Infinity isn’t a number, it’s a concept. Some infinities are bigger than others. For example, there’s an infinite amount of real values between 0 and 1 but there’s an even greater infinite amount of real values between 0 and 2.
The cardinal numbers have entered the chat. Along with the ordinal numbers, surreal numbers, extended real numbers, projective extended real numbers, wheels, and Riemenn sphere.
That’s not true, those two infinities are functionally the same. There are bigger infinities than others, or at least higher orders, but those two are the same order.
First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.
So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.
Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.
Say we have an arbitrary mapping from every integer to a real within [0, 1]:
This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:
0-> …8… ≠ 91-> …7… ≠ 82-> …7… ≠ 83-> …9… ≠ 04-> …2… ≠ 3
⋮ ⋱
0.98803… is not within the list
Therefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.
Yes, but in that case it’s the same amount. For every real x in the first interval there is a real y=2x in the second. Also for every real y in the second interval there is a real x=y/2 in the first.
“Infinity” and “number” mean different things in different contexts. In the context of set theory, its perfectly valid to talk about infinite numbers, e.g. https://en.wikipedia.org/wiki/Aleph_number
Those are very explicitly not referring to “the number infinity” though. They’re cardinalities. A number is used to represent the cardinality of the set, but that number is not the same as the set. It refers to the size of the set, not the value of the set. Many other sets could have the same cardinality.
I’m not quite sure what you mean by “value of the set”. Do you mean ordinal numbers? Yes, those are distinct from cardinal numbers, but they’re both referred to as “numbers” in set theory. My point is that the definition of “number” and “infinity” depends on which branch of math you’re working in.
Infinity isn’t a number, it’s a concept. Some infinities are bigger than others. For example, there’s an infinite amount of real values between 0 and 1 but there’s an even greater infinite amount of real values between 0 and 2.
The cardinal numbers have entered the chat. Along with the ordinal numbers, surreal numbers, extended real numbers, projective extended real numbers, wheels, and Riemenn sphere.
That’s not true, those two infinities are functionally the same. There are bigger infinities than others, or at least higher orders, but those two are the same order.
May as well go through the proofs:
First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.
So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.
Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.
Say we have an arbitrary mapping from every integer to a real within [0, 1]:
0 -> 0.89236… 1 -> 0.47389… 2 -> 0.84776… 3 -> 0.18790… 4 -> 0.90542… ⋮ ⋱This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:
0 -> …8… ≠ 9 1 -> …7… ≠ 8 2 -> …7… ≠ 8 3 -> …9… ≠ 0 4 -> …2… ≠ 3 ⋮ ⋱ 0.98803… is not within the listTherefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.
This isn’t true. Both of those sets have the same cardinality as the real numbers. Measuring infinities can be weird that way.
They are both strictly larger than the rationals, though.
Yes, but in that case it’s the same amount. For every real x in the first interval there is a real y=2x in the second. Also for every real y in the second interval there is a real x=y/2 in the first.
“Infinity” and “number” mean different things in different contexts. In the context of set theory, its perfectly valid to talk about infinite numbers, e.g. https://en.wikipedia.org/wiki/Aleph_number
Those are very explicitly not referring to “the number infinity” though. They’re cardinalities. A number is used to represent the cardinality of the set, but that number is not the same as the set. It refers to the size of the set, not the value of the set. Many other sets could have the same cardinality.
I’m not quite sure what you mean by “value of the set”. Do you mean ordinal numbers? Yes, those are distinct from cardinal numbers, but they’re both referred to as “numbers” in set theory. My point is that the definition of “number” and “infinity” depends on which branch of math you’re working in.