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Joel David Hamkins

mathematics and philosophy of the infinite

Joel David Hamkins

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Tag Archives: Saharon Shelah

Superdestructibility: a dual to Laver's indestructibility

Posted on September 25, 2011 by Joel David Hamkins
1

[bibtex key=HamkinsShelah98:Dual]

After small forcing, any <πœ…-closed forcing will destroy the supercompactness, even the strong compactness, of πœ….

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Posted in Publications | Tagged approximation-and-cover, forcing, indestructibility, large cardinals, Saharon Shelah, supercompact | 1 Reply

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Proof and the Art of Mathematics, MIT Press, 2020

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  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    Not always. Consider a diamond, where the max and min have two edges, or the one point order, where it has none.
  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    And when you refer to vertices in a partial order "with exactly one edge" are you referring instead to the minimal and maximal elements? Also, I normally think of posets as reflexive relations, but this would make all your digraphs have loops at every vertex. So I guess you want an irreflexive version of partial […]
  • Comment by Joel David Hamkins on Is every directed graph the quotient of poset where boundary nodes are identified?
    Could you clarify what you intend by the quotient of a partial order by an equivalence relation, when it is not a congruence? I guess you mean that every instance of the order relation suffices for an edge in the quotient?
  • Comment by Joel David Hamkins on Algorithms to count restricted injections
    Have you mixed up 𝑛 and π‘š in the beginning? You have 𝑓⁑(π‘Ž), where π‘Ž is from {1,…,π‘š}, but the domain of 𝑓 is said to be {1,…,𝑛}. If $n
  • Comment by Joel David Hamkins on Terminology: commonly used name for an πœ” machine?
    Of course ultimately the computational power has nothing to do with fitting the computation into finite time, but rather just the idea of making sense of a computation with infinitely many steps. So you may be interested in the BΓΌchi automata (en.wikipedia.org/wiki/B%C3%BCchi_automaton), and beyond this, the infinite time Turing machines (jstor.org/stable/2586556), which extend the operation […]
  • Comment by Joel David Hamkins on Is this theory of bottomless hierarchy, consistent?
    Ah, sorry, you had the comma before β†’ not after, namely, βˆƒπ‘¦ ∈𝐴 :πœ™, β†’.
  • Comment by Joel David Hamkins on Is this theory of bottomless hierarchy, consistent?
    I confess that I have long been a little confused by your manner of using colons and commas in formal expressions, since it is different from what I am used to in first-order logic or in type theory. For example, how am I to read the meaning of "βˆƒπ‘¦ ∈𝐴 :πœ™ β†’,"? And how are we […]
  • Comment by Joel David Hamkins on Is every external downshifting elementary embedding 𝑗 with 𝑗⁑(π‘₯) =𝑗⁑[π‘₯], an automorphism?
    Ah, yes, of course. Thanks!

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