That is meaningless until you tell me what that means.

I can tell you what I mean by decimal notation. Decimal notation is a *function* that inputs a counting number 1, 2, 3, ... and outputs the decimal digit d(n) that appears in that decimal place [to the right of the decimal point, it's always understood.]

In other words, decimal notation is a function

d : N -> {0,1,2,3,4,5,6,7,8,9}

where N is the natural numbers starting with 1.

We can define the *value* of a given decimal expression {d(n)} as d(1)/10 + d(2()/100 + d(3)/1000 + ...

where we have already developed the theory of the real numbers, infinite sequences and series, and limits. You can refer to any standard textbook on real analysis for the details.

The key point here is that *"infinity" is not a natural number*. All the natural numbers are finite, even though there are infinitely many of them. There is no such thing in standard math as the infinitieth decimal position. It's simply not defined and there's no place for it in the accepted system.

Now it's fair for you to say that you have a different theory of the real numbers and that you don't believe in real analysis as it is presently understood. I have no problem with that. Real analysis is not the last word on the mysterious nature of the infinite. It's just a clever body of ideas and techniques that have served us well. Intellectual ideas come and go. I'm not emotionally wedded to standard math if something better comes along in a a few hundred years. Or tomorrow morning. You never know what genius is out there putting the finishing touches on the next revolution.

In other words I'm open-minded about alternative frameworks for the real numbers.

But when you say that a given digit is in the "infinitieth" place, as you can see that is *meaningless in accepted mathematics*, namely the fact that the n-th digit is defined for all n in the natural numbers, and infinity is not a natural number. So I have no idea what you mean by the infinitieth digit; and if you seek to convince me that you have thought through the implications of your own ideas, then I'd ask you to simply define for me:

What do you mean by the "infinitieth" digit?