It's not really right to say one is a larger infinity - infinity is not a quantity and they are both infinite sets. What can be said is they have different cardinality.
By "size" in mathematics we mean the measure of the number of elements in the set.
Given 2 sets we can determine which set has a larger number of elements - the "size" or measure - by comparing each element from each set. Given sets A and B, we say Size(A)>Size(B) if and only if when comparing each set element by element we have left over elements from A that cannot be matched up with elements from B because all the elements of B have already been compared once and only once to elements in A. For example, for sets {1,6,7} and {2} we compare each element by element, take any from {1,6,7} and compare it to the one element in {2}, there are still elements in {1,6,7} that are uncompared. Therefore, {1,6,7}>{2}.
Cardinality literally just means the number of elements in the set, so the cardinality of {1,6,7} is 3. Notice, this is identical to the above definition I gave for "size".
Therefore, it is proper to say that even though the cardinality of the reals and integers are "infinite" the reals are strictly larger than the integers or have a bigger "size" by the above definition. This follows from Cantors diagonal argument. To distinguish these two different infinities we can call one uncountable and one countable - or assign it an Aleph number. Either way, one is strictly more than the other despite both being infinite.
It's not really right to say one is a larger infinity - infinity is not a quantity and they are both infinite sets. What can be said is they have different cardinality.
By "size" in mathematics we mean the measure of the number of elements in the set.
Given 2 sets we can determine which set has a larger number of elements - the "size" or measure - by comparing each element from each set. Given sets A and B, we say Size(A)>Size(B) if and only if when comparing each set element by element we have left over elements from A that cannot be matched up with elements from B because all the elements of B have already been compared once and only once to elements in A. For example, for sets {1,6,7} and {2} we compare each element by element, take any from {1,6,7} and compare it to the one element in {2}, there are still elements in {1,6,7} that are uncompared. Therefore, {1,6,7}>{2}.
Cardinality literally just means the number of elements in the set, so the cardinality of {1,6,7} is 3. Notice, this is identical to the above definition I gave for "size".
Therefore, it is proper to say that even though the cardinality of the reals and integers are "infinite" the reals are strictly larger than the integers or have a bigger "size" by the above definition. This follows from Cantors diagonal argument. To distinguish these two different infinities we can call one uncountable and one countable - or assign it an Aleph number. Either way, one is strictly more than the other despite both being infinite.