crank wrote:scott1328 wrote:

I am arguing with your benighted terminology. you keep saying integers are not real numbers, but they are, you keep saying other very odd things. Like 2.0000... is not the same as 2, that discrete objects cannot be counted with real numbers, they can. For some reason you seem to think that the "non-existance" of reals implies a discrete universe, it doesn't.

Or at least that is what you appear to be saying to me.

If you're doing some kind of mathematics that is restricted to integers, then to me, the 2 used there is not real, it's integer. Yes, integers are a subset of reals, but when you're doing something where reals don't apply, then you can't use 2.000..., you use 2. There's some fundamental difference when using an integer verses using a real, even if they're the same number. It's the difference between continuous and discrete, how an integer can only change by whole multiples of 1, and reals can change by infinitesimals. I apologize that I don't know the technical language, but I can't believe these distinctions aren't made in some way.

I think, in my far from qualified way, that infinite precision can't be real, from the informational aspects, which you don't agree exist, but there's a lot of mathematicians and physicists who do. Also, because of how there's a lot of physics that imply a quantized universe, I don't even know if that could technically be equivalent to 'discrete universe', though it has some real basic similarities if it isn't equivalent.

I think the problem here is that you are thinking of irrational numbers in terms of their expression/expansion rather than in terms of their position on a continuous line. All real numbers partition a line. Take for example sqrt(2) this quantity partitions the line into two subsets: set A which contains all the negative numbers and all the numbers whose square is less than 2, and set B which contains all those numbers whose squares are greater than or equal to two. If you think about it this way, you realize that it is not about calculating the decimal positions, but rather of deciding to which partition a quantity belongs.

In the same way you can see that the integer 2 partitions a line in exactly the same way the irrational sqrt(2) partitions a line. In set A are all those quantities that are less than two, in set B are all those quantities that are greater than or equal to 2.

This method called a Dedekind cut is in fact used in the definition and construction real numbers as an extension of the the rational numbers.

Given a set Q of all rational numbers, b is a real number if there is a partitioning of set Q into set A that contains all elements of Q that are less than b, and set B that contains all the elements of Q that are greater than or equal to b. If b is an element of set B then b is itself rational. If b is not an element of B then b is irrational.