If the real numbers have a strictly greater cardinality than the natural numbers... I think I have some questions. Is there any set with a cardinality strictly in between the real numbers and the natural numbers? If there is, is there a "least infinity" strictly bigger than the natural numbers? Is there a "greatest infinity" strictly smaller than the real numbers? Are the natural numbers the "smallest infinity"? Is there a "largest infinity"?
Very good question, particularly the first one is still an open problem in first theory. There is a hypothesis that the answer is no (the continuum hypothesis), there is no set with cardinality between the reals and integers. But this remains to be verified; as for the other questions, I think most of these rely on the continuum hypothesis to be answered.
@@coolmaththeorems I looked into it a bit. It looks like it has been shown that it can't be shown whether or not the continuum hypothesis is true. But if it is true, it looks like the "least infinity" strictly bigger than the natural numbers would be the size of the real numbers and the "greatest infinity" strictly smaller than the real numbers would be the size of the natural numbers. I wonder what could be said with the continuum hypothesis unanswered.
Love how you're smiling, shows how passionate you are!
Helpful but wish audio louder
This one’s a fun topic
Yes, really funky part of math hahaha
could limit cantor's argument to just [0,1] simpler
So, you basically count the uncomfortables.
yepp, a set is called countable if its bijective with the natural numbers or finite.
If the real numbers have a strictly greater cardinality than the natural numbers... I think I have some questions. Is there any set with a cardinality strictly in between the real numbers and the natural numbers? If there is, is there a "least infinity" strictly bigger than the natural numbers? Is there a "greatest infinity" strictly smaller than the real numbers? Are the natural numbers the "smallest infinity"? Is there a "largest infinity"?
Very good question, particularly the first one is still an open problem in first theory. There is a hypothesis that the answer is no (the continuum hypothesis), there is no set with cardinality between the reals and integers. But this remains to be verified; as for the other questions, I think most of these rely on the continuum hypothesis to be answered.
@@coolmaththeorems I looked into it a bit. It looks like it has been shown that it can't be shown whether or not the continuum hypothesis is true. But if it is true, it looks like the "least infinity" strictly bigger than the natural numbers would be the size of the real numbers and the "greatest infinity" strictly smaller than the real numbers would be the size of the natural numbers. I wonder what could be said with the continuum hypothesis unanswered.
Different sizes means boundaries. Kinda contradicts the popular understanding of infinity :/
Yeah infinity is a weird concept when you think about it