1
$\begingroup$

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?

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

1 Answer 1

1
$\begingroup$

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.

$\endgroup$
9
  • 4
    $\begingroup$ 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...) $\endgroup$
    – Goldstern
    Mar 2, 2020 at 23:39
  • 2
    $\begingroup$ @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) $\endgroup$
    – Goldstern
    Mar 2, 2020 at 23:45
  • 1
    $\begingroup$ @Goldstern Sure, it's better to do so, but with the product you still add $\kappa$-many reals which are whatever-generic over $V$. $\endgroup$ Mar 3, 2020 at 0:26
  • 4
    $\begingroup$ In addition to products and the iterations already mentioned, random reals admit the option of forcing with the measure algebra of $2^\kappa$. $\endgroup$ Mar 3, 2020 at 18:02
  • 1
    $\begingroup$ @HannesJakob Take the product measure induced from the uniform measure on $2$. $\endgroup$ Mar 4, 2020 at 0:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.