On Elementary Theories of Ordinal Notation Systems based on Reflection Principles

We consider the constructive ordinal notation system for the ordinal ${\epsilon_0}$ that were introduced by L.D. Beklemishev. There are fragments of this system that are ordinal notation systems for the smaller ordinals ${\omega_n}$ (towers of ${\omega}$-exponentiations of the height $n$). This systems are based on Japaridze's provability logic $\mathbf{GLP}$. They are closely related with the technique of ordinal analysis of $\mathbf{PA}$ and fragments of $\mathbf{PA}$ based on iterated reflection principles. We consider this notation system and it's fragments as structures with the signatures selected in a natural way. We prove that the full notation system and it's fragments, for ordinals ${\ge\omega_4}$, have undecidable elementary theories. We also prove that the fragments of the full system, for ordinals ${\le\omega_3}$, have decidable elementary theories. We obtain some results about decidability of elementary theory, for the ordinal notation systems with weaker signatures.Comment: 23 page

Jesus’ Principles of Breaking Barriers: A Reflection on John 4:3-42

This paper investigates Jesus’ Principles of Breaking Barriers: A Reflection on John 4:3-42 as a pivot and principle for breaking barriers from biblical perspective. Worthy of note is the fact that there existed socio-ethnic barrier between the Jews and Samaria; gender barrier imposed by Jesus’ androcentric culture and a moral barrier imposed by the Samaritan woman’s assumed behaviour. This paper using hermeneutical theory of biblical interpretation and historical insight into this Johannine text brings to the fore that, the Jewish-Samaritan barrier had ethnic, religious and political undertones which made the Samaritans and Jews to see themselves as ‘enemies’; the interaction with the Samaritan woman broke barriers and set a new agenda for relationship with ‘inferior’ gender and morally low in the human society. Herein exists principles of breaking barriers set by Jesus which are worthy of emulation in the twenty-first century which is plagued by socio-ethnic, religious, gender and moral barriers

The Physics of Bicycling

Faculty reflection on VCU Great Bike Race Book course. Course Description: The science of bicycling will be explored by measuring and observing key physical principles of the cyclist

Reflection using the derivability conditions

Reflection principles are a way to build non-conservative true extensions of a theory. However the application of a reflection principle needs a proof predicate, and the effort needed to provide this is so great as to be not really practical. We look at a possible way to avoid this effort by using, instead of a proof predicate, a predicate defined using only necessary `modal' properties. Surprisingly, we can produce powerful non-conservative extensions this way. But a reflection principle based on such a predicate is essentially weaker, and we also consider its limitations