Thomas Benjamin
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Member for 12 years, 11 months
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Last seen this week
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A Question Regarding Weak Diamond
(Perhaps the property of normal first countable spaces being "collectionwise Hausdorf" in a model of $MA_{\omega_1}$(S)[S] might have something to do with it, at least as far as $\Phi_{\omega_1}$ failing yet all Whitehead groups being free goes....)
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A Question Regarding Weak Diamond
Could one, for example, have a model of ZFC+$MA_{\omega_1}(S)[S]$+$\lnot$CH in which all Whitehead groups are free? If not, why not?
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A Question Regarding Weak Diamond
Mc'Kenney: And yet, in their paper "Locally Compact Perfectly Normal Spaces May All Be Paracompact", Larson and Tall show that (in their Theorem 3) $MA_{\omega_1}$(S)[S] implies that all Whitehead groups are free, even though $MA_{\omega_1}$(S)[S] also implies (as you point out) that $2^{\omega}$=$2^{\omega_1}$. Thus you have a situation in which $\Phi_{\omega_1}$ fails and yet all Whitehead groups are free. Why is that?
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A Question Regarding Weak Diamond
Paul McKenny: Does Larson and Todorcevic's result depend in any way, shape or form on $AD^{L(\mathscr R)}$?|
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A Question Regarding Weak Diamond
What happened to the answer? I made a hard copy of the reference the answer gave (thanks, by the way), went away from the computer to read through the reference and when I came back to the computer,the answer was gone. I had some questions to ask regarding the relation between the reference and the answer, so I would like to see the answer again.
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A Question Regarding Weak Diamond
@RamirodelaVega: Have you a reference?
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Harrington's unpublished note "The constructible reals can be anything"
@Mohammad Golshani: The actual title of the preprint is, "The constructible reals can be [almost] anything" (as the scan of the preprint shows). It seems an interesting result, but what is its significance (I don't doubt its significance, but am interested in how it relates to the results of Solovay and Jensen regarding nonconstructible sets of integers)?
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Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)
@Prof. Hamkins: Can V as characterized in your theorem contain sets nonconstructible relative to L? Do AC and GCH hold in V as characterized by your theorem? In your conception of the set-theoretic multiverse, is there a forcing extension M[G] of this V?
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weakening naive comprehension to avoid the paradoxes
@noah: when you speak of non-classical logics, do you include systems of logic where excluded middle doesn't hold? Russell's paradox (seemingly) obtains because one assumes excluded middle. The same (possibly) implies the means for extricating ones self from Curry's paradox as well...
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A question about Paraconsistent Set Theory and the Continuum Hypothesis
I am currently reading Weber's paper and the gist of his proof that not-CH holds is that |omega|<|P(omega)|<|P(omega)| holds in his system of paraconsistent set theory.
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What is the impact on Godels theorem of Paraconsistency?
@Mozibur: Which paraconsistent set theory disproves CH? Could you provide a sketch of the proof?
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A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
and the term 'object' seems to me to be a neutral term), if one takes the definition and theorem directly from ordinary recursive function theory, cannot be generated from the *-recursive functions, will lie outside the given inner model. Therefore, it is not a 'set' relative to the given inner model and therefore relative to the given inner model, a proper class. Is this essentially your line of reasoning? Please let me know.
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A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
@Francois: Since you have not responded to my comment (and perhaps with good reason, too, if it is something you think I should already know...), I will offer up a possible line of reasoning (at the risk of this line of reasoning being a 'straw man', so to speak...) that would make productivity in SO a proper class. Here goes: the *-recursive sets form the smallest inner model of SO (the model being the 'universe' of sets of ordinals). Since the productive 'object' ( I call it an object because it is to be determined whether it is to be a proper class or set,