Timeline for Show that the positive existential theory is undecidable
Current License: CC BY-SA 3.0
19 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2015 at 14:28 | vote | accept | Mary Star | ||
| Nov 21, 2015 at 15:27 | comment | added | Joel David Hamkins | The issue of $x(1)=1$ returned, in order to make the assertion positive. See the comments on my answer. | |
| Nov 21, 2015 at 14:34 | comment | added | Todd Trimble | To address any lingering concerns about the suitability of the question for MO, I'd recommend an editing of the question to give some context and/or motivation for the research interest, and maybe a light editing to increase clarity (we are dealing with the positive existential theory of $\mathbb{C}[t, e^{\lambda t}]_{\lambda \in \mathbb{C}}$ as a model of the theory of rings with a derivation operator; it sometimes helps to use English words instead of symbols). Other than that, I agree with Joel David Hamkins that this question is suitable for MO. | |
| Nov 21, 2015 at 14:08 | comment | added | Joel David Hamkins | I posted an answer. The issue of $x(1)=1$ seems unnecessary to me now. | |
| Nov 21, 2015 at 14:07 | answer | added | Joel David Hamkins | timeline score: 11 | |
| Nov 21, 2015 at 12:21 | comment | added | Mary Star | Or do you mean something else? @JoelDavidHamkins | |
| Nov 20, 2015 at 15:33 | comment | added | Mary Star | First of all $n$ has to be a constant in $\mathbb{C}[t,e^{\lambda t}]$, so $n\in \mathbb{C}$. If we want that the solution $tx'=nx$ is a polynomial we want that $n\in \mathbb{N}$, right? To get $\mathbb{Z}$ do we have to take an other differential equation? For example $tx'=-nx$? Or do we get it somehow else? @JoelDavidHamkins | |
| Nov 20, 2015 at 15:33 | comment | added | Mary Star | Yes, I meant $t-1$. Why does it stand in the polynomial ring but not in the exponential-polynomial ring? $$$$ To define $\mathbb{Z}$ in the structure $\mathbb{C}[t,e^{\lambda t}]_{\lambda \in \mathbb{C}}$ by an existential formula in the language of rings with differentiation and a constant for polynomial $t$, we want to express $n\in \mathbb{Z}$ in the structure $\mathbb{C}[t,e^{\lambda t}]$, right? @JoelDavidHamkins | |
| Nov 20, 2015 at 12:02 | comment | added | Joel David Hamkins | I don't really understand the close votes, since I think the question is interesting. Can you define $\mathbb{Z}$ in the structure $\mathbb{C}[t,e^{\lambda t}]_{\lambda\in \mathbb{C}}$ by an existential formula in the language of rings with differentiation and a constant for the polynomial $t$? | |
| Nov 20, 2015 at 2:57 | comment | added | Joel David Hamkins | But that doesn't seem true to me. For example, take $x=\frac 1ee^t$, which has $x(1)=1$. But unless I am mistaken, I don't think we have that $t-1$ divides $x-1$. (It would be true in the polynomial ring, but not in this exponential-polynomial ring.) | |
| Nov 20, 2015 at 2:08 | comment | added | Joel David Hamkins | I think you mean $t-1$ rather than $z-1$. You are saying that polynomial/exponential expression $x(t)$ has $x(1)=1$ just in case it is a multiple of the term $(t-1)$. | |
| Nov 20, 2015 at 1:35 | comment | added | Mary Star | I found now in my notes that we can express "$x(1)=1$" as "$z-1 \mid x-1$". Why does this stand? @JoelDavidHamkins Do you have an idea about the reduction? Or do we not do it as I said in my question above? | |
| Nov 20, 2015 at 0:40 | comment | added | Mary Star | No, this cannot be expressed by the language. We have to write it in a way that is allowed in the language. @JoelDavidHamkins | |
| Nov 20, 2015 at 0:36 | comment | added | Joel David Hamkins | Interesting question. Your notation $x(1)=1$ suggests that you allow to evaluate the polynomial expressions in your ring, but this is not actually mentioned in the language. Could you clarify whether this is allowed or not? | |
| Nov 19, 2015 at 23:46 | history | edited | Mary Star |
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| Nov 19, 2015 at 23:23 | review | Close votes | |||
| Nov 21, 2015 at 23:14 | |||||
| Nov 19, 2015 at 20:20 | history | edited | Mary Star | CC BY-SA 3.0 |
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| Nov 19, 2015 at 20:05 | history | edited | Mary Star | CC BY-SA 3.0 |
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| Nov 19, 2015 at 19:45 | history | asked | Mary Star | CC BY-SA 3.0 |