Ordinals and cardinals are difficult to explain rigorously without using technical notation. This video achieved this goal clearly and in a minimal time for the listener. Bravo and thank you for this piece of art !!
Helpful video, thank you. Must clarify one point. Ordinal addition and multiplication are non-commutative. omega plus omega does NOT equal 2 times omega. Two times omega is just omega. Omega plus omega is omega*2.
Ordinality represents an order type. If there's one element greater than everything (w + 1), thats different from no element being greater than everything (w), even though they both have the same number of elements.
@@Avgur_Smile Based on this video, yes. Both sets share the cardinality of the natural numbers. Since the latter is the the natural numbers excluding a single element. An infinite set reduced by a single element is still an infinite set, and since there is no extra elements that need to ordered at the end of the set, I am pretty sure it has same ordinality (which many are identifying as being called order type in rigorous context).
@@isaiahgonzalez8845 If we consider mathematics as a science and not as a religious teaching, there should be no place for statements like "I am pretty sure" in it. Proof is needed. It seems to me that in this case we have a statement that, based on the ZFC axiom system, cannot be proved or disproved according to Gödel's completeness theorem.
@@Avgur_Smile Mathematics is neither science nor religion. This is certainly not the case of incompleteness of ZFC. Consider the function f that maps {0,1,...} to {1,2,...} by adding 1 to each element, this preserves the order (as x>y implies x+1>y+1) and is a bijection from one set to the other, hence it is an order isomorphism. As there is an order isomorphism between the two ordered sets the two sets have the same order type. This is certainly within the scope of ZFC.
Ordinals and cardinals are difficult to explain rigorously without using technical notation. This video achieved this goal clearly and in a minimal time for the listener. Bravo and thank you for this piece of art !!
The explanation here is not rigorous by any stretch of the imagination.
@@angelmendez-rivera351 Ok brother it's rigorous enough for us lesser mortals
@@saiftaher2210 .i.e., it isn't rigorous, but conveniently understandable. Those aren't synonymous.
that last minute of this video was very information dense. I would love to learn more about large cardinals and forcing axioms
Thank you for these wonderfully clear videos, truly excellent!
Fascinating and mind-boggling at the same time... 👍
Lots of stuff there that I didn't know. Thank you for putting all if this together. Looking forward to the next one.
Helpful video, thank you. Must clarify one point. Ordinal addition and multiplication are non-commutative. omega plus omega does NOT equal 2 times omega. Two times omega is just omega. Omega plus omega is omega*2.
Thank you
Thank you! This makes much more sense
please make a video on the Church-Kleene ordinal!
Excellent. Thank you.
WOAH, A PROVEN TO BE UNPROVABLE!
great video...
Thanks!
I can't understand why the ordinality of the infinite set increases when you change the order of the elements?
Ordinality represents an order type. If there's one element greater than everything (w + 1), thats different from no element being greater than everything (w), even though they both have the same number of elements.
@@thedapperegg689 Do you think that ordinality of {0, 1, 2, 3, 4, ...} and ordinality of {1, 2, 3, 4, ...} are equal to omega?
@@Avgur_Smile Based on this video, yes. Both sets share the cardinality of the natural numbers. Since the latter is the the natural numbers excluding a single element. An infinite set reduced by a single element is still an infinite set, and since there is no extra elements that need to ordered at the end of the set, I am pretty sure it has same ordinality (which many are identifying as being called order type in rigorous context).
@@isaiahgonzalez8845 If we consider mathematics as a science and not as a religious teaching, there should be no place for statements like "I am pretty sure" in it. Proof is needed.
It seems to me that in this case we have a statement that, based on the ZFC axiom system, cannot be proved or disproved according to Gödel's completeness theorem.
@@Avgur_Smile Mathematics is neither science nor religion. This is certainly not the case of incompleteness of ZFC. Consider the function f that maps {0,1,...} to {1,2,...} by adding 1 to each element, this preserves the order (as x>y implies x+1>y+1) and is a bijection from one set to the other, hence it is an order isomorphism. As there is an order isomorphism between the two ordered sets the two sets have the same order type. This is certainly within the scope of ZFC.
From now on I think you better say; "Let's invent the maths!" 😉
These "higher" ordinals seem pretty useless. They are all just contained in aleph-null.