Online research in philosophy

Tarski [5] showed that for any set X, its set w(X) of well-orderable subsets has cardinality strictly greater than that of X, even in the absence of the axiom of choice. We construct a Fraenkel-Mostowski model in which there is an infinite strictly descending sequence under the relation |w (X)| = |Y|. This contrasts with the corresponding situation for power sets, where use of Hartogs' ℵ-function easily establishes that there can be no infinite descending sequence under the relation |P(X)| = |Y|.

Recently Timothy Williamson asked ‘How probable is an infinite sequence of heads?’ In this paper, I suggest the probability of an infinite sequence of heads.

• Intelligent Tasks: Finding the Next Term of a Sequence • Difference Analysis of Polynomial Sequences • Charles Babbage’s Difference Engine • Finding the Form of the Sequence. • Gaussian Elimination. • Example Application: the Pie Cutting Sequence • What has this to do with Intelligence? • What has it all to do with Consciousness (if anything)?

Assume T is a small superstable theory. We introduce the notion of a flat Morley sequence, which is a counterpart of the notion of an infinite Morley sequence in a type p, in case when p is a complete type over a finite set of parameters. We show that for any flat Morley sequence Q there is a model M of T which is τ-atomic over {Q}. When additionally T has few countable models and is 1-based, we prove that within M there is an infinite Morley sequence I, with I $\subset$ dcl(Q), such that M is prime over I.

Can associative learning take place without awareness? We explore this issue in a sequence learning paradigm with amnesic and control participants, who were simply asked to react to one of four possible stimuli on each trial. Unknown to them, successive stimuli occurred in a sequence. We manipulated the extent to which stimuli followed the sequence in a deterministic manner (noiseless condition) or only probabilistically so (noisy condition). Through this paradigm, we aimed at addressing two central issues: first, we asked whether sequence learning takes place in either condition with amnesic patients. Second, we asked whether this learning takes place without awareness. To answer this second question, participants were asked to perform a subsequent sequence generation task under inclusion and exclusion conditions, as well as a recognition task. Reaction times results show that amnesic patients learned the sequence only in the deterministic condition. However, they failed to be able to reproduce the sequence in the generation task. In contrast, we found learning for both sequence structures in control participants, but only control participants exposed to a deterministic sequence were successful in performing the generation task, thus suggesting that the acquired knowledge can be used consciously in this condition. Neither amnesic nor control participants showed correct old/new judgments in the recognition task. The results strengthen the claim that implicit learning is at least partly spared in amnesia, and the role of contextual information available for learning is discussed. © 2006 Elsevier Ltd. All rights reserved.

Sequential behavior is essential to intelligence, and it is a fundamental part of human activities ranging from reasoning to language, and from everyday skills to complex problem solving. In particular, sequence learning is an important component of learning in many task domains — planning, reasoning, robotics, natural language processing, speech recognition, adaptive control, time series prediction, financial engineering, DNA sequencing, and so on. Naturally, there are many different approaches towards sequence learning, resulting from different perspectives taken in different task domains. These approaches deal with somewhat differently formulated sequential learning problems (for example, some with actions and some without), and/ or different aspects of sequence learning (for example, sequence prediction vs. sequence recognition). Sequence learning is clearly a difiicult task. More powerful algorithms for sequence learning are needed in all of these afore-mentioned domains. It is our view that the right approach to develop better techniques, algorithms, models.

We study increasing F-sequences, where F is a dilator: an increasing F-sequence is a sequence (indexed by ordinal numbers) of ordinal numbers, starting with 0 and terminating at the first step x where F(x) is reached (at every step x + 1 we use the same process as in decreasing F-sequences, cf. [2], but with "+ 1" instead of "- 1"). By induction on dilators, we shall prove that every increasing F-sequence terminates and moreover we can determine for every dilator F the point where the increasing F-sequence terminates. We apply these results to inverse Goodstein sequences, i.e. increasing (1 + Id) (ω) -sequences. We show that the theorem every inverse Goodstein sequence terminates (a combinatorial theorem about ordinal numbers) is not provable in ID 1 . For a general presentation of the results stated in this paper, see [1]. We use notions and results concerning the category ON (ordinal numbers), dilators and bilators, summarized in [2, pp. 25-31].

We define DNA sequence by a bottom-up approach, starting with a real sequence from an actual biological sample. By providing axioms for notions of string, substring and strand, we formally define a DNA sequence, and a DNA molecule as composed of two antiparallel strands. We note that a sequence is a kind of group in which each member stands a certain relation to every other. The spatial aspects of a DNA sequence are also described.