There are two distinct notions, infinite ordinals, which are extremely important, and infinite cardinals which are more intuitive, but less useful.
Infinite ordinals can be understood by considering sequences of points on a line. The rules are you can move to the right only to add points, in discrete steps, and whenever these points reach any sort of limiting accumulation point, the accumulation point is in the set.
So you can make infinitely many steps to the right, reach an accumulation point, and make infinitely many steps again. You can have accumulation points of accumulation points, the structure can be incredibly complex. But it is easy to see, under these conditions, that moving to the left, you always hit zero after a finite number of steps. The reason is you can't reach an accumulation point when going down, because then the limiting point would have no neighbor to the right, so it wasn't produced by a step, contrary to the construction.
This construction produces the countable ordinals, and you can understand it as the partial sums of a sequence, the sequence which is the length of the steps. Cantor identified these infinite ordinal structures as important, and founded set theory to study them. He understood that ordinals allow induction -- if a property holds for the ordinal "0" and it is true that the property for ordinal a implies the property for a+1, and also that the property for all ordinals limiting to ordinal b implies the property holds at b, then the property holds for all ordinals. This is the transfinite induction which gives set theory power over arithmetic.
He also identified the notion of set cardinality, and used it to argue that the real numbers are uncountable. But he was so in love with the ordinal structure, that he was sure that the real numbers too could be given an ordinal structure. He believed that the real numbers were the size of the first uncountable ordinal, and this is the continuum hypothesis, and struggled to prove it.
Infinite ordinals can be understood by considering sequences of points on a line. The rules are you can move to the right only to add points, in discrete steps, and whenever these points reach any sort of limiting accumulation point, the accumulation point is in the set.
So you can make infinitely many steps to the right, reach an accumulation point, and make infinitely many steps again. You can have accumulation points of accumulation points, the structure can be incredibly complex. But it is easy to see, under these conditions, that moving to the left, you always hit zero after a finite number of steps. The reason is you can't reach an accumulation point when going down, because then the limiting point would have no neighbor to the right, so it wasn't produced by a step, contrary to the construction.
This construction produces the countable ordinals, and you can understand it as the partial sums of a sequence, the sequence which is the length of the steps. Cantor identified these infinite ordinal structures as important, and founded set theory to study them. He understood that ordinals allow induction -- if a property holds for the ordinal "0" and it is true that the property for ordinal a implies the property for a+1, and also that the property for all ordinals limiting to ordinal b implies the property holds at b, then the property holds for all ordinals. This is the transfinite induction which gives set theory power over arithmetic.
He also identified the notion of set cardinality, and used it to argue that the real numbers are uncountable. But he was so in love with the ordinal structure, that he was sure that the real numbers too could be given an ordinal structure. He believed that the real numbers were the size of the first uncountable ordinal, and this is the continuum hypothesis, and struggled to prove it.
