# Martin Goldstern's Papers

This page is supposed to contain a list of all my mathematical papers (about 2 dozen of them) with links to tex/dvi/ps files, plus abstracts, bibliographical references, and perhaps more. For the moment, here are only a bibtex file, and a few papers:
• In April 2000 I gave a talk

(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.)

• A nontechnical introductory paper (in German) on Gödel's completeness theorem appeared in the journal "Internationale Mathematische Nachrichten" in 2006, and a bit later a paper on Gödel's constructible universe.
• The paper:

was written jointly with Saharon Shelah. Also

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.
• A paper on uniform distribution:

was written jointly with Jörg Schmeling and Reinhard Winkler. ["Mathematische Nachrichten" 219, 2000]

• Several papers on interpolation, namely
1. My first paper in universal algebra:

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.

2. A companion paper strengthens the result to bounded lattices:

For texing it you need latex2e and eepic.sty. Also

3. Finally, in my most recent paper I managed to prove a similar result for ortholattices.
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.

For texing it you need latex2e and eepic.sty. Also

4. This paper looks at interpolation from a more general point of view:

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.

5. The old problem of order-polynomially complete lattices was almost resolved in 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.

6. A few years later, we showed that if L is an infinite opc lattice, then the cardinality of L cannot be an inaccessible cardinal. Hence:

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

7. In a "survey" paper on 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

8. In a joint paper with Shelah

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

• my most popular paper (which also happens to be my habilitation thesis)
[Israel Math Conf.Proceedings vol 6, 1992]
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.

• The paper

A Taste of Proper Forcing (pdf file)

is almost, but not quite, entirely unlike "Tools".

• My favorite paper:
[Monatshefte für Mathematik 1993]
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.