Timeline for How do working constructivists get by with out the zero product property?
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2021 at 22:04 | vote | accept | ToucanIan | ||
| Sep 16, 2021 at 16:59 | answer | added | Arno | timeline score: 3 | |
| Sep 16, 2021 at 16:55 | comment | added | Noah Schweber | Now asked at MSE. | |
| Sep 16, 2021 at 16:52 | comment | added | ToucanIan | @MattF. Thanks for your feedback! I will post the question else where. | |
| Sep 16, 2021 at 16:09 | comment | added | user44143 | @Toucanlan, since you have only provided one example, and resolving it is not close to research-level, I have voted to close this question as not representing research-level mathematics. | |
| Sep 16, 2021 at 15:55 | comment | added | ToucanIan | @NoamD.Elkies thanks for the clarification! | |
| Sep 16, 2021 at 15:52 | history | edited | ToucanIan | CC BY-SA 4.0 |
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| Sep 16, 2021 at 15:46 | comment | added | ToucanIan | @MattF. The question that is posed stands with or without an example. The example may be removed if you insist that it is not neccesary, but the substance of the question is not dependent on any one example. | |
| Sep 15, 2021 at 20:55 | comment | added | Noam D. Elkies | @ToucanIan ouch, typos: $x<1$ if $x' \leq 0$ (not $\epsilon \leq 0$), and $x > -1$ if $x' \geq 0$ (not $\epsilon \geq 0$) -- sorry! (and too late to edit the originally comment) | |
| Sep 15, 2021 at 20:33 | comment | added | user44143 | @Toucanlan, what is the point of more general techniques when you have only provided one example and it is resolved well already? | |
| Sep 15, 2021 at 18:59 | history | edited | ToucanIan | CC BY-SA 4.0 |
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| Sep 15, 2021 at 13:36 | comment | added | ToucanIan | @NoamD.Elkies If you let $\epsilon = 1/2$ how are there cases where $\epsilon \leq 0$? | |
| Sep 15, 2021 at 13:30 | comment | added | ToucanIan | @LSpice I would disagree but I guess it will be up to the moderators. | |
| Sep 15, 2021 at 13:27 | comment | added | ToucanIan | @AndreasBlass great point. Adding the assumption of apartness seems reasonable for many situations. | |
| Sep 15, 2021 at 3:23 | comment | added | Noam D. Elkies | The example of $x^2 - 4 = 0$ does not really need the "zero product property": even for a constructivist, it is true that every real $x$ satisfies $x < 1$ or $x > -1$, and then $x = -2$ or $x = 2$ respectively. (A constructive real number $x$ is basically a black box that inputs rational $\epsilon > 0$ and outputs rational $x'$ such that $|x-x'| < \epsilon$, with the outputs for different $\epsilon$ consistent with the triangle inequality. So, let $\epsilon = 1/2$, and then $x<1$ if $\epsilon \leq 0$ while $x>-1$ if $\epsilon \geq 0$.) | |
| Sep 15, 2021 at 2:14 | comment | added | user44143 | The zero-product property fails for continuous functions too: $fg=0$ does not imply $f=0$ or $g=0$. But one version of the @AndreasBlass comment is that if continuous functions $f,g$ satisfy $fg=0$ and $g-f>1/n$ then $f=0$ or $g=0$. | |
| Sep 15, 2021 at 1:54 | comment | added | Andreas Blass | Unless I'm overlooking something, it seems constructively valid that, if $xy=0$ and $x$ is apart from $y$ (meaning there's a positive rational number smaller than $|x-y|$) then $x=0$ or $y=0$. That's enough to handle your example, but I'm not sure whether there are tougher examples or methods to handle them. | |
| Sep 15, 2021 at 0:25 | review | Close votes | |||
| Sep 29, 2021 at 11:09 | |||||
| Sep 15, 2021 at 0:07 | comment | added | LSpice | This seems like a good question for our sister site MSE; but, since it does not concern research-level mathematics, it will probably be closed here. | |
| Sep 15, 2021 at 0:07 | history | edited | LSpice | CC BY-SA 4.0 |
Name of article; proofreading
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| Sep 14, 2021 at 23:58 | history | asked | ToucanIan | CC BY-SA 4.0 |