We define a direct translation from finite rooted trees to finite natural functions which shows that the Worm Principle introduced by Lev Beklemishev is equivalent to a very slight variant of the well-known Kirby-Paris' Hydra Game. We further show that the elements in a reduction sequence of the Worm Principle determine a bad sequence in the well-quasi-ordering of finite sequences of natural numbers with respect to Friedman's gapembeddability. A characterization of gap-embeddability in terms of provability logic due to Lev Beklemishev (...) is also presented. (shrink)

We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics.

The paper focuses on the semiotic principles of the organisation of ecosystems, attempting to find concepts that point to relations and not to elements. (1) Consortium (the term introduced by Johannes Reinke around 1873) can be defined as a group of organisms connected via (sign) relations, or groups of interspecific semiosic links in biocoenosis. The consortial relations include trophic and topic relations, both implying a recognition (identification) of the object by an organism involved (these, i.e., are sign relations). These relations (...) are ecologically inheritable. (2) Umwelt (the term introduced by Jakob von Uexküll around 1909) can be defined as a set of relations an organism has in an ecosystem (as in a semiosphere). The formation of an umwelt is dependent on the modelling system of the organism. (3) Biophony (the term introduced by Bernie Krause around 2000) denotes the coordination of inter- and intra-species relations in a soundscape of a biological community. This can be seen as a special case of Komposition as defined by Jakob and Thure von Uexküll. (4) Ecological code (as introduced, e.g. by Alexander Levich around 1977) can be defined as the set of (sign) relations (regular irreducible correspondences) characteristic to an entire ecosystem. We also mention the concepts of ecomones and coactones (introduced by Marcel Florkin in 1965) as the substances which are responsible for mediation of ecological inter-individual relations. All the relations as sign-relations evidently imply both a static or structuralist description (in terms of codes), and a processual description (in terms of semiosis carried on by interpretation). We conclude that all the above mentioned concepts can be viewed as conceptually connected and are suitable for semiotic description of biological communities. (shrink)