The discussion revolves around the definition of an ordered pair in set theory, specifically examining whether the definition (x,y)={x,{y}} allows for distinguishing between the first and second elements using only set theory principles.
Participants are actively questioning the definition and its implications, with some providing examples to illustrate their points. There is an acknowledgment of the complexity involved in identifying elements without prior knowledge of their identities.
Some participants reference the typical naming conventions in set theory, indicating a shared understanding of foundational concepts, while also noting the potential confusion in distinguishing elements without explicit identifiers.
reb659 said:No. But why does it matter if it is in it or not?
Dick said:You asked whether you can distinguish between x and y using set theory.
Hurkyl said:Here is an ordered pair, by this definition.
{{2}, {1}}
Can you tell me what the left element is, and what the right element is?
Remember that we're doing set theory, so x and y are variables denoting sets. There exists x and y such that (x,y)={{2},{1}}.Dick said:The definition of (x,y) is {x,{y}}, isn't it? Not {{x},{y}}.
Hurkyl said:Remember that we're doing set theory, so x and y are variables denoting sets. There exists x and y such that (x,y)={{2},{1}}.
(Also, my 1 and 2 are meant to be naming sets whose precise identity isn't relevant. The typical naming scheme here is 0={}, 1={0}, 2={0,1}, ... Mainly, I just didn't want to have millions of braces lying around in my example)