Timeline for Looking for a source for Intended Interpretation
Current License: CC BY-SA 3.0
79 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2020 at 7:23 | comment | added | AnduinWilde | "an identification of a mathematical concept and an intuitive (i.e., pre-mathematical) concept. The former is the usual theory of the integers (N) as developed for example by von Neumann in a set-theoretic context. The latter are (the totality of) the familiar numbers that human beings are familiar with before they learn anything about set theory." In fact, the latter is also mathematical concept, for before the formalization is invented, mathematics has existed for 2000 years! And many of the mathematicians for now don't even learn the logic. | |
| Mar 14, 2017 at 7:55 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 22, 2016 at 7:01 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| May 22, 2016 at 7:01 | vote | accept | Mikhail Katz | ||
| S Apr 4, 2016 at 9:16 | history | bounty ended | Mikhail Katz | ||
| S Apr 4, 2016 at 9:16 | history | notice removed | Mikhail Katz | ||
| Apr 4, 2016 at 9:16 | vote | accept | Mikhail Katz | ||
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| Apr 3, 2016 at 16:08 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Apr 2, 2016 at 15:07 | answer | added | Will Sawin | timeline score: 4 | |
| Mar 31, 2016 at 16:23 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 27, 2016 at 18:10 | answer | added | Mauro ALLEGRANZA | timeline score: 6 | |
| S Mar 27, 2016 at 15:09 | history | bounty started | Mikhail Katz | ||
| S Mar 27, 2016 at 15:09 | history | notice added | Mikhail Katz | Improve details | |
| Mar 20, 2016 at 15:51 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 16, 2016 at 17:53 | comment | added | Carl Mummert | I have deleted several comments which I think are covered by the answer that I wrote. | |
| Mar 16, 2016 at 16:24 | comment | added | Mikhail Katz | @Carl my question was not meant to challenge what you describe as a vast majority but rather to find a proper reference. | |
| Mar 16, 2016 at 16:05 | answer | added | Carl Mummert | timeline score: 13 | |
| Mar 16, 2016 at 15:55 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 16, 2016 at 15:50 | comment | added | Mikhail Katz | @Carl, Wang speaks of "ordinary" integers, implying the numbers familiar to people before they learn ZFC, whereas Shoenfield speaks of "natural numbers" which is just another term for elements of the ZFC $\mathbb{N}$. I will elaborate in the question. | |
| Mar 16, 2016 at 14:29 | comment | added | Mikhail Katz | @CarlMummert, Hajek and Pudlak on page 12 write that "0.28. Recall that N is the set of natural numbers." Shoenfield on page 23 does something similar, referring to "the obvious individuals" but without explaining what those obvious things are. Neither of these addresses my question. | |
| Mar 16, 2016 at 9:02 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| S Mar 15, 2016 at 13:31 | history | notice removed | CommunityBot | ||
| S Mar 15, 2016 at 13:31 | history | unlocked | CommunityBot | ||
| Mar 9, 2016 at 13:30 | history | notice removed | François G. Dorais | ||
| S Mar 8, 2016 at 12:45 | history | notice added | Todd Trimble | Content dispute | |
| S Mar 8, 2016 at 12:45 | history | locked | Todd Trimble | ||
| Mar 8, 2016 at 10:14 | answer | added | Semen Kutateladze | timeline score: -2 | |
| Mar 8, 2016 at 9:51 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 7, 2016 at 17:54 | comment | added | Mikhail Katz | Sounds intriguing. Can you elaborate? | |
| Mar 7, 2016 at 17:49 | comment | added | Semen Kutateladze | I think that any intended interpretation of numbers today is much more immaterial today than it was in the last century. The diversity of interpretations has taken its place. | |
| Mar 7, 2016 at 15:43 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 7, 2016 at 15:28 | comment | added | Mikhail Katz | @HansAdler, Francois seems to think that on the contrary this question is difficult to answer :-) | |
| Mar 7, 2016 at 7:59 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 6, 2016 at 19:56 | answer | added | François G. Dorais | timeline score: 16 | |
| Mar 6, 2016 at 16:43 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 6, 2016 at 16:30 | history | protected | Todd Trimble | ||
| Mar 6, 2016 at 15:20 | history | notice added | Mikhail Katz | Draw attention | |
| Mar 6, 2016 at 10:00 | comment | added | Mikhail Katz | @CarlMummert, if you get a chance format your comment as an answer keeping in mind that, as clarified in the question, I am not looking for occurrences of using the term "intended interpretation" to describe "a standard $\mathbb{N}$" but rather for occurrences of identification of it with the ordinary counting numbers. | |
| Mar 4, 2016 at 6:38 | comment | added | user12283 | I agree with Gabriel Nivasch's "I don't know why would you want a reference for such an obvious thing". The very question may very well be a sign that there is something wrong about your general approach to mathematics. With this philosophy-style(?) approach we would never get anything done. | |
| Mar 3, 2016 at 16:36 | review | Close votes | |||
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| Mar 3, 2016 at 16:11 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 3, 2016 at 14:44 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 3, 2016 at 13:03 | answer | added | Gabriel Nivasch | timeline score: 7 | |
| Mar 3, 2016 at 12:38 | history | reopened |
Mikhail Katz Joonas Ilmavirta Andrey Rekalo Todd Trimble |
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| Mar 3, 2016 at 12:38 | comment | added | Todd Trimble | I am casting a last vote to reopen, because I think the reason for closure as "unclear what you're asking" is not particularly apt. An answer has been provided in comments and it seems (based on the discussion) that it will be accepted as answering the question. | |
| Mar 3, 2016 at 10:33 | comment | added | Gabriel Nivasch | Let us continue this discussion in chat. | |
| Mar 3, 2016 at 10:33 | comment | added | Mikhail Katz | @GabrielNivasch could you possibly provide the section numbers? I was able to locate a second edition but not the third edition. | |
| Mar 3, 2016 at 10:31 | comment | added | Gabriel Nivasch | I mean 62 of the book numbering, not of the PDF document numbering | |
| Mar 3, 2016 at 10:30 | comment | added | Mikhail Katz | @GabrielNivasch I find neither "interpretation" nor "model" on page 62 in the second edition. | |
| Mar 3, 2016 at 10:26 | comment | added | Gabriel Nivasch | Third edition, 2010 | |
| Mar 3, 2016 at 10:25 | comment | added | Mikhail Katz | @GabrielNivasch, thanks. I see that there are several editions of this. Which edition are your page numbers from? | |
| Mar 3, 2016 at 10:22 | comment | added | Gabriel Nivasch | I don't know why would you want a reference for such an obvious thing, but here it is: Rautenberg's "A concise introduction to mathematical logic". On page 62 he says that "interpretation" = "model", on page 105 he defines $\mathcal N=(\mathbb N, 0, S, +, \cdot)$, and on page 106 he calls $\mathcal N$ the standard model. (You might be able to download the book as PDF from Springer through your institution's library.) | |
| Mar 3, 2016 at 10:03 | history | edited | Mikhail Katz |
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| Mar 3, 2016 at 8:55 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 3, 2016 at 8:07 | review | Reopen votes | |||
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| Mar 3, 2016 at 7:55 | history | edited | Mikhail Katz |
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| Mar 2, 2016 at 23:12 | history | closed |
Andrés E. Caicedo Wolfgang user1688 Marco Golla Alexey Ustinov |
Needs details or clarity | |
| Mar 2, 2016 at 16:24 | history | edited | Mikhail Katz | CC BY-SA 3.0 |
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| Mar 2, 2016 at 15:49 | comment | added | Mikhail Katz | All Schagrin now, but both Wang and Schagrin are signed at the bottom. | |
| Mar 2, 2016 at 15:47 | comment | added | Mikhail Katz | @EmilJeřábek well this goes to show that it is better to have a traditional reference :-) Do you have one in mind by any chance? | |
| Mar 2, 2016 at 15:45 | comment | added | Emil Jeřábek | Now it’s all Wang for me. Sounds like content dynamically generated from a database gone bananas. | |
| Mar 2, 2016 at 15:44 | comment | added | Mikhail Katz | Somebody seems to be playing with that page as we speak. My browser currently identifies first page with Schagrin, second with Wang, and third with Schagrin again. | |
| Mar 2, 2016 at 15:42 | comment | added | Emil Jeřábek | Interesting. For me, it’s exactly the opposite (the first page Schagrin, the next two Wang). | |
| Mar 2, 2016 at 15:39 | comment | added | Mikhail Katz | @EmilJeřábek, thanks for your comment. The article in EB is in three pages. The first page is attributed to Hao Wang in my browser, and the second and third to Schagrin. | |
| Mar 2, 2016 at 15:38 | comment | added | Emil Jeřábek | The linked Britannica page is terribly broken in all browsers I tried, nevertheless it appears to credit the text to Morton L. Schagrin rather than Wang. | |
| Mar 2, 2016 at 15:36 | comment | added | logicute | Computation and logic in the real world, proceedings of Computability in Europe (CIE) 2007. It's mostly a recap of some intuition about Tennenbaum's theorem though. | |
| Mar 2, 2016 at 15:14 | comment | added | Mikhail Katz | @logicute, it seems that Paula Quinon posted this at philpapers already in 2006 but where did the article appear? | |
| Mar 2, 2016 at 15:08 | comment | added | Mikhail Katz | @logicute, that sounds promising. Who is it by? Why don't you format this as an answer? | |
| Mar 2, 2016 at 15:05 | comment | added | logicute | Doesn't "The Intended Model of Arithmetic. An Argument from Tennenbaum’s Theorem" fit the bill? | |
| Mar 2, 2016 at 14:58 | comment | added | Mikhail Katz | @logicute, right, and I am looking for a reference that reproduces Wang's characterisation in a more traditional venue. | |
| Mar 2, 2016 at 14:56 | comment | added | logicute | Wang's sentence describes the intended model of arithmetic, not what an intended model is supposed to be. So it is consistent with "always". | |
| Mar 2, 2016 at 14:40 | comment | added | Mikhail Katz | @logicute, when you say "always" you seem to ignore the sentence I reproduced from Wang. Wang proposes a characterisation of an intended model (I didn't claim that it is a mathematical definition). | |
| Mar 2, 2016 at 14:31 | comment | added | logicute | Not at all, hence the "I think". If it was unclear, I meant that 'intended model' seems to have always been taken for its intuitive value and not as a mathematical concept. But it seems that "the model of the second-order theory" seems like a good candidate for a definition. | |
| Mar 2, 2016 at 14:06 | comment | added | Mikhail Katz | @logicute Do you have a source for that? | |
| Mar 2, 2016 at 13:19 | history | edited | Mikhail Katz |
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| Mar 2, 2016 at 13:01 | review | Close votes | |||
| Mar 2, 2016 at 23:12 | |||||
| Mar 2, 2016 at 12:58 | comment | added | logicute | Standard model is also 'intended model', which as a general notion is not a mathematical one (Hodges does not give a definition, and Skolem probably not). However I think the right way to define standard model (at least in the case of Peano arithmetic) is to define the standard model as the model of the second-order theory. | |
| Mar 2, 2016 at 10:14 | history | edited | Mikhail Katz |
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| Mar 2, 2016 at 9:00 | history | asked | Mikhail Katz | CC BY-SA 3.0 |