0:25 Referring to this statement "finite means countable and infinite obviously means uncountable", I would like to add a comment. Thanks to Georg Cantor, it is now a standard practice in mathematics to categorize sets into two main types: countable sets - encompassing both finite and infinite sets - and uncountable sets. The idea of a countable infinite set might seem strange at first because we cannot "count" an infinite set in the usual sense of finishing the count. However in mathematics, the term countable has a specific or specialized meaning. A set is countable if its elements can be indexed by the natural numbers. In other words, a countable set can be placed into a sequence, such as (x1,x2,x3,…,xn) or (x1,x2,x3,…). Cantor's great discovery was that there exist sets which cannot be indexed by the natural numbers. For example Cantor showed the set of real numbers cannot be put in a sequence (x1,x2,x3,…) without necessarily missing elements of the reals. The empty set is a special case: it is a countable set, as it contains no elements, and can trivially be indexed.
This is such a well taught and understandable course that it would also work for high schoolers
happy teacher's day mam ... you are the best teacher I have ever seen in this world ...
0:25 Referring to this statement "finite means countable and infinite obviously means uncountable", I would like to add a comment.
Thanks to Georg Cantor, it is now a standard practice in mathematics to categorize sets into two main types: countable sets - encompassing both finite and infinite sets - and uncountable sets.
The idea of a countable infinite set might seem strange at first because we cannot "count" an infinite set in the usual sense of finishing the count. However in mathematics, the term countable has a specific or specialized meaning. A set is countable if its elements can be indexed by the natural numbers. In other words, a countable set can be placed into a sequence, such as (x1,x2,x3,…,xn) or (x1,x2,x3,…).
Cantor's great discovery was that there exist sets which cannot be indexed by the natural numbers. For example Cantor showed the set of real numbers cannot be put in a sequence (x1,x2,x3,…) without necessarily missing elements of the reals.
The empty set is a special case: it is a countable set, as it contains no elements, and can trivially be indexed.
Thank you for sharing this video. This was very helpful.
holy cow this was so helpful! Thanks!
happy teacher's day
Thank you!
❤