Ordinal numbers.

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Giuseppe Peano says 1 is the smallest natural number, while John von Neumann says that 0 is the smallest natural number?
Yes. In the past, or even nowadays, 0 as a number is not quite natural in many people's eyes. However, when mathematicians want to give a vigorous definition for ordinal numbers, they figure out that 0 should be the starting point.

What is the vigorous definition for ordinal numbers?
An ordinal number α is a set of ordinal numbers that are smaller then α.

So 3={0,1,2}, 5={0,1,2,3,4} are ordinal numbers?
True. All natural numbers are ordinal numbers. Indeed the set

ω={0,1,2,3,...}
is also an ordinal number. It is the smallest infinite ordinal number.

The second smallest infinite ordinal is {0,1,2,...,ω}?
Yes, it is ω+1. Note that it is different from 1+ω which equals to ω. The definition of addition is like this--
{0,1,2,...}+{0,1,2,...}={0,1,2,..., 0,1,2,...},
where the symbols with different colors are considered different, and that the elements are listed in the order.
By renaming the elements, we see that
3+2={0,1,2}+{0,1}={0,1,2}+{3,4}={0,1,2,3,4}=5.
The addition for finite ordinals is the usual addition of natural numbers.
However, we see that
ω+1={0,1,2,...}+{0}={0,1,2,...}+{ω}={0,1,2,...,ω} but
1+ω={0}+{0,1,2,3,...}={0}+{1,2,3,...}=ω.
The two are different as ω+1 has a largest element but ω does not.

I doubt we still ignore 0 with we deal with ordinal numbers in daily language?
Well, we have invented a word "zeroth", denote that something should come earlier before the first object in a series. For example, the zeroth chapter or the zeroth law of thermodynamics.