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)

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

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)

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 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)

This article reviews the strengths and limitations of five major paradigms of medical computer-assisted decision making (CADM): (1) clinical algorithms, (2) statistical analysis of collections of patient data, (3) mathematical models of physical processes, (4) decision analysis, and (5) symbolic reasoning or artificial intelligence (Al). No one technique is best for all applications, and there is recent promising work which combines two or more established techniques. We emphasize both the inherent power of symbolic reasoning and the promise of artificial intelligence (...) and the other techniques to complement each other. (shrink)