And the only way out of using hyperhyper reals is to use the epsilon-delta definition of continuity on hyperreal functions, which defeats the purpose of using nonstandard analysis in the first place. So once you've committed to hyperreals, you do need hyperhyper reals for this proof.
I'm only just now beginning to properly understand hyperreals so I'll take a stab at it.
f_N(x) is not a real function so we have yet to define what it being continuous means. Suppose we use the same definition as we used for real function. Then h\sim r \implies f_N(h)\sim f_N(r), right?
This...
I meant "safer". Anyway, so what you mean is we define the sum to be 1/2 because it "extends" properly from previous know formulae? Like how we define 0! = 1, even though the factorial definition is for positive integers only?
That is illegal yet as well. The geometric progression works only if 0<x<1. As it appears that there is no legal way to prove this, would it not be better to say the sum is undefined?
Why not? From what I just read, Planck time is, theoretically, the smallest time-difference measurable. If things are only defined by what's measurable, then wouldn't it be pointless to bring in the concept of other universes when we can only observe one?
Edit:
Do you have a way of explaining...
So my first example wasn't perfect, but are you implying that time and distances are also discrete as well?
If in one universe an atom decays at 00:00 and in another universe the atom decays at 00:01. Shouldn't there be infinite universes where it decays somewhere in between? We probably can't...
If everything that can happen will happen, then there are an infinite number of universes.
Assuming all universes were identical and the only difference is when someone was asked to think of an integer n, a positive real number less than 1, or one thing in an infinite set. If in all the...
The same can be said for googolplex.
I think you've absolutely underestimated the size of GN by the way you keep trying to compare it to something we can conceptualize. No sort of simple layman's comparison is possible.
"However, firstly, are you sure that n!<n^n?"
n! = n \cdot (n-1) \cdot (n-2) \ldots 2 \cdot 1
n^n = n \cdot n \cdot n \ldots n \cdot n
They both have the same number of terms in the product but after the first term, subsequent terms in n! are always less than the corresponding terms in...
I don't think we can have a square-root function because by definition square-roots have double-values. So \sqrt{x} refers to the principle square-root which can indeed be a function.
Those are the same questions. The different question would be "What is the value of x if x = \sqrt{4}" or "What is the principle square root of 4?".
"What is the square root of 4?"
- The square-root of 4 is the number that was squared to produce 4. It could be 2 or it could be -2. So the...
The problem with what you're aiming for is the number will be so huge, you won't even be able to describe how huge it is. Just from reading the wikipedia page, g_{1}, which is the first step for a sequence of 64 terms that are growing unimaginably faster than exponentiation, is already too big...