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.
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.
Interpolation in {0,1}-lattices,
For texing it you need latex2e and eepic.sty. Also
Interpolation in ortholattices,
I managed to prove a similar result for 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,
a joint work with Saharon Shelah. ["Algebra Universalis", 39, 1998.](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.
This paper is being written jointly with Saharon Shelah
(who is a bit unhappy that the solution turned out to be so easy...)
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
I review some of these results and add a few new points. I also try to make propaganda for the idea that set theory (and mathematical logic in general) can be useful in other areas of mathematics.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 L^{2 } and L^{3} have different antichain conditions, i.e., that for some infinite cardinal kappa there is an [incomparability-]antichain of cardinality in L^{3}, but not in L^{2}? Or how about L^{17} vs L^{16}?
There are several variants of the question, and the answer is in general yes:
Again
Tools for your forcing construction
You cannot tex this file, as you need some input files, and anyway chances are that your particular dialect of amstex will not be the one this paper wants. Anyway, here are:In this paper I present some known and some new preservation theorems for forcing, mainly about countable support iteration.
A Taste of Proper Forcing (Postscript file)
is almost, but not quite, entirely unlike "Tools".
An Application of Shoenfield's Absoluteness Theorem to the Theory of Uniform Distribution
Again,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
(available at the arXiv).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.
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