Zhang Jing
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Registered User
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I'm an undergraduate majoring in Mathematics and Computer Science in National University of Singapore.
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May 19 |
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Non-standard model of the domination principle What is the forcing language here (sorry for getting into messy details)? For example, what is $(s,j)\Vdash \bar{n}\in X$? In the normal context of strings, $\sigma\Vdash \bar{n}\in X \leftrightarrow \sigma(n)=1$ (I got from Odifreddi's book). In addition, my intention was to preserve the first-order universe so that $B\Sigma_2^0$ is still false. But I am not sure whether the forcing mentioned here would alter the first order universe. Thanks! |
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May 18 |
revised |
Non-standard model of the domination principle added 12 characters in body |
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May 18 |
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Non-standard model of the domination principle @François: Thanks! Can I just have it clarified what it means to say $g\geq f$ in the definition of poset? Since I am interested in the non-standard universe, do you think the same argument goes through (it occurs to me so). |
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May 17 |
revised |
Non-standard model of the domination principle added 693 characters in body |
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May 17 |
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Non-standard model of the domination principle Yeah the first property was what I was asking about. |
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May 16 |
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Non-standard model of the domination principle Indeed. That was poorly phrased. I was thinking about the confinality in the ordinal (in order to define a dominating function if any). In this case, it is indeed bounding schemes that may be helpful. |
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May 16 |
revised |
Non-standard model of the domination principle deleted 127 characters in body; added 47 characters in body |
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May 16 |
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Non-standard model of the domination principle @Jason: Sorry for its being poorly phrased. I am actually looking for some non-standard model in which RCA_0 and Domination principle hold. My guess of the universe being a regular cardinal is not a characterization for sure because one could easily produce a counter-example. To be exact, I would love some examples of non-standard models in which the principle holds. |
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May 16 |
revised |
Non-standard model of the domination principle deleted 8 characters in body |
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May 16 |
asked | Non-standard model of the domination principle |
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Apr 4 |
revised |
Cohesive sets with degree below some non-high 1-generic degrees? added 9 characters in body |
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Mar 17 |
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Indices of r.e. sets Thanks! I think I should remove that! |
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Mar 17 |
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Indices of r.e. sets deleted 627 characters in body |
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Mar 16 |
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Indices of r.e. sets @François: Exactly. However, I was trying to adapt this to prove something else as explained in the EDIT. I think there is a hole in this argument though. |
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Mar 16 |
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Indices of r.e. sets added 629 characters in body |
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Mar 15 |
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Indices of r.e. sets @Joel: An algorithm I had in mind was: Suppose e is given, given input x, run $phi_e(e)$. If it converges, take the value $y$ and if x is among the first $p(y)$ elements from A, halt. We could code the description of the program to get a Goedel number which will have the desired property. But the problem is the potential use of A is infinite in the program, I am not sure whether it's okay to claim the program is recursive in A and further it has domain $W_{g(e)}$ instead of $W_{g(e)}^A$? |
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Mar 15 |
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Indices of r.e. sets @François: I agree on the possibility. But it is still not clear for me how to produce such index given the unknown status of $\phi_e(e)$. |
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Mar 15 |
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Indices of r.e. sets @Joel: Since A is effectively immune, A is not possible to be c.e, since if so, A is the subset of itself and the cardinality is not bounded. I mean the members of A in the natural number order. |
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Mar 15 |
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Indices of r.e. sets @Emil: Well, but the problem is whether the r.e. index could be found recursively in A. If yes, is it possible to exhibit such program? |
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Mar 15 |
asked | Indices of r.e. sets |
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Mar 5 |
awarded | ● Fanatic |
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Feb 19 |
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$\Sigma_1^0-COH$? @François: Oh Thanks! I will check the references first! |
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Feb 19 |
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$\Sigma_1^0-COH$? Then $\bar D$ would be a cohesive set for $<R_k: k\in \omega>$ since $\bar D \subset \bar R_k$ for all $k\in \omega$. But why $\bar D \subset ^* X$ or $\bar D\subset ^* \bar X$ holds? Making $D$ cofinite might help. Another thing I noticed was for the first bullet you phrased $X$ as $A-computable$, but $\Sigma_1^0$ actually states that for any collection of sets such that each set is r.e. in A, there exists a cohesive set for this collection. Thus I suppose there does exists a uniform listing but it is a listing of all r.e. sets in A. |
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Feb 19 |
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$\Sigma_1^0-COH$? I guess I did not fully see the last two characterizations. What would the cohesive set be like? Thanks! |
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Feb 18 |
awarded | ● Commentator |
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Feb 18 |
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$\Sigma_1^0-COH$? Thanks for the information. I've been thinking about the relation between $Σ_1^0-COH$ and $RT_2^2$ since $Σ_1^0-COH$ is stronger than $COH$. Would $RT_2^2$ also imply $Σ_1^0-COH$? |
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Feb 16 |
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$\Sigma_1^0-COH$? edited title |
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Feb 16 |
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$\Sigma_1^0-COH$? Thanks! Should be $\Sigma^0_1$ |
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Feb 16 |
asked | $\Sigma_1^0-COH$? |
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Feb 11 |
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Cohesive sets with degree below some non-high 1-generic degrees? @Adam: Thanks for the note. But I believe in your definition of $W$ it should be $\Phi^\sigma(n)=\varphi_n(n)$ |
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Feb 9 |
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Cohesive sets with degree below some non-high 1-generic degrees? @Adam: Thank you! I would take a look at that paper! Is it also mentioned there why no 1-generic set could compute a DNC function? |
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Feb 8 |
revised |
Cohesive sets with degree below some non-high 1-generic degrees? edited tags |
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Feb 7 |
asked | Cohesive sets with degree below some non-high 1-generic degrees? |
