7,417 research outputs found
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
On what I do not understand (and have something to say): Part I
This is a non-standard paper, containing some problems in set theory I have
in various degrees been interested in. Sometimes with a discussion on what I
have to say; sometimes, of what makes them interesting to me, sometimes the
problems are presented with a discussion of how I have tried to solve them, and
sometimes with failed tries, anecdote and opinion. So the discussion is quite
personal, in other words, egocentric and somewhat accidental. As we discuss
many problems, history and side references are erratic, usually kept at a
minimum (``see ... '' means: see the references there and possibly the paper
itself).
The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The
other half, concentrating on model theory, will subsequently appear
Spectra of Monadic Second-Order Formulas with One Unary Function
We establish the eventual periodicity of the spectrum of any monadic
second-order formula where:
(i) all relation symbols, except equality, are unary, and
(ii) there is only one function symbol and that symbol is unary
Elementary submodels in infinite combinatorics
The usage of elementary submodels is a simple but powerful method to prove
theorems, or to simplify proofs in infinite combinatorics. First we introduce
all the necessary concepts of logic, then we prove classical theorems using
elementary submodels. We also present a new proof of Nash-Williams's theorem on
cycle-decomposition of graphs, and finally we improve a decomposition theorem
of Laviolette concerning bond-faithful decompositions of graphs
Ultrafilter convergence in ordered topological spaces
We characterize ultrafilter convergence and ultrafilter compactness in
linearly ordered and generalized ordered topological spaces. In such spaces,
and for every ultrafilter , the notions of -compactness and of
-pseudocompactness are equivalent. Any product of initially
-compact generalized ordered topological spaces is still initially
-compact. On the other hand, preservation under products of certain
compactness properties are independent from the usual axioms for set theory.Comment: v. 2: some additions and some improvement
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