Cohen's final solution of the problem of the cardinality of the
continuum within
ZFC
gave rise to
Boolean models.
Scott forecasted in 1969:
We must ask whether there is any interest
in these nonstandard models aside from the independence proof;
that is, do they have any mathematical interest?
The answer must be yes, but we cannot yet give a really good
argument.^{1}
Takeuti
coined the term “Boolean valued analysis” for applications of the models
to analysis.
The
Farkas Lemma, also known as the Farkas–Minkowski Lemma,
plays a key role in linear programming and the relevant areas of optimization.The aim of this talk is to demonstrate how Boolean valued analysis may
be applied to simultaneous linear inequalities with operators.
This particular theme is another illustration of the deep and powerful
technique of “stratified validity” which is characteristic of
Boolean valued analysis.
We definitely feel truth, but we cannot define truth properly.
That is what
Tarski explained to us in the 1930s.
We pursue truth by way of proof, as wittily phrased by
Mac Lane.
Model theory evaluates and counts truth and proof.
The chase of truth not only leads us close to the truth we pursue but also enables
us to nearly catch up with many other
instances of truth which we were not aware nor even foresaw at the start
of the rally pursuit. That is what we have learned from
the Boolean models elaborated in the 1960s by Scott,
Solovay,
and
Vopěnka.

[1]
Kjeldsen T. H.,
Different motivations and goals in the historical development of the theory of systems of linear inequalities.
Arch. Hist. Exact Sci., 56:6, 459–538 (2002).

[2]
Floudas C. A. and Pardalos P. M. (eds.),
Encyclopedia of Optimization.
Springer (2009).

[3]
Scott D.,
Boolean Models and Nonstandard Analysis.
Applications of Model Theory to Algebra, Analysis, and Probability, 87–92.
Holt, Rinehart, and Winston (1969).

[4]
Takeuti G.,
Two Applications of Logic to Mathematics.
Iwanami Publ. & Princeton University Press (1978).

[5]
Kusraev A. G. and Kutateladze S. S.,
Boolean Valued Analysis.
Kluwer Academic Publishers (1999).

[6]
Kusraev A. G. and Kutateladze S. S.,
Introduction to Boolean Valued Analysis.
Nauka Publishers (2005) [in Russian].
This is an abstract of a talk at the
Maltsev Centennial in Novosibirsk
on August 27, 2009.
As regards technicalities see
arXiv.org and
Optimization Online.
Slides are available in
PDF.
Footnote:
^{1}
At the time, I was disappointed that no one took
up my suggestion. And then I was very surprised
much later to see the work of Takeuti and his
associates. I think the point is that people have
to be trained in Functional Analysis in order to
understand these models. I think this is also
obvious from your book and its references. Alas,
I had no students or collaborators with this kind
of background, and so I was not able to generate
any progress.
(From
Dana Scott's Letter of April 29, 2009 to S.S. Kutateladze.)
Dana Scott
with the participants of the seminar of the
Laboratory of Functional Analysis
at Novosibirsk on August 28, 2009.