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April 13th, 2017, 05:51 PM   #11
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Originally Posted by romsek View Post
oh give me a break.
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.

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Originally Posted by romsek View Post
I'm speaking within the context of this problem to a student grappling with the concept of absolute value.
EXACTLY why you should be precise.

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Originally Posted by romsek View Post
This is not the place to discuss whether or not infinity is a real number or not.
Why misinform someone who's already a little confused about the real numbers? Why not set OP up to succeed, instead of planting false information in his or her mind? If they're learning about absolute value, this is exactly when they need to start to understand what a real number is.

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.
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April 13th, 2017, 06:10 PM   #12
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Originally Posted by Maschke View Post
<minor rant>
And while I'm at it, y'all are raising my blood pressure writing $\{x : \text{foo}\}$, leading yet another generation of students to be confused about unrestricted comprehension. This includes the tutorial linked earlier in this thread which starts off with this erroneous example. Frege and Russell had this conversation over a century ago, let's all learn something from it.
</minor rant>
Can you link the Frege and Russell conversation you are referring to? Or explain the difference between $\{x : \text{foo}\}$ and $\{x | \text{foo}\}$? I don't actually know, and can't find reference of it anywhere.
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April 13th, 2017, 06:32 PM   #13
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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.
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April 13th, 2017, 06:32 PM   #14
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Originally Posted by Joppy View Post
Can you link the Frege and Russell conversation you are referring to? Or explain the difference between $\{x : \text{foo}\}$ and $\{x | \text{foo}\}$? I don't actually know, and can't find reference of it anywhere.
Oh yes certainly, happy to explain this.

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:
Originally Posted by Gottlob Frege
A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.
Great stuff, isn't it? Frege is a forgotten genius, he's credited with inventing universal and existential quantification. But today he's mostly remembered for being wrong about unrestricted comprehension, the formation of sets based on only predicates.

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
Thanks from Joppy

Last edited by Maschke; April 13th, 2017 at 06:47 PM.
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April 13th, 2017, 06:59 PM   #15
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Originally Posted by Maschke View Post
Oh yes certainly, happy to explain this.

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.
Honestly you can be as much of a pedant as you like, it doesn't bother me. The same goes for your blood pressure.

Quote:
Originally Posted by Maschke View Post
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 had his book all ready to be published, he was in the final phases of publication. 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:



Great stuff, isn't it? Frege is a forgotten genius, he's credited with inventing universal and existential quantification. But today he's mostly remembered for being wrong about unrestricted comprehension, the formation of sets based on only predicates.

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. In fact we have simply described the real numbers. The axiom of specification makes Russell's paradox disappear.

https://en.wikipedia.org/wiki/Axiom_..._specification

https://en.wikipedia.org/wiki/Naive_set_theory
Thanks for this. I was not aware of Frege's work.

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].
What do you mean it didn't take off? As i understand it, type theory solves the problem of Russell's paradox, which as far as set theory is concerned, is all we care about. Epimenides paradox among others is not resolved through type theory, but this isn't the concern.
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April 13th, 2017, 07:10 PM   #16
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Originally Posted by Joppy View Post

What do you mean it didn't take off? As i understand it, type theory solves the problem of Russell's paradox, which as far as set theory is concerned, is all we care about. Epimenides paradox among others is not resolved through type theory, but this isn't the concern.
Russell's type theory didn't get picked up by the mainstream. Zermelo's axioms of set theory (ZFC) won the twentieth century. I don't know any more than that about Russell's type theory.

Last edited by Maschke; April 13th, 2017 at 07:15 PM.
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