# On Gross Spaces

@article{Shelah1995OnGS, title={On Gross Spaces}, author={Saharon Shelah and Otmar Spinas}, journal={arXiv: Logic}, year={1995} }

A Gross space is a vector space E of infinite dimension over some field F, which is endowed with a symmetric bilinear form � : E 2 → F and has the property that every infinite dimensional subspace U ⊆ E satisfies dimU ⊥ < dimE. Gross spaces over uncountable fields exist (in certain dimen- sions) (see (G/O)). The existence of a Gross space over countable or finite fields (in a fixed dimension not above the continuum) is independent of the axioms of ZFC. This was shown in (B/G), (B/Sp) and (Sp2…

## 4 Citations

Cardinal invariants of the continuum and combinatorics on uncountable cardinals

- MathematicsAnn. Pure Appl. Log.
- 2006

The Spectrum of the Γ-Invariant of a Bilinear Space☆

- Mathematics
- 1997

Abstract To every symmetric bilinear space (X, φ) of regular uncountable dimension κ, an invariant Γ(X, φ) ∈ P (κ)/ F (κ) (where F (κ) is the club filter) can be assigned. We prove that in…

Historic iteration with aleph_epsilon-support

- Computer Science
- 1996

One aim of this work is to get a universe in which weak versions of Martin axioms holds for some forcing notions of cardinality aleph_0, aleph_1 and aleph_2 while on aleph_2 club, the ``small''…

EVADING PREDICTORS WITH CREATURES

- Mathematics
- 1998

We continue the theory of evasion and prediction which was introduced by Blass and developed by Brendle, Shelah, and Laflamme. We prove that for arbitrary sufficiently different f, g ∈ ωω, it is…

## References

SHOWING 1-10 OF 38 REFERENCES

Iterated forcing in quadratic form theory

- Mathematics
- 1992

In [Sp1] and [B/Sp] it has been shown that the existence of quadratic spaces of uncountable dimension over finite or countable fields sharing the property that every infinite dimensional subspace has…

Independence and consistency proofs in quadratic form theory

- MathematicsJournal of Symbolic Logic
- 1991

These properties have been considered first in [G/O] in the process of investigating the orthogonal group of quadratic spaces, and it has been shown there (in ZFC) that over arbitrary uncountable fields (**)-spaces of unccountable dimension exist.

Strong negative partition relations below the continuum

- Mathematics
- 1991

PROOF. We use the idea in the proof of the Engelking-Karlowitz theorem. Assume tha t p A. Let { A a : a < A} be different subsets of p. Assume tha t c witnesses Pr(~) . Pu t G = {(w,g) :w e [p]<o,,…

Further cardinal arithmetic

- Mathematics
- 1996

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S≤ℵ0(κ), ⊆) for κ real valued measurable (Section 3),…

Souslin forcing

- MathematicsJournal of Symbolic Logic
- 1988

Abstract We define the notion of Souslin forcing, and we prove that some properties are preserved under iteration. We define a weaker form of Martin's axiom, namely , and using the results on Souslin…

TOOLS FOR YOUR FORCING CONSTRUCTION

- Mathematics
- 1992

A preservation theorem is a theorem of the form: "If hP�,Q� : � < �i is an iteration of forcing notions, and every Qsatisfies ' in V P� , then Psatisfies '." We give a simplified version of a general…

Vive la Différence I: Nonisomorphism of Ultrapowers of Countable Models

- Mathematics
- 1992

We show that it is not provable in ZFC that any two countable elementarily equivalent structures have isomorphic ultrapowers relative to some ultrafilter on ω.

Set theory - an introduction to independence proofs

- MathematicsStudies in logic and the foundations of mathematics
- 1983

The Foundations of Set Theory and Infinitary Combinatorics are presented, followed by a discussion of easy Consistency Proofs and Defining Definability.