/sci/ - Science & Math

>>12382920
>Have you found any examples of non-computable numbers
Lol. I wouldn't claim such foolishness. It's you who is claiming such numbers exist, and that in fact they are the vast majority of numbers. To me, a "non-computable number" is a meaningless phrase. It doesn't refer to anything, so using the term is pointless.
>I will continue to suppose that addition of non-computables still works, since you can add the sequences of the two numbers, and that sum is still convergent (Rudin also proved this).
You can continue to suppose whatever you want. You can continue to suppose that every infinitely large gnome is made up of a sequence of donuts, if that appeals to you. Just don't pretend any of it is real or even makes sense.
Also Rudin defined the reals in terms of Dedekind cuts, not Cauchy sequences. And for Dedekind cuts there is no way to add them.
>It would help if you've read his book, but it's not necessary.
Oh I have, and more advanced analysis texts as well. It's a good book.

>>12379305
My bad. I meant an algorithm for determining whether a given sequence is Cauchy (satisfies the Cauchy criterion).

>>11670498
>>11670500
If you disagree, point out a flaw in my reasoning.

>>11608907
You could show that the difference between the are of the inscribed polygon and the circle tends to zero by fitting the circle into another polygon and showing that the difference of the area of the two polygons tends to 0 as the number of sides increases. There are other strategies to prove it though (for example, using squares).

>>11597560
Possible 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.