Now as you are counting, you are following a principle of probability where you want to always have a 0 put into the place-value before a 1 is put there, and once a place value is all filled up with 1s you begin a new place-value. For example, if the place-value is 3, then you start it with 100, next you turn the first zero into a 1 for 101, next you turn the second 0 into a 1 with the first 1 untouched, producing 110, and finally all the 0s are turned to 1 before going to the 4th place value.
Now the above list counted in decimals is this:
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Now, here is my question for the bright student. Take the decimal number 1*10^603, a 1 with 603 zeroes after it, and now say it is a binary number, and now here is the question, what decimal number does the binary number of 1 with 603 zeroes represent?
For example, the decimal number 10^4 is a binary number of 10000 and it represents the number 16.
But quiz questions aside, I am getting much closer to a logical answer to why infinity borderline must be of a form of 1 with a lot of trailing zeroes such as 1*10^603.
It has to do with Galois Group theory and Algebra. It has to do with the fact that when you have any system of numbers that are discrete or having empty space between successor numbers, requires what is called a "units measure".
The Naturals are discrete and its units measure is 1.
The Reals and Rationals with a borderline at infinity of 1*10^603 becomes a discrete set with empty spaces that require a "units measure".
For example, 10 can be an infinity borderline because it creates 0.1 as a units measure. But say someone chose 12 to be infinity, then we have 1/12 as 0.083333.. and that cannot act as a "units measure".
So we need pi to tell us what exponent power for the infinity borderline and it is Floor-pi*10^603 and once we have that 10^603, we fall back to 1*10^603 as the true borderline, for it furnishes us with a units measure of 1*10^-603.