would mostowski have liked it ...niwinski/haifa/mirna.pdf · the kojman-shelah method is in fact...

Would Mostowski have liked it?

Some Perspectives in Mathematical Logic

Mirna Džamonja, University of East Anglia

Norwich, UK

samedi 12 octobre 13

In this hommage to Andrzej Mostowski, whom I did not have the pleasure to know, I use the opportunity to acknowledge the great contribution of the Polish mathematicians to the worldwide mathematics, in particular Mostowski’s generation which saw some really hard times.


Theorem. (Dž. + Kunen 1993, Fundamenta Mathematicae)

Under CH there is a compact hereditarily separable and hereditarily Lindelöf space with a measure of an uncountable type.

Ken Kunen, University of Wisconsin-Madison


Mostowski started his work in topology, working with Kuratowski. In fact he finished his thesis with Tarski and moved to logic.


Logic

(Set theory and Model theory)

used in Topology and Analysis


Theorem (Malliaris-Shelah, annoucement PANS 2013, submitted) p=t.

p=pseudointersection numbert=tower number

The proof uses models of ZFC, ultrapowers: model theory, set theory, forcing and absolutness

and topology. «General topology meets model theory: on p and t»


t is the smallest size of a «tower»: a sequence of infinite subsets of N decreasing mod finite, with no infinite

pseudo-intersection.

p is the smallest size of a family of infinite subsets ofN in which every non-empty finite subfamily has an

infinite intersection, but the family does not have an infinite pseudo-intersection.

pseudo-intersection= a set contained mod finite in each member of the family


strict order property =) SOPn+1 =) SOPn =) not simple =)not stable

(1) classification theory of Shelah

(2) assymptotic classification theory by Keisler’s order (compare regular ultrapowers): the main result of

Malliaris-Shelah

SOP 3 =) C⇤ �maximality () SOP 2 =) SOP 1 =) not simple

(3) The Malliaris-Shelah proof, uses ultrapowers and models of ZF, was known before under GCH (Dž.-Shelah 2004) by a much inferior proof

SOP 2 =) C⇤ �maximality

(4) Open questions: which implications below are true?

Connection: from ultrapowers to cardinal invariants


Theorem (Shelah, Usvyatsov 2008) Suppose that ✓ and � are two regular

cardinals with @2 ✓ < ✓+ < �, and that (8 < �)@0 < �.Then the minimal number of Banach spaces of of density � needed to embed

such spaces isometrically is 2

✓. In particular, if 2

✓ > � then there is no

isometrically universal Banach space �.

Universal Banach spaces


Ingredients: serious model theory, Robinson theories, strict order property. POINT: can give a model theoretic definition of ``isometry``.

The model theoretic approach does not apply to isomorphisms. At all!

Theorem (Kojman, Shelah 1992) Suppose that ✓ and � are two regular

cardinals with @2 ✓ < ✓+ < �.Then the minimal number of linear orders of size � needed to embed all

linear orders of size � is 2

✓.


The Kojman-Shelah method is in fact set-theoretic, it was one of the first uses of pcf theory, in particular club guessing.

Well, if model theory did not help us with isomorphisms, let us go back to the set-theoretic origin and try to go directly to the isomorphisms.


Theorem Suppose that ✓ and � are two regular cardinals with @2 ✓ <✓+ < �, and that (8 < �)@0 < �.

Then

(1) the minimal number of spaces of the form C(St(A)) of density � needed to

embed all Banach spaces of the form C(St(B)) of density � very positively

is 2

✓. In particular, if 2

✓ > � then there is no very-positively universal

space C(St(A)) of density �.

(2) if 2

✓ > � then there are at least cf(2

✓) pairwise non-very positively isomor-

phic Banach spaces C(K) of density �.

submitted 2013

Open question: is the universality number under isomorphisms always the same as that under isometries?


Theorem (Bankston 2009) The continua X on which every semi-monotone mapping from X

onto X is monotone, are exactly the locally connected continua.

Def. A continuum is a connected compact Hausdorff space.

CONTINUUA THEORY


A mapping f:X--> Y is monotone if the preimage of every subcontinuum is a subcontinuum.

A mapping f:X--> Y is semi-monotone if for every subcontinuum K of Y there is a subcontinuum C of

X with f[C]=K and for every U open containedin K, the preimage of U is a subset of C.


Perspectives


Logic is the process of modelling a thought (human, mathematical, lingustic, computer ...)

Mathematical logic is a subject bridging mathematics and logic. We need both of them- let us not neglect either.

We should not be afraid of our techniques. Beauty in mathematics (and in life?) can be reached only by making our hands dirty.


.... A. Mostowski, Fundamenta 1955 (cited 11 times)


would mostowski have liked it ...niwinski/haifa/mirna.pdf · the kojman-shelah method is in fact...

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