24 Jul 2015, 8:55 pm
25 Jul 2015, 5:25 pm
Rudin wrote:
Infinity is an interesting concept there is more than one type of infinity. Some are bigger and smaller than others.
k+∞=∞
k*∞=∞
That infinity is called Aleph-1 and is the smallest infinity.
Imagine the universe is a sphere that has a infinite radius (Aleph 1) and a volume of 4/3* π*r^3 if you multiply the radius by 17,000,000,000,000,000,000,000 i.e have V=(4/3* π*(17,000,000,000,000,000,000,000r)^3) the volume is still equivalent.
An example of an infinite set is the natural numbers 1,2,3,4,5... there is an infinite number of elements in this set Aleph 1 is called the cardinality of the set (number of elements). The integers, fractions and algebraic numbers have the same cardinal.
But the real numbers i.e 1.23344356575675788....., 2.212564545454445......, pi, e, phi, 1, -1, -0.93817239121829312363712361873... etc. have a cardinality of Aleph 2.
Another set with a cardinality of aleph 2 is the complex numbers they are the same size as the real numbers.
The power set of set is the number of subsets in said set.
Example:
S={1,2,3,4,5...}
P(S)={1},{2},{3},..., {1,2}, {1,3}, {1,4},..., {2,1}, {2,3},..., {1,2,3}, {1,2,4},... a much larger set in case you didn't notice.
The cardinality function is denoted by card (S) where S is a set. card (P(s))=2^(card(S)) so if S has cardinality aleph-1 then P(s) has cardinality Aleph-2.
Very interesting stuff. You should read more about it it is very interesting especially the brilliant man who discoevred it.
k+∞=∞
k*∞=∞
That infinity is called Aleph-1 and is the smallest infinity.
Imagine the universe is a sphere that has a infinite radius (Aleph 1) and a volume of 4/3* π*r^3 if you multiply the radius by 17,000,000,000,000,000,000,000 i.e have V=(4/3* π*(17,000,000,000,000,000,000,000r)^3) the volume is still equivalent.
An example of an infinite set is the natural numbers 1,2,3,4,5... there is an infinite number of elements in this set Aleph 1 is called the cardinality of the set (number of elements). The integers, fractions and algebraic numbers have the same cardinal.
But the real numbers i.e 1.23344356575675788....., 2.212564545454445......, pi, e, phi, 1, -1, -0.93817239121829312363712361873... etc. have a cardinality of Aleph 2.
Another set with a cardinality of aleph 2 is the complex numbers they are the same size as the real numbers.
The power set of set is the number of subsets in said set.
Example:
S={1,2,3,4,5...}
P(S)={1},{2},{3},..., {1,2}, {1,3}, {1,4},..., {2,1}, {2,3},..., {1,2,3}, {1,2,4},... a much larger set in case you didn't notice.
The cardinality function is denoted by card (S) where S is a set. card (P(s))=2^(card(S)) so if S has cardinality aleph-1 then P(s) has cardinality Aleph-2.
Very interesting stuff. You should read more about it it is very interesting especially the brilliant man who discoevred it.
Georg Cantor
one of history beautiful minds 
25 Jul 2015, 5:29 pm
Adamantium wrote:
Fnord wrote:
This is also called a Type 1 Multiverse in Max Tegmark's Taxonomy of Universes.
Source: Wikipedia Article on Multiverse
Wikipedia wrote:
"... an infinite universe will contain an infinite number of Hubble volumes, all having the same physical laws and physical constants. In regard to configurations such as the distribution of matter, almost all will differ from our Hubble volume. However, because there are infinitely many, far beyond the cosmological horizon, there will eventually be Hubble volumes with similar, and even identical, configurations. ... Given infinite space, there would, in fact, be an infinite number of Hubble volumes identical to ours in the Universe ..."
Source: Wikipedia Article on Multiverse
Which logically means an infinite number of Fnords and infinities of near-Fnord variants in nearly identical hubble volumes...
Something entertaining to think about.
Dog help us all
(if there is more than one Fnord)
25 Jul 2015, 6:32 pm
slave wrote:
Rudin wrote:
Infinity is an interesting concept there is more than one type of infinity. Some are bigger and smaller than others.
k+∞=∞
k*∞=∞
That infinity is called Aleph-1 and is the smallest infinity.
Imagine the universe is a sphere that has a infinite radius (Aleph 1) and a volume of 4/3* π*r^3 if you multiply the radius by 17,000,000,000,000,000,000,000 i.e have V=(4/3* π*(17,000,000,000,000,000,000,000r)^3) the volume is still equivalent.
An example of an infinite set is the natural numbers 1,2,3,4,5... there is an infinite number of elements in this set Aleph 1 is called the cardinality of the set (number of elements). The integers, fractions and algebraic numbers have the same cardinal.
But the real numbers i.e 1.23344356575675788....., 2.212564545454445......, pi, e, phi, 1, -1, -0.93817239121829312363712361873... etc. have a cardinality of Aleph 2.
Another set with a cardinality of aleph 2 is the complex numbers they are the same size as the real numbers.
The power set of set is the number of subsets in said set.
Example:
S={1,2,3,4,5...}
P(S)={1},{2},{3},..., {1,2}, {1,3}, {1,4},..., {2,1}, {2,3},..., {1,2,3}, {1,2,4},... a much larger set in case you didn't notice.
The cardinality function is denoted by card (S) where S is a set. card (P(s))=2^(card(S)) so if S has cardinality aleph-1 then P(s) has cardinality Aleph-2.
Very interesting stuff. You should read more about it it is very interesting especially the brilliant man who discoevred it.
k+∞=∞
k*∞=∞
That infinity is called Aleph-1 and is the smallest infinity.
Imagine the universe is a sphere that has a infinite radius (Aleph 1) and a volume of 4/3* π*r^3 if you multiply the radius by 17,000,000,000,000,000,000,000 i.e have V=(4/3* π*(17,000,000,000,000,000,000,000r)^3) the volume is still equivalent.
An example of an infinite set is the natural numbers 1,2,3,4,5... there is an infinite number of elements in this set Aleph 1 is called the cardinality of the set (number of elements). The integers, fractions and algebraic numbers have the same cardinal.
But the real numbers i.e 1.23344356575675788....., 2.212564545454445......, pi, e, phi, 1, -1, -0.93817239121829312363712361873... etc. have a cardinality of Aleph 2.
Another set with a cardinality of aleph 2 is the complex numbers they are the same size as the real numbers.
The power set of set is the number of subsets in said set.
Example:
S={1,2,3,4,5...}
P(S)={1},{2},{3},..., {1,2}, {1,3}, {1,4},..., {2,1}, {2,3},..., {1,2,3}, {1,2,4},... a much larger set in case you didn't notice.
The cardinality function is denoted by card (S) where S is a set. card (P(s))=2^(card(S)) so if S has cardinality aleph-1 then P(s) has cardinality Aleph-2.
Very interesting stuff. You should read more about it it is very interesting especially the brilliant man who discoevred it.
Georg Cantor
one of history beautiful minds Yes. He had Bipolar Disorder which was most likely caused by harassment from his colleagues which regarded his work in set theory.
25 Jul 2015, 6:38 pm
eric76 wrote:
naturalplastic wrote:
zer0netgain wrote:
Or the mindbender that a subset of infinity is itself infinite even though it's smaller. 
The set of odd numbers is smaller than the set of whole numbers. But both are infinity.
It is very easy to map the set of whole numbers onto the odd numbers and the odd numbers onto the whole numbers.
Therefore, there is a 1:1 correspondence between the two sets and thus both have the same number of elements.
Consider this:
The odd numbers are the positive integers not divisible by 2: { 1, 3, 5, 7, 9, ... }
The Whole numbers are the postive integers and 0: { 0, 1, 2, 3, 4, 5, ... }
Let O be the set of odd numbers and W be the set of whole numbers.
Define f :W->O by f(x)=2x+1 for all x in W. Since for every x in W, there is an f(x) in W, we have that |O| >= |W|.
Define g:O->W by g(y)=(y-1)/2 for all y in O. Since for every y in O, there is a g(y) in W, we have that |O|<=|W|.
Thus, |O|=|W|.
Yes exactly. The person the OP quoted is actually wrong, there is the same number of each.
Take {1,2,3,4...} and {1,3,5,7,...}. Two sets are equal if there are the same number of elements or as eric76 said there is a 1-1 correspondence between the elements in each set i.e.
1 2 3 4 . . .
1 3 5 7 . . .
Same number in each. Cantor's diagonal proof shows the rationals (fractions + and -) are equivalent in cardinal to the integers and natural numbers.
22 Jun 2016, 5:28 am
Ban-Dodger
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Tollorin
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Joined: 14 Jun 2009
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23 Jun 2016, 2:13 pm
eric76 wrote:
naturalplastic wrote:
zer0netgain wrote:
Or the mindbender that a subset of infinity is itself infinite even though it's smaller.
The set of odd numbers is smaller than the set of whole numbers. But both are infinity.
It is very easy to map the set of whole numbers onto the odd numbers and the odd numbers onto the whole numbers.
Therefore, there is a 1:1 correspondence between the two sets and thus both have the same number of elements.
Consider this:
The odd numbers are the positive integers not divisible by 2: { 1, 3, 5, 7, 9, ... }
The Whole numbers are the postive integers and 0: { 0, 1, 2, 3, 4, 5, ... }
Let O be the set of odd numbers and W be the set of whole numbers.
Define f :W->O by f(x)=2x+1 for all x in W. Since for every x in W, there is an f(x) in W, we have that |O| >= |W|.
Define g:O->W by g(y)=(y-1)/2 for all y in O. Since for every y in O, there is a g(y) in W, we have that |O|<=|W|.
Thus, |O|=|W|.
Even then, there is still infinite that are bigger that other. Take the natural numbers; 1,2,3,4,5,6... and so on. There is a infinity of natural numbers. Yet, only by adding a decimal; 0.1, 0.2, 0.3, 0.4, 0.5, .... and so on, you are multiplying that infinite by ten. And you can multiply this result by ten a infinite number of times. There is a infinity of infinite.
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