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Q1. What is the consistency strength of the failure of square on singular cardinals?

Q2. What are known as partial results in this direction?

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    $\begingroup$ To put emphasis on everything defeats the purpose of having that option available. Thus, I removed it. $\endgroup$
    – user9072
    Commented Nov 8, 2013 at 11:49

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One place to start reading is this:

James Cummings and Sy-David Friedman, "$\square$ On the Singular Cardinals". The Journal of Symbolic Logic Vol. 73, No. 4 (Dec., 2008), pp. 1307-1314.

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    $\begingroup$ Squares, mice, silver machines. What's next, hamsters? $\endgroup$
    – Andrej Bauer
    Commented Nov 8, 2013 at 14:27
  • $\begingroup$ @Andrej, also Amoeba and other Creatures. But what about sheaves, germs, and so on? And also diamonds, club suites, and other gems. $\endgroup$
    – Asaf Karagila ♦
    Commented Nov 8, 2013 at 14:29
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    $\begingroup$ Daggers, pistols and hand grenades too! $\endgroup$
    – François G. Dorais
    Commented Nov 8, 2013 at 14:50
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    $\begingroup$ @AndrejBauer I wish. It is unfortunate notation, a bad joke taken too far. $\endgroup$
    – Andrés E. Caicedo
    Commented Nov 8, 2013 at 16:13
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    $\begingroup$ Don't forget the paper "A new construction of a non-constructible $\Delta^1_3$ subset of $\omega$" by Jensen and Johnsbraten. $\endgroup$
    – saf
    Commented Nov 27, 2013 at 19:22

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