# Are there forcing notions adding $\kappa$ random, sacks, prikry, or Mathias reals?

This is my first question here, so if I am doing things incorrectly, please let me know. Now on to the question:

The forcing notion $$Fn(\kappa,2)$$, which constists of partial functions from $$\kappa$$ to $$2$$ with finite support, ordered by reverse inclusion, is known to add $$\kappa$$ new cohen reals.

Are there similare forcing notions adding $$\kappa$$ new random, sacks, prikry, or Mathias reals?

• Misread this at first as to asking about such forcings for cardinal $\kappa$, instead of adding $\kappa$ many reals. Mar 2, 2020 at 19:08
• "Notion", rather than "notation". Mar 3, 2020 at 4:05

The reason $$Fn(\kappa,2)$$ works for adding $$\kappa$$ new cohen reals is, because there exists a bijection between $$\kappa$$ and $$\kappa\times\omega$$, therefore the forcing $$Fn(\kappa,2)$$ is isomorphic (as a partial order) to $$Fn(\kappa\times\omega,2)$$ which itself is isomorphic to a Finite Support iteration of Length $$\kappa$$ of Cohen Forcing.
Similarly, for any forcing of your desired type (be it random, sacks, ... forcing), you could use a Finite Support iteration of length $$\kappa$$ to obtain your desired forcing. If you need an introduction to Forcing Iterations, Jechs book on Set Theory is a good start.
• A finite support iteration of non-ccc forcings, such as Mathias forcing or Sacks forcing, will (after $\omega$ many steps) always collapse $\omega_1$. This is usually an undesirable feature, and this is a reason why countable support iteration was invented. (But CS iterations has its own problems...) Mar 2, 2020 at 23:39
• @NoahSchweber For example, if you take the product of two Mathias forcings, it will lose some of the interesting properties that a single Mathias forcing or a CS iteration of Mathias forcing has, such as the Laver property. (The product of two dominating forcing notions, or of three unbounded forcing notions, will add a Cohen real.) - Also, both FS and CS products of infinitely many Mathias forcings will collapse $\aleph_1$. - Of course there are other produts (e.g. mixed support) Mar 2, 2020 at 23:45
• @Goldstern Sure, it's better to do so, but with the product you still add $\kappa$-many reals which are whatever-generic over $V$. Mar 3, 2020 at 0:26
• In addition to products and the iterations already mentioned, random reals admit the option of forcing with the measure algebra of $2^\kappa$. Mar 3, 2020 at 18:02
• @HannesJakob Take the product measure induced from the uniform measure on $2$. Mar 4, 2020 at 0:51