231 research outputs found
Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse
We work with symmetric extensions based on L\'{e}vy Collapse and extend a few
results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her
P.h.d. thesis. We also observe that if is a model of ZFC, then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -distributive and is -complete.
Further we observe that if is a model of ZF + , then
can be preserved in the symmetric extension of in terms of
symmetric system , if
is -strategically closed and is
-complete.Comment: Revised versio
Notions of Relative Ubiquity for Invariant Sets of Relational Structures
Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers w as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on w. For example, in every sense of relative ubiquity considered here, the set of dense linear orderings on w is ubiquitous in the set of linear orderings on w
LOGICAL FOUNDATIONS OF PHYSICS. PART I. ZENO’S DICHOTOMY PARADOX,SUPERTASKS AND THE FAILURE OF CLASSICAL INFINITARY REASONING
Several variants of Zeno’s dichotomy paradox are considered, with the objective of exploring the logical foundations of physics. It is shown that Zeno’s dichotomy paradox leads to contradictions at the metamathematical (as opposed to formal) level in the basic classical infinitary reasoning that is routinely used in theoretical physics. Both Newtonian mechanics and special relativity theory suffer from these metamathematical inconsistencies, which occur essentially because the classical refutation of the
dichotomy paradox requires supertasks to be completed. In previous papers, Non-Aristotelian Finitary Logic (NAFL) was proposed as a logical foundation for some of the basic principles of quantum mechanics, such as, quantum superposition and entanglement. We outline how the finitistic and paraconsistent reasoning used in NAFL helps in resolving the metamathematical inconsistencies that arise from the dichotomy paradox
- …