Timeline for Looking for a source for Intended Interpretation
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 22, 2016 at 7:01 | vote | accept | Mikhail Katz | ||
| Mar 27, 2016 at 13:26 | comment | added | Mikhail Katz | Carl, concerning your comment that for the vast majority of mathematicians, the natural numbers are the ordinary/intuitive/informal counting numbers, and all those terms are synonymous: I tend to agree with this. I am precisely looking for a reference making this identification of intuitive numbers with $\mathbb{N}$ explicitly. | |
| Mar 17, 2016 at 17:27 | comment | added | Carl Mummert | "One can easily breeze through Kleene thinking that he is talking about the ZF semantic realisation of the syntax of the theory, rather than the familiar II. " That's a feature of the exposition - it seems maybe quixotic to look for something else. Most mathematical logic books are written in a way that is agnostic about whether the metatheory is formal or informal. They can be read both ways, depending on the taste of the reader. So one reader may view every result in e.g. Monk's logic book as a formal theorem of ZFC, while another may view the results as facts about a platonic universe. | |
| Mar 17, 2016 at 14:09 | comment | added | Mikhail Katz | ...Kleene seems to slip into the usual meaning of the term interpretation (say s-interpretation) when he describes the intuitive side as the set of natural numbers. The usual meaning of interpretation involves interpreting a (syntactic) theory in a (semantic) model, often against a ZF backdrop. I do find the Kleene reference somewhat useful, but it does not seem to be careful enough in distinguishing between i-interpretation and s-interpretation. Understanding the issue of the i-interpretation apparently should not be an issue of a majority (even overwhelming majority) vote. | |
| Mar 17, 2016 at 14:08 | comment | added | Mikhail Katz | Carl, user @logicute above mentioned sources arguing in favor of the identification in question. These are the article by Paula Quinon and Konrad Zdanowski, as well as the Halbach and Horsten reference they cite. There are some others. What the intended interpretation (or i-interpretation for short) involves is the identification of the natural numbers (a mathematical entity), on the one hand, and as Wang calls them the ordinary ones (intuitive entity), on the other. To understand the i-interpretation one therefore needs to distinguish the two... | |
| Mar 17, 2016 at 10:56 | comment | added | Carl Mummert | Of course, for the vast majority of mathematicians, the natural numbers are the ordinary/intuitive/informal counting numbers, and all those terms are synonymous. Only an author who wanted to make a distinction between two of them would worry about which word was used, and "natural number" is the most common one. So finding authors who go out of their way to emphasize the fact that two words mean the same thing may doom the search from the outset. The reason the "standard" model is "standard" is exactly that most mathematics is carried out in the standard model, rather than some other model. | |
| Mar 17, 2016 at 7:36 | comment | added | Mikhail Katz | @Todd, collection is a generic term that does not have the ZF-connotations of set. Note also that Kleene describes them as natural numbers rather than Wang's ordinary ones. One can easily breeze through Kleene thinking that he is talking about the ZF semantic realisation of the syntax of the theory, rather than the familiar II. | |
| Mar 16, 2016 at 18:22 | comment | added | Todd Trimble | At this point I can only agree with Carl's last comment (his last sentence) that "set" does have pre-formal, intuitive meaning and certainly does not have to refer to a technical term. I'm also worried at this point that you're never going to get an answer that satisfies you because the rules of the game, as far as I can make them out, seem very strict. Would the word "collection" be an acceptable substitute for "totality" in your last comment? | |
| Mar 16, 2016 at 18:08 | comment | added | Mikhail Katz | @Todd, I only presume to take it upon myself to give lessons to Kleene because you asked me to :-) At any rate he could have said instead "So for the intended interpretation, the variables range over the familiar/intuitive/informal numbers 0,1,2,…, i.e. the latter totality is the domain." Notice that I replaced "set" which is a technical term in ZF by an informal term "totality" and also replaced "natural" by a choice of three possible adjectives that would make it clear that his righthandside is an intuitive entity rather than giving the impression that he is talking about set-theoretic seman | |
| Mar 16, 2016 at 17:44 | comment | added | Carl Mummert | A challenge for me is that Wang and Kleene both appear to be perfectly clear to me (I am also confused about a Wikipedia-like focus on finding a source that literally uses particular words). Kleene, in particular, identifies intended interpretations as using our informal or semiformal concepts as a model of a formal system, and views the standard model of arithmetic as such - so it seems that he clearly says that the intended interpretation of arithmetic uses our informal or semiformal notion of natural number to model system $N$. Don't the informal natural numbers form a set, informally? | |
| Mar 16, 2016 at 17:22 | comment | added | Todd Trimble | @katz I'm back to not understanding what it is you are after. Just for kicks, please tell me what Kleene ought to have said that would satisfy you. | |
| Mar 16, 2016 at 17:13 | comment | added | Mikhail Katz | @ToddTrimble, but he seems to describe what he referred to earlier as the "informal side" by the words "this set is the domain". In the end he is no clearer than Wang. | |
| Mar 16, 2016 at 17:11 | comment | added | Todd Trimble | @katz So what would be wrong with that? That would be the formal side of the connection. | |
| Mar 16, 2016 at 16:53 | comment | added | Mikhail Katz | Kleene uses the term "informal" which is promising but then he writes "i.e. this set is the domain" which seems to indicate that he is back to the set-theoretic model interpreting the syntax of the theory. @Todd | |
| Mar 16, 2016 at 16:24 | comment | added | Todd Trimble | Based on some offline discussions I've been having with the OP, I imagine it's the Kleene reference which comes closest to providing what he's looking for, since it requires no further interpolation from the reader to supply words suggesting the connection between the informal and the formal (even if that may seem tacitly understood in other sources). The glosses at the end of the answer make some useful distinctions. | |
| Mar 16, 2016 at 16:12 | history | edited | Carl Mummert | CC BY-SA 3.0 |
added 29 characters in body
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| Mar 16, 2016 at 16:05 | history | answered | Carl Mummert | CC BY-SA 3.0 |