>>11761169Possible objections you might have:
1. Not all numbers have a sequential decimal expansion of the form f:N->{digits}. Evidently, an expression like 0.0...1 cannot be made into such a sequence because for every position in a natural sequence there are finitely many terms coming before it.
To make sense of an expression like 0.0...1 rigorously, you could instead define it as an ordinal sequence f:(w+1)->digits. w+1 is ordinal obtained by taking the natural numbers and adding one element to the end that is larger than every other element. (Look up ordinals here:
https://en.wikipedia.org/wiki/Ordinal_number)
So 0.0....1 could be interpreted as f(0)=0, f(1)=".", f(2)=...=f(n)=0 and f(w)=1.
Now the problem still arises because not only you've gained a lot more numbers in this way, they would not be considered to be numbers by mathematicians because you cannot add, subtract them.
To illustrate this:
What's 10* 0.0...1? Intuitively, we shift the decimal point by 1 (do you have other suggestions?). But that would result in exactly the same representation, hence the same number 0.0...1! And we cannot have that because if 10*0.0...1 = 0.0...1, subtracting 0.0...1 we find 9*0.0...1=0 and so if we assume 0.0...1!=0, we can divide by it to find a blatant contradiction 9=0. So we see that even allowing nonstandard decimal sequences doesn't solve the problem: we need to be able to do arithmetic on numbers in expected ways and assuming 0.99...!=1 always leads us to a contradiction!
2. Not all numbers have decimal representations. In that case, there's not a lot to say here on my part except to ask what do you mean by a number then? Because in all these discussions a prevaling implicit assumption has been that numbers mostly ARE their decimal representations. "What's 0.999..? Obviously it's the number you get by writing 0 and 9999 repeating": there is no notion that it's just a notation that represents some number: it's a number itself.
cont.
