Formal numbers, 11 Essence of Set Theory

[@epsilon7758]

Your channel is a masterclass ! Do more video pls we need it

[@azoshin]

Nice explanation of the Axiom of infinity. Loved learning it 👍.

[@nathn.k]

10:23 isn't omega already part of the well orderings in P(ω × ω)?

[@notmyname7698]

7:06 What if we don't remove the further ordinal numbers?

[@notmyname7698]

Like, what if we remove just any arbitrary crazy sets?

[@procdalsinazev]

@@notmyname7698 If you select only ordinals from the set guaranteed by axiom of infinity, then the result could be any set of ordinal numbers containing zero, and being closed to ordinal successor. It could be for example any limit ordinal -- omega, omega*2, omega^omega, omega1+omega, etc. But it doesn't even need to be an ordinal since it could have "holes", it could be for example the set of all ordinal numbers alpha such that alpha < omega, or omega+42 < alpha < omega*2.

[@notmyname7698]

​@@procdalsinazevI mean why did we remove the further ordinal numbers if we want omega1?

[@procdalsinazev]

@@notmyname7698 The main reason is to have control over what we have. Without the removal, you don't know what the set is -- it could be omega, and it could be something bigger. If we do the filtering step, we know we are dealing with omega. Regarding the construction of omega1 -- if you follow the construction exactly with a bigger set, two things could go wrong -- either the set could contain some ordinals bigger than omega1, so after the final union with this set, you will have extra elements. There is also the other thing that could go wrong -- the set could be uncountable, so you construct wrong (too big) ordinal numbers in the translation process from a well-ordered set into an ordinal.

[@nathn.k]

10:08 But how can we know that the set of well orders of the natural numbers created is the set of *all* well orders of the natural numbers?

[@procdalsinazev]

The set of all well orders of natural numbers was constructed at 8:40. It uses the axiom of separation which gives us all subset of a given subset satisfying certain property. Every well-order is an element of P(ω × ω), so the axiom of separation provides the set of all well orders on ω.

[@nathn.k]

@@procdalsinazev But how can we know that "Every well-order is an element of P(ω × ω)"

[@procdalsinazev]

@@nathn.k By definition, an order on S is a subset of the cartesian product S x S satisfying certain properties, as discussed in chapter 10a.

[@nathn.k]

@@procdalsinazev oh yeah you're right I forgot