Cardinals and ordinals, part 1

[@jaltoorey4445]

Wonderful thank you!

[@BelegaerTheGreat]

5:00 but if i take {1, 3} subswt will it fitst be "1"?

[@AG-sp8tc]

Great video. Helped me understand Von Neumann ordinals. But what is puzzling me is that in elementary set theory we defined a set as an UNORDERED collection. Here in higher mathematics we use the same word "set" for ORDERED collections. Basically we are calling sequences sets. Is there a different definition of set in higher mathematics that allows ordered sets?

[@annaclarafenyo8185]

Since ordinals extend notions of proof by induction, you can define addition on ordinals by induction, extending addition on the integers. This addition definition doesn't end up commutative, 1+ omega is just omega right back, while omega+1 is what you defined. But ordinal addition is in every proper sense 'ordinary addition', because it extends the definition by induction.

[@warwolt]

Fantastic video

[@NoNTr1v1aL]

Amazing video!

[@lucifersomora9068]

I Know this is Silly, but an Easyish way to Understand could be Like Playing a Game where You Make Ex. A Sword, and Every Sword You Make has a Tier. Out of making an Infinity of of them, all of the Higher Tiers are Set Off to the Side for when Your Done Making Infinitely Many of them. The Cardinal is how many You Made, but the Well Ordered Ordinality was Differentiated by the Tier.

[@lucifersomora9068]

Side Note: Doesn't that Now Blend Ordinality and Probability?

[@brett_zesty]

why are you capitalizing like that 🧐

[@lucifersomora9068]

@@brett_zesty Something I Started in Highschool Where I Give Emphasis on Any Word with a Capital Beginning.

[@tokajileo5928]

no background music please

[@cryonim]

nah, it became wildly more interesting than a generic classroom lecture thanks to it. If you want generic textbook explaination, wiki has your back.

[@sikhachakraborty3359]

❤​@@cryonim

[@shahinjahanlu2199]

Thx

[@Zero-es-natural]

I like your video but I am against using the term "ordinality". There is a better term that is actually used by set theorists: "order type".

[@angelmendez-rivera351]

The term "order type" is used by order theorists, not set theorists.

[@Zero-es-natural]

@@angelmendez-rivera351 Maybe it is used by order-theorists but I assure you that set theorists use it, as I am one I certify it.

[@angelmendez-rivera351]

Calling {0, 1, 2, 4, ..., 3} "longer" than {0, 1, 2, 3, 4, ...} is incorrect. Length requires a well-defined measure space. What you should say is that {0, 1, 2, 4, ..., 3} has an order type larger than {0, 1, 2, 3, 4, ...}.

[@warwolt]

Eh, the video was prefaced with the word length being used only analogously, and that it was in fact ordinality that was being spoken about

[@angelmendez-rivera351]

@@warwolt The preface is not helpful. Using the terminology incorrectly is bound to mislead and confuse people. That is not education.

[@warwolt]

@@angelmendez-rivera351 a good student pays attention

[@angelmendez-rivera351]

@@warwolt Paying attention is not going to do anything about literally using the incorrect terminology. I have no idea what part of this is difficult to comprehend.

[@warwolt]

@@angelmendez-rivera351 lol