda-value the different
sequentialisations of a computation can be distinguished operationally. We
prove a version of the Genericity Lemma stating that unsolvable terms are
generic and can be replaced by arbitrary terms of equal or greater order.
... Sierpinski showed that the Continuum Hypothesis (CH) holds if and only if there are sets A, B ⊆ R 2 such that A ∪ B = R 2 and for any a, b ∈ R the sections A a = {y : (a, y) ∈ A} and B b = {x : (x, b) ∈ B} are countable [9]. In [10,11] Törnquist and Weiss studied many Σ 1 2 definable versions of some equivalent forms of CH which happen to be equivalent to "all reals are constructible". ...
We study $\Sigma^1_2$ definable counterparts for some algebraic equivalent
forms of the Continuum Hypothesis. All turn out to be equivalent to "all reals
are constructible".
... There exists an anti-chain of size c in (P(N) * , ⊆). Although this fact is well known (see, for example, [24]), we give a brief argument for the reader's convenience. ...
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).
... Then τ (C|·) is measurable and, by the dominated convergence theorem, (L * µ)(C) = x∈X τ (C|x) dµ(x) for any Radon measure µ on X ; hence, G α also has the property. However, by the above proposition C2.7, G 0 has the property, so by transfinite induction [27] [116] all the G α have the property. ...
Bayesian networks and their accompanying graphical models are widely used for
prediction and analysis across many disciplines. We will reformulate these in
terms of linear maps. This reformulation will suggest a natural extension,
which we will show is equivalent to standard textbook quantum mechanics.
Therefore, this extension will be termed "quantum". However, the term "quantum"
should not be taken to imply this extension is necessarily only of utility in
situations traditionally thought of as in the domain of quantum mechanics. In
principle, it may be employed in any modeling situation, say forecasting the
weather or the stock market--it is up to experiment to determine if this
extension is useful in practice. Even restricting to the domain of quantum
mechanics, with this new formulation the advantages of Bayesian networks can be
maintained for models incorporating quantum and mixed classical-quantum
behavior. The use of these will be illustrated by various basic examples.
Parrondo's paradox refers to the situation where two, multi-round games with
a fixed winning criteria, both with probability greater than one-half for one
player to win, are combined. Using a possibly biased coin to determine the rule
to employ for each round, paradoxically, the previously losing player now wins
the combined game with probability greater than one-half. Using the extended
Bayesian networks, we will formulate and analyze classical observed, classical
hidden, and quantum versions of a game that displays this paradox, finding
bounds for the discrepancy from naive expectations for the occurrence of the
paradox. A quantum paradox inspired by Parrondo's paradox will also be
analyzed. We will prove a bound for the discrepancy from naive expectations for
this paradox as well. Games involving quantum walks that achieve this bound
will be presented.
We study a differential game of kind of several pursuit points and one evasion point moving along the edges of a regular simplex of dimension d. It is assumed that maximum magnitude of velocity of evader is twice as much as the maximum magnitudes of velocities of pursuers. An exact mathematical formulation of the problem is given by introducing special classes of strategies adapted for games on graphs. It is proved that if the number of pursuers is greater than ⌊d∕2⌋ + 1, then the game is completed in favor of pursuers, otherwise in favor of the evader.KeywordsDifferential gameMany pursuersEvaderGames on graphsStrategyControl
Set theory is a branch of mathematics with a special subject matter, the infinite, but also a general framework for all modern mathematics, whose notions figure in every branch, pure and applied. This Element will offer a concise introduction, treating the origins of the subject, the basic notion of set, the axioms of set theory and immediate consequences, the set-theoretic reconstruction of mathematics, and the theory of the infinite, touching also on selected topics from higher set theory, controversial axioms and undecided questions, and philosophical issues raised by technical developments.
In this paper, a differential game of kind of several pursuers and one evader is studied. All the players move only along the edges of a simplex of dimension [Formula: see text]. The maximal speed of each pursuer is less than that of the evader. If the state of a pursuer coincides with the state of the evader, then pursuit is completed. An exact mathematical formulation of the problem is given by introducing special classes of strategies adapted for games on graphs. Sufficient conditions for completion of pursuit and possibility of evasion are obtained. In the case where the simplex is regular we obtained a condition. If this condition is satisfied, then pursuit can be completed, else evasion is possible.
From different areas of mathematics, such as set theory, geometry, transfinite arithmetic or supertask theory, in this book more than forty arguments are developed about the inconsistency of the hypothesis of the actual infinity in contemporary mathematics. A hypothesis according to which the uncompletable lists, as the list of the natural numbers, exist as completed lists. The inconsistency of this hypothesis would have an enormous impact on physics, forcing us to change the continuum space-time for a discrete model, with indivisible units (atoms) of space and time. The discrete model would be a great simplification of physical theories, including relativity and quantum mechanics. It would also suppose the solution of the old problem of change, posed by the pre-Socratics philosophers twenty-seven centuries ago.
We propose generalized versions of strong equity and Pigou-Dalton transfer principle. We study the existence and the real valued representation of social welfare relations satisfying these two generalized equity principles. Our results characterize the restrictions on one period utility domains for the equitable social welfare relation (i) to exist; and (ii) to admit real-valued representations.
In fluid mechanics, infinitesimal masses \({\mathrm d}M\), called fluid parcels, occupy infinitesimal volumes \({\mathrm d}V\) that have well-defined positions \(\vec {r}(t)\) in some fixed frame of reference, similar to point masses in classical point mechanics. The fluid parcels fill some connected volume in space homogeneously (tearing fluids are not treated here), where space is the Euclidean vector space \(\mathbb {R}^3\).
We present sufficient conditions for the non-representability of a class of lexicographic preferences. The proof of non-representability builds on the classical Debreu (1954) argument and its usefulness in generating counter examples is demonstrated through a series of examples. JEL Codes: D01, D11
A topology τ on a monoid S is called shift-continuous if for every a, b ∈ S the two-sided shift S → S, x ↦ axb, is continuous. For every ordinal α ≤ ω, we describe all shift-continuous locally compact Hausdorff topologies on the α-bicyclic monoid Bα. More precisely, we prove that the lattice of shift-continuous locally compact Hausdorff topologies on Bα is anti-isomorphic to the segment of [1, α] of ordinals, endowed with the natural well-order. Also we prove that for each ordinal α the α + 1-bicyclic monoid Bα+1 is isomorphic to the Bruck extension of the α-bicyclic monoid Bα.
Some collections of submodules of a module defined by certain conditions are studied. A generalization of the notion of radical (preradical) filter is considered. We study the form of filters of semisimple modules and direct sums.
We study an infinite countable iteration of the natural product between ordinals. We present an "effective" way to compute this countable natural product; in the non trivial cases, the result depends only on the natural sum of the degrees of the factors, where the degree of a nonzero ordinal is the largest exponent in its Cantor normal form representation. Thus we are able to lift former results about infinitary sums to infinitary products. Finally, we provide an order-theoretical characterization of the infinite natural product.
In this paper we consider a semitopological α-bicyclic monoid B α and prove that it is algebraically isomorphic to a semigroup of all order isomorphisms between the principal upper sets of ordinal ω α. We prove that for every ordinal α for every (a, b) ∈ B α if a or b is a non-limit ordinal then (a, b) is an isolated point in B α. We show that for every ordinal α < ω + 1 every locally compact semigroup topology on B α is discrete. However we construct an example of a non-discrete locally compact topology τ lc on B ω+1 such that (B ω+1 , τ lc) is a topological inverse semigroup. This example shows that there is a gap in [16, Theorem 2.9], where is stated that for every ordinal α there is only discrete locally compact inverse semigroup topology on B α. In this paper all topological spaces are assumed to be Hausdorff. By N we shall denote the set of all positive integers. A semigroup S is called inverse if for every x ∈ S there exists a unique y ∈ S such that xyx = x and yxy = y. Later such an element y will be denoted by x −1 and will be called the inverse of x. A map inv : S → S which assigns to every s ∈ S its inverse is called inversion. By ω we denote the first infinite ordinal. A topological (inverse) semigroup is a topological space together with a continuous semigroup operation (and an inversion, respectively). Obviously, the inversion defined on a topological inverse semigroup is a homeomorphism. If S is a semigroup (an inverse semigroup) and τ is a topology on S such that (S, τ) is a topological (inverse) semigroup, then we shall call τ an (inverse) semigroup topology on S. A semitopological semigroup is a topological space together with a separately continuous semigroup operation. Let f be a map between two partial ordered sets (A, ≤ A) and (B, ≤ B), then we shall call f a monotone if for every a, b ∈ A if a ≤ A b then f (a) ≤ B f (b). We shall call f an order isomorphism if f is monotone bijection and its inverse map f −1 is also monotone. For a partially ordered set (A, ≤), for an arbitrary X ⊂ A and x ∈ A we write: 1) ↓X = {y ∈ A : there exists x ∈ X such that y ≤ x}; 2) ↑X = {y ∈ A : there exists x ∈ X such that x ≤ y}; 3) ↓x = ↓{x}; 4) ↑x = ↑{x}; 5) X is a lower set if X = ↓X; 6) X is an upper set if X = ↑X; 7) X is a principal lower set if X = ↓x for some x ∈ X; 7) X is a principal upper set if X = ↑x for some x ∈ X.
We study density requirements on a given Banach space that guarantee the existence of subsymmetric basic sequences by extending Tsirelson's well-known space to larger index sets. We prove that for every cardinal $\kappa$ smaller than the first Mahlo cardinal there is a reflexive Banach space of density $\kappa$ without subsymmetric basic sequences. As for Tsirelson's space, our construction is based on the existence of a rich collection of homogeneous families on large index sets for which one can estimate the complexity on any given infinite set. This is used to describe detailedly the asymptotic structure of the spaces. The collections of families are of independent interest and their existence is proved inductively. The fundamental stepping up argument is the analysis of such collections of families on trees.
In this paper we consider the $\alpha$-bicyclic monoid $\mathcal{B}_{\alpha}$ as a semitopological semigroup and prove that it is algebraically isomorphic to the semigroup of all order isomorphisms between the principal upper sets of ordinal $\omega^{\alpha}$. Also we prove that for every ordinal $\alpha$ for every $(a,b)\in \mathcal{B_{\alpha}}$ if $a$ or $b$ is a non-limit ordinals then $(a,b)$ is an isolated point in $\mathcal{B}_{\alpha}$ and for every ordinal $\alpha<\omega+1$ every locally compact semigroup topology on $\mathcal{B}_{\alpha}$ is discrete. However we construct an example of non-discrete locally compact topology $\tau_{lc}$ on $\mathcal{B}_{\omega+1}$ such that $(\mathcal{B}_{\omega+1},\tau_{lc})$ is a topological inverse semigroup. This example shows that there is a gap in \cite[Theorem~2.9]{Hogan-1984}, where author stated that for every ordinal $\alpha$ there is only discrete locally compact inverse semigroup topology on $\mathcal{B_{\alpha}}$.
Most of the problems encountered so far have involved convex sets, or other sets with considerable intrinsic geometric structure. Nevertheless, one can study the geometry of much more general objects, for example, sets that are just closed or Lebesgue measurable, or even completely arbitrary. This chapter contains a selection of problems which, at least at first glance, are of such a general nature.
It is shown how the logical status of axioms and of the imaginary number i are isomorphic. Logical meanings for various number forms are discussed. Exponentials of 2 and of e can be shown to refer to discontinuous and continuous designations for their exponents, respectively. The equal symbol in equations can be mapped to transfinite ordinals, so that exponentials of such transfinite quantities labelled with i, can refer to the set of all sets of physical situations to which an axiom, in the form of an equation, applies. This may be useful in interpreting and unifying Psi and Electromagnetic wave functions.
The building blocks of mathematics are sets, yet no one should say what a set is. An axiomatic treatment of set theory postulates the existence of certain undefined or primitive objects, called sets, together with symbols and axioms governing their use. A graphic analogy can be made with geometry where there are given undefined objects called points, lines, and planes, together with a collection of axioms relating these objects to each other. An axiom of projective geometry is P: If L
1and L
2are distinct lines, there is one and only one point P on both L
1and L
2. Here the relation “P is on L
1” is undefined. Another axiom is P*: If P
1and P
2are distinct points, there is one and only one line L on both P
1and P
2. Any collection of objects which, when properly named as “points” and “lines”, satisfy the axioms of geometry, serve as points and lines, equally as well as any other such collection. It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug (David Hilbert, quoted by Weyl [44, p. 635]). Nevertheless, from the axioms we soon discover that some objects in a geometry are not “points” and some objects are not “lines”. Analogously in the theory of sets, after we have named our candidates for sets, and listed the undefined symbols and the axioms governing them, we find that not everything in sight is a set. For example, as Cantor observed, there is no such set as “the set of all sets”.
A presentation is made showing how imaginary numbers, exponentials, and transfinite ordinals can be given logical meanings that are applicable to the definitions for the axioms of Quantum Mechanics (QM). This is based on a proposed logical definition for axioms which includes an axiom statement and its negation as parts of an undecidable statement which is forced to the tautological truth value: true. The logical algebraic expression for this is shown to be isomorphic to the algebraic expression defining the imaginary numbers ± i (V-l). This supports a progressive and Hegelian view of theory development. This means that thesis and antithesis axioms in the QM theory structure which should be carried along at present could later on be replaced by a synthesis to a deeper theory prompted by subsequently discovered new experimental facts and concepts. This process :ould repeat at a later time since the synthesis theory axioms would then be considered as a lew set of thesis statements from which their paired antithesis axiom statements would be derived. The present epistemological methods of QM, therefore, are considered to be a good way of temporarily leapfrogging defects in our conceptual and experimental knowledge until a deeper determinate theory is found.
These considerations bring logical meaning to exponential forms like the Psi and wave functions. This is derives from the set theoretic meaning for simple forms like 2 which is blown to be the set of all subsets of the (discrete) set, A. The equal symbol in equations which are axioms, and all its other symbols, can be mapped to a transfinite ordinal, [maginary exponential forms (like e*″) can be shown to stand for the (continuous) set of all subsets or the set of all experimental situations (which thus includes arbitrary sets of experimental situations) which are based on the axiom, 0, a transfinite ordinal.
We study a transfinite generalization of the ordinal (Hessenberg) natural sum
obtained by taking suprema at limit stages. We show that a transfinite natural
sum differs from the more usual transfinite ordinal sum only for a finite
number of iteration steps. We show that the transfinite natural sum of a
sequence of ordinals can be obtained as a "mixed sum" (in an order-theoretical
sense) of those ordinals, in fact, it is the largest mixed sum which satisfies
a finiteness condition, relative to the ordering of the sequence.
Let me start with the following quotation from Mostowski: Tarski, in oral discussions, has often indicated his sympathies with nominalism. While he never accepted the ‘reism’ of Tadeusz Kotarbiński, he was certainly attracted to it in the early phase of his work. However, the set-theoretical methods that form the basis of his logical and mathematical studies compel him constantly to use the abstract and general notions that a nominalist seeks to avoid. In the absence of more extensive publications by Tarski on philosophical subjects, the conflict appears to have remained unresolved.2 My aim in this paper is to throw light on this cognitive conflict or dissonance of Tarski between his nominalistic and empiricistic sympathies and his “Platonic” mathematical practice as well as why he was so parsimonious in expressing his philosophical views.
This paper examines the problem of aggregating infinite utility streams with a social welfare function that respects the Hammond Equity and Weak Pareto axioms. The paper provides a complete characterization of domains (of the one period utilities) on which such an aggregation is possible. A social welfare function satisfying the Hammond Equity and Weak Pareto axioms exists on precisely those domains which are well-ordered sets in which the elements of the set are ordered according to the decreasing magnitude of the numbers belonging to the set. We show through examples how this characterization can be applied to obtain a number of results in the literature, as well as some new ones.
As far as algebraic properties are concerned, the usual addition on the class
of ordinal numbers is not really well behaved; for example, it is not
commutative, nor left cancellative etc. In a few cases, the natural Hessemberg
sum is a better alternative, since it shares most of the usual properties of
the addition on the naturals.
A countably infinite version of the natural sum has been used in a recent
paper by V\"a\"an\"anen and Wang, with applications to infinitary logics. We
provide an order theoretical characterization of this operation. We show that
this countable natural sum differs from the more usual infinite ordinal sum
only for an initial finite "head" and agrees on the remaining infinite "tail".
Various kinds of infinite mixed sums of ordinals are discussed.
The aim of this paper is to investigate the relationship between the Abelian groups G with Ext ( G , G ( κ ) ) = 0 ${\operatorname{Ext}(G,G^{(\kappa )}) = 0}$ and those satisfying Ext ( G , G ( κ ) ) ≅ Ext ( G , G ) ( κ ) ${\operatorname{Ext}(G,G^{(\kappa )})\cong \operatorname{Ext}(G,G)^{(\kappa )}}$ for various cardinals κ. In particular, we will give the structure of self-Ext-small groups. Our results answer several open problems, and give new proofs of known theorems concerning ⊕ $\oplus $ -splitters, tilting groups and finitely faithful S -groups.
We characterize the abelian groups G for which Ext(G, -) commutes with direct sums.
An open problem posed by John H. Conway in [2] was whether one could, on his system of numbers and games, ‘… define operations of addition and multiplication which will
restrict on the ordinals to give their usual operations’. Such a definition for addition was later given in [4], and this paper will show that a definition also exists for multiplication. An ordinal exponentiation operation is also
considered.
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