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I'm following the Jech's Multiple Forcing for a seminar group and I intend to show how to add one or some reals to extensions.

I studied Solovay's model and I can see why learning how to add random reals is an important tool, as well as adding Cohen reals, etc.

My doubt is concerned to reals like Sacks, Pikry-Silver, Mathias, Laver and Grigorieff.

Where were these reals relevant to solve questions related to consistency results?

Thank you!

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Each of these notions was introduced to solve specific problems. The references in Jech's book are a good place to start. The results that the book itself includes show some of these applications. If these do not suffice, you need to specify in which way these examples are not satisfactory, and what instead it is that you are after. – Andrés E. Caicedo Mar 16 '14 at 16:23
(Typo: Pikry should be Prikry.) – Andrés E. Caicedo Mar 16 '14 at 16:24
I think this is a fine question, and I would think that someone could post an answer explaining in a sentence or two the principal features of each of these notions of genericity. – Joel David Hamkins Mar 16 '14 at 17:57

You might want to take a look in Andreas Blass' chapter of the Handbook of Set Theory. In particular section 11 and 11.10, which include summaries of various generic reals, and their properties (as well a nice table).

It should be added that the various generic reals are used to separate many of the characteristics of the continuum (e.g. iterating Sacks real forcing for $\omega_2$ steps will force the continuum to be $\omega_2$, but many of the characteristics will remain $\omega_1$, as the aforementioned table indicates).

Blass, A. "Combinatorial Cardinal Characteristics of the Continuum", Handbook of set theory, (eds. Foreman M., Kanamori A.) pp. 395–489, Springer 2010.

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