Timeline for Kuratowski's definition of ordered pairs
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 21, 2011 at 8:48 | comment | added | user13113 | One formulation of what I mean is as a class of ordered tuples (satisfying the property that makes it a function), where a class is (as usual) defined by a logical predicate. | |
| May 20, 2011 at 22:58 | comment | added | Hans-Peter Stricker | @Hurkyl: What exactly is a function in your setting? Do you mean a formula, an "algorithm", or a set (of ordered tuples)? Or is there no difference? | |
| Apr 24, 2011 at 17:52 | comment | added | Amit Kumar Gupta | Or are you suggesting a language that takes additional things as primitive? If so, is the payoff in additional expressive power enough to make up for the loss of simplicity? Maybe these questions are too broad for a comment thread, but what the heck. | |
| Apr 24, 2011 at 17:50 | comment | added | Amit Kumar Gupta | @Theo, one appeal of set theory as a foundations is that while it's simple in the sense that it only takes membership as primitive, thus allowing for a cleaner analysis of the foundations, it's expressive power is such that it can interpret most of mathematics, thus for the mathematician who wants to take "ordered pair" as a primitive notion, he or she can do so without raising new foundational questions. Are you suggesting a language that doesn't take membership as primitive? If so, how would you address concerns that membership ought to be a primitive notion? (continued) | |
| Apr 24, 2011 at 17:33 | comment | added | Amit Kumar Gupta | @Joel, I think he's saying that the concept of ordered pair, as well as every other concept in mathematics, is better interpreted in some language/theory other than the language of sets/ZFC. | |
| Apr 24, 2011 at 11:20 | comment | added | Joel David Hamkins | Theo, regarding your first comment, is it your view that it isn't necessary to define an ordered pair concept in set theory? Or are you saying that it isn't necessary or desirable to do set theory at all? | |
| Apr 24, 2011 at 2:52 | comment | added | Theo Johnson-Freyd | element of $(x,y)$ that contains $0$. For option 3, again, you say: if your pair contains a unique element $a$, then $a = \{x\} = \{\{x\}\}$, so $x = \{x\}$ and return $\operatorname{first} = \operatorname{second} = a$; otherwise, the pair contains precisely two elements, (at least) one of which is a singleton; if precisely one is a singleton, you know what to do, but if both are singletons and different, you have to think. You're OK if in your foundations, the equations $a = \{b\}$ and $b = \{a\}$ together imply $a=b$, and not otherwise. | |
| Apr 24, 2011 at 2:45 | comment | added | Theo Johnson-Freyd | So the point is from 2 you can recover the pair via the algorithm: $(x,y)$ has exactly one element that itself has precisely one element; this element is $\operatorname{first}(x,y)$. If $(x,y)$ has more than one element, then it has precisely one element with precisely two elements; the element thereof that is not $\operatorname{first}(x,y)$ is $\operatorname{second}(x,y)$. In 4, you use a similar algorithm: For $\operatorname{first}$, $(x,y)$ has an element that contains $0$, and at most one other element; if $\{0\} \in (x,y)$, return $0$; else return the other element in the (continued) | |
| Apr 24, 2011 at 2:39 | comment | added | Theo Johnson-Freyd | @Hurkyl: I was trying to work out that example, and you got there first! | |
| Apr 24, 2011 at 2:33 | comment | added | user13113 | Actually, definition 1 itself depends on the axiom of foundation to be a good model of ordered pairs. If x satisfies x={{x,y}, z}, then by definition 1: ({x,y}, z) = { {x,y}, { {x,y}, z } } = { {x,y}, x } = (x, y) | |
| Apr 24, 2011 at 2:26 | comment | added | Theo Johnson-Freyd | I think this answer basically hits the mark. To the OP: It's the 21st century, so why do you care how to define ordered pairs in some set theory, when there are better languages available for doing mathematics? | |
| Apr 24, 2011 at 1:49 | history | answered | user13113 | CC BY-SA 3.0 |