9 research outputs found
The formal verification of the ctm approach to forcing
We discuss some highlights of our computer-verified proof of the
construction, given a countable transitive set-model of , of
generic extensions satisfying and
. Moreover, let be the set of instances
of the Axiom of Replacement. We isolated a 21-element subset
and defined
such that for every
and -generic , implies , where is Zermelo set theory
with Choice.
To achieve this, we worked in the proof assistant Isabelle, basing our
development on the Isabelle/ZF library by L. Paulson and others.Comment: 20pp + 14pp in bibliography & appendices, 2 table
Applicable Mathematics in a Minimal Computational Theory of Sets
In previous papers on this project a general static logical framework for
formalizing and mechanizing set theories of different strength was suggested,
and the power of some predicatively acceptable theories in that framework was
explored. In this work we first improve that framework by enriching it with
means for coherently extending by definitions its theories, without destroying
its static nature or violating any of the principles on which it is based. Then
we turn to investigate within the enriched framework the power of the minimal
(predicatively acceptable) theory in it that proves the existence of infinite
sets. We show that that theory is a computational theory, in the sense that
every element of its minimal transitive model is denoted by some of its closed
terms. (That model happens to be the second universe in Jensen's hierarchy.)
Then we show that already this minimal theory suffices for developing very
large portions (if not all) of scientifically applicable mathematics. This
requires treating the collection of real numbers as a proper class, that is: a
unary predicate which can be introduced in the theory by the static extension
method described in the first part of the paper
Play Among Books
How does coding change the way we think about architecture? Miro Roman and his AI Alice_ch3n81 develop a playful scenario in which they propose coding as the new literacy of information. They convey knowledge in the form of a project model that links the fields of architecture and information through two interwoven narrative strands in an “infinite flow” of real books
Safety and Reliability - Safe Societies in a Changing World
The contributions cover a wide range of methodologies and application areas for safety and reliability that contribute to safe societies in a changing world. These methodologies and applications include: - foundations of risk and reliability assessment and management
- mathematical methods in reliability and safety
- risk assessment
- risk management
- system reliability
- uncertainty analysis
- digitalization and big data
- prognostics and system health management
- occupational safety
- accident and incident modeling
- maintenance modeling and applications
- simulation for safety and reliability analysis
- dynamic risk and barrier management
- organizational factors and safety culture
- human factors and human reliability
- resilience engineering
- structural reliability
- natural hazards
- security
- economic analysis in risk managemen
Play Among Books
How does coding change the way we think about architecture? Miro Roman and his AI Alice_ch3n81 develop a playful scenario in which they propose coding as the new literacy of information. They convey knowledge in the form of a project model that links the fields of architecture and information through two interwoven narrative strands in an “infinite flow” of real books