In his note [5] Hausner states a simple combinatorial principle, namely: $$(H)\left\{ {\begin{array}{*{20}c} {if f is a function a non - empty finite set \sigma into itself, p a} \\ {prime, f^p = i_\sigma and \sigma _0 the set of fixed points of f, then } \\ {\left| \sigma \right| \equiv \left| {\sigma _0 } \right|(mod p).} \\\end{array}} \right.$$ .He then shows how this principle can be used to prove:Fermat's little theorem,Cauchy's theorem for finite groups,Lucas' theorem for binomial numbers.Letε=(0,1, ...),ℱ (...) 1−1 the family of all one-to-one functions from a subset ofε intoε andℳ 1−1 the family of all p.r. (i.e., partial recursive) one-to-one functions from a subset ofε intoε. A subsetα of ε is finite, ifα is not equivalent to a proper subset under a function inℱ 1−1. The setα is calledisolated, if it is not equivalent to a proper subset under a function inℳ 1−1. An isolated set can also be defined as a subset ofε which has no infinite r.e. (i.e., recursively enumerable) subset. While every finite set is isolated, there are $c = 2^{\aleph _0 }$ infinite sets which are isolated; these sets are calledimmune. It is the purpose of this paper to generalize (H) to a principle (H*) for isolated sets and to show how (H*) can be used to prove generalizations of Fermat's little theorem and Cauchy's theorem for finite groups. We have been unable to generalize Lucas' theorem in a similar manner. (shrink)

A nonnegative interger is called a number, a collection of numbers a set and a collection of sets a class. We write ε for the set of all numbers, o for the empty set, N(α) for the cardinality of $\alpha, \subset$ for inclusion and $\subset_+$ for proper inclusion. Let α, β 1 ,...,β k be subsets of some set ρ. Then α' stands for ρ-α and β 1 ⋯ β k for β 1 ∩ ⋯ ∩ β k . For (...) subsets α 1 ,..., α r of ρ we write: $\alpha_\sigma - \{x \in v \ (\nabla i) \lbrack i \in \sigma \Rightarrow x \in \alpha_i\rbrack\} \text{for} \sigma \subset (1, \ldots, r),\\ s_i = \sum \{N(\alpha_\sigma) \mid \sigma \subset (1,\ldots, r) \& N(\sigma) = i\}, \text{for} 0 \leqq i \leqq r$ . Note that α 0 = v, hence s 0 = N(v). If the set v is finite, the classical inclusion-exclusion principle (abbreviated IEP) states $(a) N(\alpha_1 \cup \cdots \cup \alpha_r) = \sum^r_{t=1} (-1)^{t-1} s_t = \sum_{o \subset_+\sigma \subset (1,\ldots,r)}$ (b) N(α' 1 ⋯ α' r ) = ∑ r t=0 (-1) t s t = ∑ (-1) N(σ) N (α σ ). In this paper we generalize (a) and (b) to the case where α 1 ,⋯, α r are subsets of some countable but isolated set v. Then the role of the cardinality N(α) of the set α is played by the RET (recursive equivalence type) Req α of α. These generalization of (a) and (b) are proved in § 3. Since they involve recursive distinctness, this notion is discussed in § 2. In § 4l we consider a natural extension of "the sum of the elements of a finite set σ" to the case where σ is countable. § 5 deals with valuations, i.e., certain mappings μ from classes of isolated sets into the collection λ of all isols which permit us to further generalize IEP by substituting μ(α) for $\operatorname{Req} \alpha$. (shrink)

This paper deals primarily with countable, simple, connected graphs and the following two conditions which are trivially satisfied if the graphs are finite: (a) there is an edge-recognition algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to decide whether they are adjacent, (b) there is a shortest path algorithm, i.e., an effective procedure which enables us, given two distinct vertices, to find a minimal path joining them. A graph $G = \langle\eta, \eta\rangle$ with η as set (...) of vertices and η as set of edges is an α-graph if it satisfies (a); it is an ω-graph if it satisfies (b). G is called r.e. (isolic) if the sets ν and η are r.e. (isolated). While every ω-graph is an α-graph, the converse is false, even if G is r.e. or isolic. Several basic properties of finite graphs do not generalize to ω-graphs. For example, an ω-tree with more than one vertex need not have two pendant vertices, but may have only one or none, since it may be a 1-way or 2-way infinite path. Nevertheless, some elementary propositions for finite graphs $G = \langle\nu, \eta\rangle$ with $n = \operatorname{card}(\nu), e = \operatorname{card}(\eta)$ , do generalize to isolic ω-graphs, e.g., n - 1 ≤ e ≤ n(n - 1)/2. Let N and E be the recursive equivalence types of ν and η respectively. Then we have for an isolic α-tree $G = \langle\nu, \eta\rangle: N = E + 1$ iff G is an ω-tree. The theorem that every finite graph has a spanning tree has a natural, effective analogue for ω-graphs. The structural difference between isolic α-graphs and isolic ω-graphs will be illustrated by: (i) every infinite graph is isomorphic to some isolic α-graph; (ii) there is an infinite graph which is not isomorphic to any isolic ω-graph. An isolic α-graph is also called a twilight graph. There are c such graphs, c denoting the cardinality of the continuum. (shrink)

In this paper we generalize the pigeonhole principle by using isols as our fundamental counting tool.

In this paper we discuss the following contributions to recursion theory made by John Myhill: two sets are recursively isomorphic iff they are one-one equivalent; two sets are recursively isomorphic iff they are recursively equivalent and their complements are also recursively equivalent; every two creative sets are recursively isomorphic; the recursive analogue of the Cantor–Bernstein theorem; the notion of a combinatorial function and its use in the theory of recursive equivalence types.

In their 2007 paper, Swierstra and Rip identify characteristic tropes and patterns of moral argumentation in the debate about the ethics of new and emerging science and technologies (or “NEST-ethics”). Taking their NEST-ethics structure as a starting point, we considered the debate about tissue engineering (TE), and argue what aspects we think ought to be a part of a rich and high-quality debate of TE. The debate surrounding TE seems to be predominantly a debate among experts. When considering the NEST-ethics (...) arguments that deal directly with technology, we can generally conclude that consequentialist arguments are by far the most prominently featured in discussions of TE. In addition, many papers discuss principles, rights and duties relevant to aspects of TE, both in a positive and in a critical sense. Justice arguments are only sporadically made, some “good life” arguments are used, others less so (such as the explicit articulation of perceived limits, or the technology as a technological fix for a social problem). Missing topics in the discussion, at least from the perspective of NEST-ethics, are second “level” arguments—those referring to techno-moral change connected to tissue engineering. Currently, the discussion about tissue engineering mostly focuses on its so-called “hard impacts”—quantifiable risks and benefits of the technology. Its “soft impacts”—effects that cannot easily be quantified, such as changes to experience, habits and perceptions, should receive more attention. (shrink)

In their 2007 paper, Swierstra and Rip identify characteristic tropes and patterns of moral argumentation in the debate about the ethics of new and emerging science and technologies (or “NEST-ethics”). Taking their NEST-ethics structure as a starting point, we considered the debate about tissue engineering (TE), and argue what aspects we think ought to be a part of a rich and high-quality debate of TE. The debate surrounding TE seems to be predominantly a debate among experts. When considering the NEST-ethics (...) arguments that deal directly with technology, we can generally conclude that consequentialist arguments are by far the most prominently featured in discussions of TE. In addition, many papers discuss principles, rights and duties relevant to aspects of TE, both in a positive and in a critical sense. Justice arguments are only sporadically made, some “good life” arguments are used, others less so (such as the explicit articulation of perceived limits, or the technology as a technological fix for a social problem). Missing topics in the discussion, at least from the perspective of NEST-ethics, are second “level” arguments—those referring to techno-moral change connected to tissue engineering. Currently, the discussion about tissue engineering mostly focuses on its so-called “hard impacts”—quantifiable risks and benefits of the technology. Its “soft impacts”—effects that cannot easily be quantified, such as changes to experience, habits and perceptions, should receive more attention. (shrink)

In transfinite arithmetic 2n is defined as the cardinality of the family of all subsets of some set v with cardinality n. However, in the arithmetic of recursive equivalence types 2N is defined as the RET of the family of all finite subsets of some set v of nonnegative integers with RET N. Suppose v is a nonempty set. S is a class over v, if S consists of finite subsets of v and has v as its union. Such a (...) class is an intersecting class over v, if every two members of S have a nonempty intersection. An IC over v is called a maximal IC , if it is not properly included in any IC over v. It is known and readily proved that every MIC over a finite set v of cardinality n ≥ 1 has cardinality 2n-1. In order to generalize this result we introduce the notion of an ω-MIC over v. This is an effective analogue ot the notion of an MIC over v such that a class over a finite set v is an ω-MIC iff it is an MIC. We then prove that every ω-MIC over an isolated set v of RET N ≥ 1 has RET 2N-1. This is a generalization, for while there only are χ0 finite sets, there are ϰ isolated sets, where c denotes the cardinality of the continuum, namely all the finite sets and the c immune sets. MSC: 03D50. (shrink)

We present here a direct elementary construction of continuous utility functions on perfectly separable totally preordered sets that does not make use of the well-known Debreu’s open gap lemma. This new construction leans on the concept of a separating countable decreasing scale. Starting from a perfectly separable totally ordered structure, we give an explicit construction of a separating countable decreasing scale, from which we show how to get a continuous utility map.

Rare genetic diseases collectively impact millions of individuals in the United States. These patients and their families share many challenges including delayed diagnosis, lack of knowledgeable providers, and limited economic incentives to develop new therapies for small patient groups. As such, rare disease patients and families often must rely on advocacy, including both self-advocacy to access clinical care and public advocacy to advance research. However, these demands raise serious concerns for equity, as both care and research for a given disease (...) can depend on the education, financial resources, and social capital available to the patients in a given community. In this article, we utilize three case examples to illustrate ethical challenges at the intersection of rare diseases, advocacy and justice, including how reliance on advocacy in rare disease may drive unintended consequences for equity. We conclude with a discussion of opportunities for diverse stakeholders to begin to address these challenges. (shrink)