21 Oct 2015, 5:18 pm
21 Oct 2015, 5:41 pm
21 Oct 2015, 6:19 pm
Avogadro constant, or 6.022 x 10^23 (or 602 sextillion and 200 quintillion). It's used in chemistry to represent the amount of atoms or molecules in one mole (which, in turn, is used to determine how many atoms/molecules are in one gram), so that's cool. 
21 Oct 2015, 10:27 pm
22 Oct 2015, 6:31 am
22 Oct 2015, 9:18 am
22 Oct 2015, 6:39 pm
23 Oct 2015, 11:18 am
23 Oct 2015, 2:18 pm
GoofyGreatDane wrote:
Rudin wrote:
GoofyGreatDane wrote:
Aleph-null : the cardinality of the set of natural numbers.
A cardinality is like the "number of elements in the set". For any set of finite cardinality, adding an element generates a set with a larger cardinality. {} has cardinality zero, while {a} has cardinality of one.
Aleph-null is the cardinality of the set of positive integers. As an infinite cardinal , it has some interesting properties.
One interesting property is that aleph-null + 1 =aleph-null , adding one element to an infinite set, or even a countably infinite number of elements to the set, does not change the cardinality of the set. So the set of even numbers , the set of all rational numbers, and the set of integers are of the same size. One set that is guartanteed to be larger than a given infinite set is the set of all subsets of the set.
Another interesting property of infinite cardinals is the continuum hypothesis. The generalized continuum hypothesis states that there is no set with cardinality between the that of an infinite set and the set of all subsets of of that set. It would imply that there is no set with cardinality between that of the natural numbers and that of the real numbers. This would make sense but has never been proven. And more interestingly, not only has this never been proven- but its also been shown to be independent of ZFC axiomatic set theory. This means that both the hypothesis and its negation are equally "valid " in ZFC- there is no way to construct a set in ZFC that you can prove has an intermediate cardinality, but you can't prove that there is no set with such an intermediate cardinality either.
A cardinality is like the "number of elements in the set". For any set of finite cardinality, adding an element generates a set with a larger cardinality. {} has cardinality zero, while {a} has cardinality of one.
Aleph-null is the cardinality of the set of positive integers. As an infinite cardinal , it has some interesting properties.
One interesting property is that aleph-null + 1 =aleph-null , adding one element to an infinite set, or even a countably infinite number of elements to the set, does not change the cardinality of the set. So the set of even numbers , the set of all rational numbers, and the set of integers are of the same size. One set that is guartanteed to be larger than a given infinite set is the set of all subsets of the set.
Another interesting property of infinite cardinals is the continuum hypothesis. The generalized continuum hypothesis states that there is no set with cardinality between the that of an infinite set and the set of all subsets of of that set. It would imply that there is no set with cardinality between that of the natural numbers and that of the real numbers. This would make sense but has never been proven. And more interestingly, not only has this never been proven- but its also been shown to be independent of ZFC axiomatic set theory. This means that both the hypothesis and its negation are equally "valid " in ZFC- there is no way to construct a set in ZFC that you can prove has an intermediate cardinality, but you can't prove that there is no set with such an intermediate cardinality either.
The set of algebraic reals is surprisingly equivalent to the integers, but much, much more dense.
You're only 12 and you understand about denseness, countability, and algebraic numbers? You must be a genius.
Rudin is very impressive, I agree.
Most 12 year olds wouldn't understand even 1% of what Rudin does.
23 Oct 2015, 3:49 pm
slave wrote:
GoofyGreatDane wrote:
Rudin wrote:
GoofyGreatDane wrote:
Aleph-null : the cardinality of the set of natural numbers.
A cardinality is like the "number of elements in the set". For any set of finite cardinality, adding an element generates a set with a larger cardinality. {} has cardinality zero, while {a} has cardinality of one.
Aleph-null is the cardinality of the set of positive integers. As an infinite cardinal , it has some interesting properties.
One interesting property is that aleph-null + 1 =aleph-null , adding one element to an infinite set, or even a countably infinite number of elements to the set, does not change the cardinality of the set. So the set of even numbers , the set of all rational numbers, and the set of integers are of the same size. One set that is guartanteed to be larger than a given infinite set is the set of all subsets of the set.
Another interesting property of infinite cardinals is the continuum hypothesis. The generalized continuum hypothesis states that there is no set with cardinality between the that of an infinite set and the set of all subsets of of that set. It would imply that there is no set with cardinality between that of the natural numbers and that of the real numbers. This would make sense but has never been proven. And more interestingly, not only has this never been proven- but its also been shown to be independent of ZFC axiomatic set theory. This means that both the hypothesis and its negation are equally "valid " in ZFC- there is no way to construct a set in ZFC that you can prove has an intermediate cardinality, but you can't prove that there is no set with such an intermediate cardinality either.
A cardinality is like the "number of elements in the set". For any set of finite cardinality, adding an element generates a set with a larger cardinality. {} has cardinality zero, while {a} has cardinality of one.
Aleph-null is the cardinality of the set of positive integers. As an infinite cardinal , it has some interesting properties.
One interesting property is that aleph-null + 1 =aleph-null , adding one element to an infinite set, or even a countably infinite number of elements to the set, does not change the cardinality of the set. So the set of even numbers , the set of all rational numbers, and the set of integers are of the same size. One set that is guartanteed to be larger than a given infinite set is the set of all subsets of the set.
Another interesting property of infinite cardinals is the continuum hypothesis. The generalized continuum hypothesis states that there is no set with cardinality between the that of an infinite set and the set of all subsets of of that set. It would imply that there is no set with cardinality between that of the natural numbers and that of the real numbers. This would make sense but has never been proven. And more interestingly, not only has this never been proven- but its also been shown to be independent of ZFC axiomatic set theory. This means that both the hypothesis and its negation are equally "valid " in ZFC- there is no way to construct a set in ZFC that you can prove has an intermediate cardinality, but you can't prove that there is no set with such an intermediate cardinality either.
The set of algebraic reals is surprisingly equivalent to the integers, but much, much more dense.
You're only 12 and you understand about denseness, countability, and algebraic numbers? You must be a genius.
Rudin is very impressive, I agree.
Most 12 year olds wouldn't understand even 1% of what Rudin does.
Thank you. I appreciate it.
23 Oct 2015, 4:01 pm
24 Oct 2015, 1:51 am
24 Oct 2015, 7:43 am
24 Oct 2015, 3:02 pm
24 Oct 2015, 3:10 pm
24 Oct 2015, 3:41 pm
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