# Boolean Models andSimultaneous Inequalities

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].

### 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.)

### Vladikavkaz Math. J., 2009, Vol. 11, No. 3, 44–50 +http://arxiv.org/abs/0907.0060

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On 28 Aug 2009, 07:20.