The subtle definition of the ordered pair (x,y).

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[PixelSergey]

I love these sorts of "trivial-seeming" proofs. Great to see them opened up!

[yyeeeyyyey]

When using this definition, higher order tuples should be defined recursively, that is, (a, b, c) is defined as ((a, b), c), and so on.

Note that the definition (a, b, c) = { {a}, {a, b}, {a, b, c} } does not work, even though it looks like a natural generalization, as shown in the following example (assume a and b are distinct):

(a, a, b) = { {a}, {a, a}, {a, a, b} } = { {a}, {a}, {a, b} } = { {a}, {a, b} } = { {a}, {a, b}, {a, b} } = { {a}, {a, b}, {a, b, b} } = (a, b, b)

that is, the definition (a, b, c) = { {a}, {a, b}, {a, b, c} } would imply the equality (a, a, b)=(a, b, b) which is unwanted for ordered pairs.

Very interesting definition nonetheless. It's always interesting how the equality {a, b}={a} when a=b affects calculations in unintuitive ways.

[Nikolas_Davis]

Very interesting! This reminds me of how you can define a natural number (or rather, its ordinal), as the set of all natural numbers preceding it. It may seem perverse to our human intuition (we conceive numbers as a model, by analogy), but in an axiomatic system that only knows of the set as a building block, there is little choice - everything has to be built from sets.

I remember how weird this felt when I was first introduced to it. To my mind, the set is a secondary object, a collection of things that are *not* (necessarily) sets (and this is in fact how naive set theory defines them, or rather doesn't define them). But in axiomatic set theory, it is the *set* that is primary, and all other objects are constructs made of sets. Takes a while to get used to :-)

[andrewlitfin]

Fun fact: in view of Separation you can weaken Pairing to get the same theory. Specifically, rather than Pairing being "For all x, for all y, there exists z such that for all w, w in z iff w=x or w=y", you can weaken it to "For all x, for all y, there exists z such that x in z and y in z" and then use Separation with formula p(w)="w=x or w=y" to separate out the subset containing *only* x and y. (I learned this from Kunen's "Set Theory" section I.4)

[francocosta]

Wow, i've never found a texbook with the proof(only the definition), is so elegant and clear. Thanks a lot!

[jimmymontoki]

Thank you so much. I had seen that definition in a book and didn't understood it. Now it's way more clear

[duncanw]

This is actually a pretty intuitive way to define an ordered object from a collection of unordered ones; you're just distinguishing the element that's first by pairing it with the whole object. I see the generalization to n-tuples too I think: you'd just have a subset of the unordered collection corresponding to the k-th element of the tuple where the k-th element is the one not found in all the smaller sets in the collection.

It's a pretty important construction too. If I recall correctly set theorists these days define functions as subsets of the Cartesian product (maybe a power set is involved?) in a subtle way that always escaped me...

[mueezadam]

Great concept to showcase. I remember coming across this more precise definition in Munkres’ Topology.

[angel-ig]

Nice! I was wondering this the other day and then forgot about it. Thanks for making it clear now!

[synaestheziac]

I don’t think I’ve ever heard the word ‘doubleton’, but I love it

[thefranklin]

I love sets so much it hurts sometimes

[punditgi]

Great videos! These are a good place to start!

[LeeDeVilleMath]

Excellent... I was just prepping a lecture on exactly this so love to see it. You've definitely highlighted the key ideas in a really nice way. I'll send my students this way when the topic comes up!

[barbietripping]

Prof be reading my mind again. Been reading and writing about this

[natepolidoro]

Very nice video, I enjoy this kind of foundational stuff.

[hypergaming]

I learned first time set theory, Nicely explained.

[abrahammekonnen]

I don't usually comment on videos(and I'm sorry for that), but I really enjoyed this video and I hope you'll have more like it.
I'll add some questions I have about the material later.

[peterdecupis]

Such kind of "trivial" demonstrations are very common in modern axiomatic Set Theories; in the case of point, Mick is treating the Kuratowski definition, that is mainly employed in the magnificent GB Theory, which distinguishes pure Classes and Set classes: the first ones can be only subjects of the fundamental transitive action (i.e. pure Classes can only contain), whilst the second ones can be both subjects and objects (i.e. sets can contain and be contained). An axiom that is implicitely employed in the video is the extensional definition of the equality relation "="; this states that a pair of symbols a, b represent the same container-class (i.e. we shall pose: a=b) iff for any containable set x: x belongs to a iff x belongs to b.
In the GB theory the "Couple" Axiom that Mick cited states that for any pair of symbols representing containable sets a, b the container Class {a, b}, that can be built via Separation axiom (throgh the well-posed proposition p(x)= "x=a VEL x=b"), is a proper Set, i.e. {a, b} may be contained by other classes or sets. The unicity of this non-order couple-set is a consequence of the extensional definition of the equality.
The GB Theory, provided with the Choice Axiom, allows a rigorous foundation of Ordinal Numbers; for instance,
finite ordered n-plets are defined as functions of finite ordinal domain n; this definition can be generalized to infinite ordered sequences which are represented by functions with a transfinite ordinal domain.

[antoniomello]

Just a little remark: Michael showed why the sets {a, b} and {{a}, {a, b}} exist by using the Axiom of Pairing, but to actually ensure the existence of the latter, you must first also show why {a} exists. For this there is another axiom of set theory that states the existence of the singleton {x} given a set x.

[Fine_Mouche]

3:30 why it's not, if a = b : {a, b}={a, a} ? because for me it's weird that {a, b}={a}, because it's mean that a set of 2 elements = a set of 1 elements, it's little like saying 1=2 for me.