April 13th, 2017, 05:51 PM  #11  
Senior Member Joined: Aug 2012 Posts: 1,783 Thanks: 482  Sorry, I'm not able to retract. You are mathematically wrong, which in this particular case I don't care about; and pedagogically wrong, which is the real problem here. Quote:
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ps  And I see you did succeed in misinforming the OP. How can you defend this? Don't mean to get on a soapbox but you can see from what you wrote and from the OP's response that you imparted a falsehood just as they're learning what a real is. Sorry I just can't dial back my concern here. Gotta go with what I wrote. Last edited by Maschke; April 13th, 2017 at 06:02 PM.  
April 13th, 2017, 06:10 PM  #12  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,545 Thanks: 518 Math Focus: Yet to find out.  Quote:
 
April 13th, 2017, 06:32 PM  #13 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,545 Thanks: 518 Math Focus: Yet to find out. 
Such good points here. I have to say it's a real eye sore when someone replies to another in need of help, but doesn't bother to consider what that person may or may not already know based on how they have worded the question and approached the topic. i.e., the response to a question is correct, but is at a level of depth that does not concern the OP, nor aids in the OP's understanding of what is going on. Rather, it leads them on a goose chase into a new world of notation and concepts that potentially make things worse. On the other hand, there are responses which are incorrect, yet depending on what the OP already knows, may be beneficial. Or perhaps not. It may be that OP is at the level of understanding where he/she may be confused by some sloppy math. Which leads me to laugh about statements regarding "pedagogically wrong" or "pedagogically right", there is simply no such thing. Learning and teaching are subjective experiences, and is something that cannot be standardized or catered to all people. 
April 13th, 2017, 06:32 PM  #14  
Senior Member Joined: Aug 2012 Posts: 1,783 Thanks: 482  Quote:
I just want to say for the record that my picky little pedantry got out of its cage again and I do apologize for letting it loose, but it does that from time to time. Sorry for the notational confusion. The distinction is not between $\{x : \text{foo}\}$ and $\{x  \text{foo}\}$, which are synonymous (and imprecise). Rather, the correct formulation is $\{x \in \mathbb R : P(x)\}$ where $P$ is a unary predicate; that is, a statement that ranges over elements of the reals and returns a value of true or false for each real that you plug in. The point is that we have to say on the left side of the colon or vertical bar what set $x$ ranges over. Back in the 1890's Gottlob Frege said that every predicate forms a set. There's the set of even numbers, the set of giraffes, the set of this, the set of that. If you have a predicate $P$ you can form the set $\{x : P(x)\}$. The colon or the vertical bar are interchangeable. The story is that Frege was in the final phases of publication of his big book of mathematical logic. Just at that moment, Bertrand Russell said, Hey, what if $P$ is the predicate $x \notin x$? If that is allowed to define a set, then that set both is and isn't a member of itself, contradiction. Frege's entire system of set theory is busted. Frege stopped the presses and put in an afterword after finding out about Russell's paradox. Frege famously wrote: Quote:
Russell's solution was to invent type theory, but this didn't take off and didn't achieve mindshare at the time. [It's gaining contemporary mindshare though]. Instead, the axiom schema of replacement was adopted, which says that if $X$ is some existing set, and $P$ is a predicate, then we can form the set $\{x \in X : P(x)\}$. In other words a predicate can only be used on the elements of an existing set. See how the Russell paradox goes away now. Suppose we form the set $\{x \in \mathbb R : x \notin x\}$. Well is $3 \in 3$? No. Is $\pi \in \pi$? No. So in fact every real number satisfies that predicate. We've just described the real numbers! The axiom of specification makes Russell's paradox disappear. https://en.wikipedia.org/wiki/Russell%27s_paradox https://en.wikipedia.org/wiki/Axiom_..._specification https://en.wikipedia.org/wiki/Naive_set_theory https://en.wikiquote.org/wiki/Gottlob_Frege Last edited by Maschke; April 13th, 2017 at 06:47 PM.  
April 13th, 2017, 06:59 PM  #15  
Senior Member Joined: Feb 2016 From: Australia Posts: 1,545 Thanks: 518 Math Focus: Yet to find out.  Quote:
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April 13th, 2017, 07:10 PM  #16  
Senior Member Joined: Aug 2012 Posts: 1,783 Thanks: 482  Quote:
Last edited by Maschke; April 13th, 2017 at 07:15 PM.  

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