If you put a person for every rational number it would look the same as the bottom track (the rational numbers are dense) and if you put a person for every irrational number it would also look the same (because the irrational numbers are dense) - but the irrational numbers are uncountable and so are a larger infinity than the rational numbers, which are merely countable infinite. The density of the rational can be derived from the Archimedean property if the rationals (there is no largest rational and you can invert any rational, the exercise is left for the reader), the density of the irrational follows from the density of the rationals (simply use any irrational divisor and that between any rational x<y we know there exists at least one n such that x<n<y). The uncountability of the irrational follows from the fact that the reals are uncountable and the countable union of countable sets are themselves countable, i.e. the union of the rationals and irrational is uncountable therefore at least one of the rationals and irrational are uncountable, we know the rationals are countable by Cantors enumeration therefore it must be that the irrational are uncountable.
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.
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If you put a person for every rational number it would look the same as the bottom track (the rational numbers are dense) and if you put a person for every irrational number it would also look the same (because the irrational numbers are dense) - but the irrational numbers are uncountable and so are a larger infinity than the rational numbers, which are merely countable infinite. The density of the rational can be derived from the Archimedean property if the rationals (there is no largest rational and you can invert any rational, the exercise is left for the reader), the density of the irrational follows from the density of the rationals (simply use any irrational divisor and that between any rational x<y we know there exists at least one n such that x<n<y). The uncountability of the irrational follows from the fact that the reals are uncountable and the countable union of countable sets are themselves countable, i.e. the union of the rationals and irrational is uncountable therefore at least one of the rationals and irrational are uncountable, we know the rationals are countable by Cantors enumeration therefore it must be that the irrational are uncountable.
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.