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Conducting Miller-Urey Experiments
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A version of -Miller forcing.

Heike Mildenberger1, Saharon Shelah2

  • 1Mathematisches Institut, Abteilung für Math. Logik, Albert-Ludwigs-Universität Freiburg, Ernst-Zermelo-Straße 1, 79104 Freiburg im Breisgau, Germany.

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|October 30, 2020
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Summary
This summary is machine-generated.

We introduce a new version of Miller forcing on uncountable cardinals. This forcing collapses the cardinal to a smaller one and adds a Cohen real, even under weaker assumptions.

Keywords:
Forcing with higher perfect trees

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Area of Science:

  • Set Theory
  • Large Cardinal Theory

Background:

  • Miller forcing is a technique used in set theory.
  • Investigating the properties of forcing notions on uncountable cardinals is crucial.

Purpose of the Study:

  • To analyze a specific variant of Miller forcing.
  • To determine its effects on cardinal collapse and the addition of Cohen reals.

Main Methods:

  • Utilizing a modified Miller forcing construction.
  • Applying set-theoretic axioms and forcing techniques.

Main Results:

  • The forcing collapses the uncountable cardinal $\kappa$ to $\aleph_1$.
  • It introduces a $\aleph_1$-Cohen real.
  • These results hold under weaker set-theoretic assumptions.

Conclusions:

  • The studied forcing is effective in collapsing cardinals and generating Cohen reals.
  • The findings contribute to understanding the flexibility of forcing in set theory.