"Most helpful math resources on the web" - via MathOverflow

This is great, here is a list of lots of great online math content.

"Soft analysis, hard analysis, and the finite convergence principle"

Tao's excellent article illustrates the comfort of deductive analysis. (Thanks FredJeffries.)

Why I would refer to it here is because he talks about for example the simple cases about for example that the rational numbers are dense in the real numbers but not having equal properties due their densities as irrationals. Another could follow that there isn't density of inexhaustible monotone condition that has a limit.

Then what's really great is that he actually knows real analysis and explains that.

I posted this as an answer and while it is a point about the development of the argument, that's not a discussion forum.

"Your set F is not cofinal. None of your f_n dominates g:x->x^2". – Robin Chapman

For each $x E N$, $\exists m >> n$ s.t. $f_m > g(x)$. Let $\mathcal{F}\g(x)$ be $f_n E F: f_n(x) > g(x)$. It is so that F\g(x) is non-empty, for each x E N, for each g_x E G. Via induction, for g(x+1) there exists m > n such that f_m in F\g(x+1). Then, where that is not so, there does not exist m >> n E N.

It's the statement: ! Exists m >> n.

So, if F\g(0) is non-empty, and F\g(k+1) is non-empty for each k, then there doesn't exist x s.t. F\g(x) is non-empty (unless rather induction fails). For each x, F\g(x) is non-empty, for any g.

"How many orders of infinity are there?" - via MathOverflow

Let F = {f_n(x) = nx} for n E N. For any growth function g in the collection of all growth functions G, there is an integer n such that f_n(x) > g(x) for each x. Proof: else there wouldn't exist natural n > g(x). Of course, for any function that is increasing with anything higher than a multiplicative constant, there exists y > x such that g(y) > f_n(y), but there also exists already m >> n s.t. f_m(y) > g(y), and f_m E F. Then F is cofinal but countable.

Orders of infinity: the 'infinitärcalcül' of Paul Du Bois-Reymond

This is compatible.

The mathematics inherent in the statement apply.

It's obvious to me that the natural/unit equivalency function is a function from the natural integers to the unit interval of real numbers.

Here is a short proof about the rationals and irrationals surjecting onto each other.

A function surjects the rational numbers onto the irrational numbers

I wrote some design plans for parallel software.

seq_reader, SeqReader.

Implementing V-Real

V-Real is a software virtual reality system that basically stores everything in terms of objects in points in space.

While that is now the external legacy API, internal mechanics and force models are implemented in terms of natural surface terms with modern data representations.

Basically towards implementation of bibliographies of surface and matter representations in voxel space, vari-scale effects are modeled (modelled) among external datagraphic references.

Then, there is ready cloning in Monte Carlo and MCMC for system smoothing through parallel local minima detection. Those then balancing the time effects in the serial, computational resources mutually minimize in product maintenance effects, maintaining error term off cut boundaries.

With then generally available solvers across layers V-Real is an industrial strength systemic modeling tool.

Then for this, part of the idea is to maintain the geometries in hi-definition. Most of the art geometry is in the triangles exported, so it would be good to get internal data representations from the running program. Thus it includes runtime analysis runtimes.

Really it is key then to model the local data representations, then import and export in the 3-D.

Towards that there should be the general multi-layer integrative dataspace over integrating data.

For that then is a study of the integrative mathematics of the three-dimensional dataspace. For that, statistics are built into the datatypes, so then types of data from different sources are gathered together and build statistics together.

With a scaleable multi-user graphical interface, V-Real is targeted towards application.

Then, to do that, I need the statistics and the runtimes. Also, there is the computational geometry machinery, which includes mathematics and the scalar description of algorithms.