Mengenlehre: Hierarchie der Unendlichkeiten
(Set
Theory: Hierarchy of Infinities)
at the ÖMG-Lehrerfortbildungstag (an event
organized by the Austrian Mathematical Society, for the further
education of high
school math teachers).
In this talk I presented a few elementary facts about the
notion of "cardinality".
The talk will be published by the ÖMG.
(pdf,
dvi and Postscript
files are also available.)
was written jointly with Saharon Shelah. Also
Metric, fractal dimensional and Baire Results on the
Distribution of Sequences and Subsequences
was written jointly with Jörg Schmeling and Reinhard Winkler.
["Mathematische Nachrichten" 219, 2000]
Interpolation of monotone functions in lattices
This paper has appeared in Algebra Universalis.
Interpolation in {0,1}-lattices,
For texing it you need latex2e and eepic.sty. Also
Interpolation in ortholattices,
For texing it you need latex2e and eepic.sty. Also
Most algebras have the interpolation property
.
I show here that if we fix a set A and a type ("language",
"signature") of algebras, the set of algebras with carrier set A that do not
have a strong interpolation property is [more or less] meager in a suitable
topology.
Order-polynomially complete lattices
must be large,
(For texing it you need latex2e).
Also a
In this paper we show that if there is an infinite lattice L
with the property that all order preserving functions from L^n to L
are induced by lattice-theoretic polynomials (i.e., "L is o.p.c."),
then the cardinality
of L must be an inaccessible cardinal.
There are no infinite
order polynomially complete lattices after all
Isn't that wonderful? The most wonderful thing about it is that
it does not make the previous paper about opc lattices
obsolete.
The essential part of the proof just takes one page. We then show that
this result cannot be proved without (some version of)
the axiom of choice.
Also
Lattices, Interpolation, and Set Theory
Also
Antichains in products of linear orders
we answer a question of
Haviar and Ploscica:
If L is a linear order, is it possible that two
different finite powers of L, say L2 and L3
have different antichain conditions, i.e., that for some infinite
cardinal kappa there is
an [incomparability-]antichain of cardinality in L3, but
not in L2? Or how about L17 vs
L16?
There are several variants of the question, and the
answer is in general yes:
Again
Tools for your forcing construction
In this paper I present some known and some new preservation
theorems for forcing,
mainly about countable support iteration.
A Taste of Proper Forcing
(pdf file)
is almost, but not quite, entirely unlike "Tools".
An Application of
Shoenfield's Absoluteness Theorem to the Theory of Uniform
Distribution
To my own surprise, I gave in this paper an application of a
theorem from set theory to a question in "normal" (also known as:
"naive") mathematics: Let X be the set of functions from the natural
numbers to the natural numbers, ordered pointwise. Assume that
(B_x: x in X) is a monotone family of measure zero sets (with some
nice definability properties). Then the union of all those
(uncountably many!) sets B_x is still of measure zero.
The Complexity of Fuzzy Logic
In this paper I investigate Lukasiewicz' infinite valued logic.
I
show that the set of 1-validities (i.e. formulas that have value 1 in
every fuzzy model) is a Pi-0-2 complete set.
I wrote this paper in 1994, after listening to an excellent
course on fuzzy logic by Petr Hajek.
A few months later I found out that Mathias Ragaz had proved
(but not published) the same result about 15 years earlier.
A clone on a set A is a family of finitary functions which contains all the
projections and is closed under composition. (In other words: the set
of term functions of some universal algebra over A.)
The family of all clones on a given set A forms a complete algebraic
lattice. Many interesting things are known about this lattice if
A is finite (for example a classification of all coatoms), but much less is
known for infinite sets A.
We investigate various families of coatoms in this lattice; it turns
out that if the cardinality of A is weakly compact, then a nice structure
can be found, and in (almost) all other cases we can show a "nonstructure"
result.
I show that every lattice can be imbedded into a
kappa-order polynomially complete lattice. kappa-opc means that
every monotone function can be interpolated my a polynomial
on any set of size kappa. Here, kappa is an arbitrary
(infinite)
cardinal number. This means that it is difficult to distinguish
between an arbitrary monotone
function and a polynomial function.
This paper will appear in Algebra Universalis in late 2000 or early 2001.
I show that (for every cardinal kappa, possibly infinite),
every ortholattice has an orthoextension
where every function can be interpolated
on any set of size kappa.
This paper is being written jointly with Saharon Shelah
(who is a bit unhappy that the solution turned out to be so easy...)
[Israel Math Conf.Proceedings vol 6, 1992]
[Monatshefte für Mathematik 1993]
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