In Spivak's Calculus, he provides a """rigorous

>>8958696
Set inclusion/difference gives the ordering.

>>8958701
Could you explain this further? I'm completely lost.

>>8958696
>Can one of you explain how this is significant or rigorous.
He is showing that if you have 2 ordered pairs which are equal to each other they are made up out of the same members, it is written right there on the page.
It is obviously important to prove this, else the definition wouldn't make much sense.

>rigorous
It obviously is rigorous, there are no uncertainties and the definition is clear and unambiguous.

>significant
You need ordered pairs to construct the rational numbers and other things but the necessity for such a thing should be obvious.

>>8958715
The fuck? What's so hard to get about [math] \{a\} \subset \{a,b\} [/math]

>>8958715
>>8958728
One more hint: [math] \{a,b\} \eq \{b,a\} [/math].
You should figure the rest out by yourself (if you can't, I suggest you study something else.)

can this be extended to triplets and n-tuples
with (a,b,c)->{{a},{a,b},{a,b,c}}
(a1,a2,...,an)->{{a1},{a1,a2}...,{a1,a2,...,an}}
using induction on the same argument?

>>8958721
No, it's rigorous in the sense that it is set theoretical, i.e. it is based on more general primitive notions.

>>8958739
What do you think?

>>8958749
I think that you're a bad teacher.

>>8958749
probably, its been a while since i did anything rigorous

>>8958728
>>8958721
>>8958735

Okay, maybe this is a better way to ask:

Why wouldn't defining (a,b) as {{a},{b}} suffice?

>>8958764
>{{a},{b}}
then (1,2)=(2,1)

>>8958696
The x value is an element of the set containing the x coordinate. The y value is an element of the set containing the y coordinate.

>>8958696
>The definition and accompanying theorem seem arbitrary to me.

That's what makes it rigorous.

>>8958764
Because {{a},{b}} = {{b},{a}}. (Math is not for you. Please study something else, for your own good also.)

>>8958753
And I think you know nothing about teaching or learning.

File: 1496688145738.png (1MB, 1027x684px) Image search: [Google]

1MB, 1027x684px

>>8958744
>it is based on more general primitive notions.
This.
Mathematics is based on set theory.
He formulates [math](a,b)[/math] in set theoretic form using [math]\{[/math] and [math]\}[/math].
From this definition we see that [math](a,b)\neq (b,a)[/math] without using the concept of left and right, but the fact that these are different sets.

>>8958778
>That's what makes it rigorous.
Brainlet.

>>8958788
>Mathematics is based on set theory.
It isn't. This is just the contemporary paradigm.

>>8958696
Is Spivak's calculus a replacement for a good Analysis textbook?

>>8958795
It's an old meme.

>>8958794
What are you talking about?
Clearly mathematics make sense on it's own.
Logic and Set theory are needed so that everything is clear and there are no contradictions.
If you don't know Logic and Set theory you're a pleb.

>>8958768
Ohhh, I see it now. A follow-up question:

(1) What would be the consequence of defining (a,b) as {{a,b},{b}} ?

(2) Continuing from this rigorous construction of the ordered pair, if

A_n = {a_n}
B_n = {a_n,b_n}

would slope would be defined as ([the element of B1 that isn't A1, unless a1=b1] - [the element of B2 that isn't A2, unless a2=b2])/(A1 - A2) ?

>>8958801
Hello silly undergrad. Mathematicians had been doing mathematics for millennia before set theory. It's a 20th century paradigm to model every mathematical construct using set theory.
>Logic and Set theory are needed so that everything is clear and there are no contradictions.
This is not even wrong.

>>8958808
>(1) What would be the consequence of defining (a,b) as {{a,b},{b}} ?
The difference is immaterial.

>>8958815
You're not saying anything new.
Stop being a smartass and playing on semantics, it make you look only dumber.
Going back on my work faggot.

>>8958826
You got put in your place, accept it and shut it.

>>8958815
>It's a 20th century paradigm to model every mathematical construct using set theory.

What's the alternative?

>>8958844
Category theory. Or any other mathematical theory that is sufficiently general to model the others. (You're asking a philosophical question.)

>>8958844
Homotopy type theory.

>>8958795
It's an interesting approach to calculus that's between analysis and calculus. It's miles better if the professor complements it.

you want to create a mathematical structure. it will be a box which contains two things. the box containing x and y will be denoted (x,y). the property which you want to be satisfied is that (x,y) = (a,b) if and only if x = a and y = b. that's it. now take this idea and try to encode it into mathematics using only sets and the membership operator. the (x,y) := { {x},{x,y} } is just one very concrete way how to create such structure.

>>8958844
No need for an alternative. Set theory works well.

>>8958865
>Set theory works well.
If only.

The important part of and ordered pair is that it is ordered obviously. Try proving the definition actually has that property.

First element is the element with just one element. Second element is the element of the two element element without the first element. Third element is the element in three element element which is neither first or second element. And so on. Sets are not ordered but can provide an order like this.

>>8958784
Too late. I already have a 4.0

>>8958786
Wrong. Try again.