We now know of a number of ways of developing Real Analysis on a basis of abstraction principles and second-order logic. One, outlined by Shapiro in his contribution to this volume, mimics Dedekind in identifying the reals with cuts in the series of rationals under their natural order. The result is an essentially structuralist conception of the reals. An earlier approach, developed by Hale in ...

We investigate the frontline of Gödel’s incompleteness theorems’ proofs and the links with computability. The Gödel incompleteness phenomenon Gödel’s incompleteness theorems [Göd31, SFKM+86] are milestones in the subject of mathematical logic. Apart from Gödel’s original syntactical proof, many other proofs have been presented. Kreisel’s proof [Kre68] was the first with a model-theoretical flav...

We formalise results from computability theory in the theorem prover Isabelle/HOL. Following the textbook by Boolos et al, we formalise Turing machines and relate them to abacus machines and recursive functions. We “tie the knot” between these three computational models by formalising a universal function and obtaining from it a universal Turing machine by our verified translation from recursiv...

Aim of the paper is to revise Boolos’ reinterpretation of second-order monadic logic in terms of plural quantification ([4], [5]) and expand it to full second order logic. Introducing the idealization of plural acts of choice, performed by a suitable team of agents, we will develop a notion of plural reference. Plural quantification will be then explained in terms of plural reference. As an app...

One reason why Lucas and Penrose should not be faulted for continuing to believe in their well-known Gödelian arguments against computationalism lies in the lack of an adequate consensus on the term ‘computation’. For instance, Boolos, Burgess and Jeffrey (2003: Computability and Logic, 4th ed. CUP, p37) define a diagonal function, d, any value of which can be computed effectively, although the...

Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. The fundamental concepts of neutrosophic set, introduced by Smarandache in [30, 31, 32] and Salama et al. in [4-29]. In Geographical information systems (GIS) there is a need to model spatial regions with indeterminate boundary and under indeterminacy. In this paper the structure of some cl...

Dear Editor, Recently, a paper was published (Salama et al., 2008) describing the effects of electromagnetic fields from a mobile phone on testicular functions in New Zealand White (NZW) rabbits. We have a number of concerns; the two most important ones deal with the exposure conditions and the biological effects respectively. Exposure was carried out by a GSM mobile phone at 800 MHz in ‘standb...

Since the world is full of indeterminacy, the neutrosophics found their place into contemporary research. The fundamental concepts of neutrosophic set, introduced by Smarandache in [7, 8]. Recently, Salama et al. in [14, 15, 16, 32 ], introduced the concept of correlation coefficient of neutrosophic data. In this paper, we introduce and study the concepts of correlation and correlation coeffici...