Timeline for Defining cones and Turing cones
Current License: CC BY-SA 3.0
23 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 21, 2016 at 11:51 | vote | accept | Maxtimax | ||
| Dec 20, 2016 at 15:45 | comment | added | Asaf Karagila♦ | @Stefan: Generally, if $x\subseteq L$, then $L[x]=L(x)$. I don't know how Ralf defined the Baire space, but I suspect that it was done with functions from $\omega$ to $\omega$. So the same principle applies. | |
| Dec 20, 2016 at 15:44 | comment | added | Stefan Mesken | @Asaf Yes, I'm aware. However, since $x \in \mathcal{N}$ is not a set of ordinals in Ralf's book, this seemed more relevant to OP. The underlying reason though, that $L[x] = L(x)$, is - as you hinted - that it can be recursively coded as a subset of $\omega$. | |
| Dec 20, 2016 at 15:41 | history | edited | Maxtimax | CC BY-SA 3.0 |
edited title
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| Dec 20, 2016 at 15:39 | comment | added | HeinrichD | It is not necessary to mention any names. Just include the mathematical objects that you talk about. (This is a general rule.) | |
| Dec 20, 2016 at 15:38 | comment | added | Maxtimax | Heinrich: you're right. I don't know if the new title is any better.. What do you suggest ? | |
| Dec 20, 2016 at 15:33 | history | edited | Maxtimax | CC BY-SA 3.0 |
edited title
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| Dec 20, 2016 at 15:33 | comment | added | Asaf Karagila♦ | @Stefan: For sets of ordinals the two models coincide. | |
| Dec 20, 2016 at 15:31 | comment | added | HeinrichD | Please consider choosing a more neutral title. Imagine you are the author of a textbook, would you be happy with a public question (which is more a statement, actually) on the internet "[your name] seems to make a mistake"? | |
| S Dec 20, 2016 at 15:29 | history | suggested | Stefan Mesken | CC BY-SA 3.0 |
improved formatting
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| Dec 20, 2016 at 15:18 | review | Suggested edits | |||
| S Dec 20, 2016 at 15:29 | |||||
| Dec 20, 2016 at 15:11 | answer | added | Stefan Mesken | timeline score: 4 | |
| Dec 20, 2016 at 15:01 | comment | added | Stefan Mesken | Also note that $L[x]$ is not the least inner model containing $x$ - the least inner model containing $x$ is $L(x)$. $L[x]$, in contrast, is the least inner model closed under $y \mapsto y \cap x$. However, for $x \in \mathcal{N}$, these notions agree, i.e. $L[x]= L(x)$. | |
| Dec 20, 2016 at 14:19 | comment | added | Maxtimax | The mistake was essentially that $\sigma$ is not in $\mathcal{N}$ and I hadn't thought of the fact that the coding of strategies through elements of $\mathcal{N}$ was recursive, and that therefore it went smoothly both with Turing degrees and constructibles. | |
| Dec 20, 2016 at 14:14 | review | Close votes | |||
| Dec 21, 2016 at 8:38 | |||||
| Dec 20, 2016 at 14:12 | comment | added | Asaf Karagila♦ | Maxtimax, what seems to be the mistake, though? Could you perhaps state it clearly? | |
| Dec 20, 2016 at 14:11 | comment | added | Asaf Karagila♦ | @Todd: Generally, for a set of ordinals $A$, $L[A]$ is the smallest transitive model of ZFC containing all the ordinals such that $A$ is an element of the model. It can be also be constructed in a similar fashion to $L$ by iterated definable power sets by augmenting the language to include a predicate which we interpret as $A$ (intersected with each stage of the construction). So if $x$ is a real number, or a subset of $\omega$, then $L[x]$ is the least such model with $x$ inside of it. | |
| Dec 20, 2016 at 13:59 | comment | added | Simon Thomas | It's recursive. | |
| Dec 20, 2016 at 13:59 | comment | added | Maxtimax | Thomas, that's true but is the coding necessarily in $L[x]$ ? | |
| Dec 20, 2016 at 13:53 | comment | added | Simon Thomas | Strategies $\sigma$ are easily coded by elements of the Baire space and are usually identified with the corresponding element. (In a similar fashion, elements of the Baire space are usually called reals.) | |
| Dec 20, 2016 at 13:52 | comment | added | Maxtimax | Sorry, it's the smallest inner model containing $x$, where $x \in \mathcal{N}$ (actually the definition is for any $x$, but here it's in the Baire Space). The more rigorous definition goes through the same construction as for $L$ (the constructible universe) except the first step is with $cl(x)$, the transitive closure of $x$. | |
| Dec 20, 2016 at 13:46 | comment | added | Todd Trimble | What is $L[x]$, for people who don't have Jech's book in front of them? | |
| Dec 20, 2016 at 13:40 | history | asked | Maxtimax | CC BY-SA 3.0 |