How is infinity defined?

A set is infinite if it isn't finite. And a set X is finite, if it is empty or if there exists a one-on-one correspondence $X\rightarrow \{1,...,n\}$ for a natural number n.

ok and why can we say that there are more reals than naturals . I mean they are both infinite. I have seen cantors diagonal argument.

ok and why can we say that there are more reals than naturals . I mean they are both infinite. I have seen cantors diagonal argument.
There are different sizes of infinity. Two sets are said to have equal cardinality (=equal size) if there exists a one-on-one correspondence between them. A set A is said to have less or equal cardinality than B if there exists an injection $A\rightarrow B$. So a set A has strictly less cardinality if there exists an injection $A\rightarrow B$ but there does not exists a bijection $A\rightarrow B$.

So, it is very easy to see why the naturals have less cardinality than the reals. Indeed, consider

$$\mathbb{N}\rightarrow \mathbb{R}:~n\rightarrow n$$,

this is an injection. So the cardinality of N is less (or equal!!) to the cardinality of R. But, in fact, the cardinality is striclty less. For that, we need to show that there does not exist a bijection between N and R, and this is what Cantor's diagonal argument shows.

Ok I see , I am very much enjoying this conversation .

Ok I see , I am very much enjoying this conversation .
I'm glad you find this forum informative!

You may want to read "The pea and the sun" by Wapner. It has some very informative things on infinity and it's paradoxes...

thanks for the recommendation