Continuum Analysis Fundamentals (draft)

[Ross A. Finlayson]

\section{Introduction}

Usual formalisms in finite combinatorics and
also the resulting mathematics from geometry
finds continuum analysis central and ubiquitous.

TeX is a computerized typesetting system originated
and specialized for often usual document contents
and diagrams and in structured and semantic content
settings.

Mathematical proving systems are often computerized
inference and result establishment settings, their
often tractable symbolic form involves in a usual
document framework what sorts would follow, the usual
language of TeX and its normative computer language
MIX, among mathematical proving systems, that results
from the document as template, here what are among
renderings, usual maintenance of inline rendering
the collections and organizations of the mathematical
proving systems establishments, or their values.

Continuum Analysis fundamentals are formalizations,
the axiomatics of continuum objects are shown to
be of central forms and including for reasons of motion.

The usual (and standard) mathematical formalizations
are same, definitions of continuity are mathematical
definitions about the continuity in space, fields,
signals, and in spaces and functions, and about
sets and parts and relations, their value representations
and conditions as mathematical structures as
maintaining and established "continuity", by definition.
Referenced are ZF (Zermelo-Fraenkel, modern) set theory
and usual terms from real analysis (standard), function
theory, ordering theory, then for forms: usual modern,
standard statements as above the combined representations
for mathematical proving systems.

A primary object in continuum analysis is a constant
or object called "infinity" that in systems of numerical
value and arithmetic evaluates as "greater than" than
any named, finite, or bounded value. "Infinity" is
written as "$\infty$".

As a constant, definitions of establishment of
relations of arithmetic forms of infinity, have
that suitable organizations or structures,
maintain the usual and exact representation
of the "infinite value", in terms of the
arithmetical constants $0$, $1$.

In the quotient space as about the multiplicative identity and additive successor, to $0$, as a constant for ratio
for value, the term "$\infty$" is not in the space of
multiplicative results about 1, but, does maintain
being its own multiplicative inverse, as if it were
(in the space, or as that its product is from the quotient space).

\[\frac{\infty}{\infty} = 1 \]

Infinity is similarly "centralized" as about the origin.

\[\infty-\infty = 0 \]

The "value representations" of the systems of value
can only write the constant "$\infty$" as maintains
that in its expression, and that is an otherwise
valueless constant, instead would most establish
the reference of form of term, for "not equals
infinity" or for "less-than" ($'<'$) then always writing
infinity in terms of "greater-than" ($'<'$) (or for example
in systems "less-than").

This way the definition of arithmetic as from fundamentals,
sees that the most usual or direct maintenance of relations
of value in the value space, as the necessarily continuous
space of relations is so maintained, the continuous or
real value space, has that "$\infty$", the constant,
maintains a valid expression, for its expression of
constant, not necessarily empty, value relations in types.

The continuous value relation in the terms looks as:

\[ \frac{1}{\infty} = 0 \]

that clearly the resulting use of the constant, is
invalidated where a) $\infty$ doesn't exist in the
value space, via any necessarily direct (or mutual) inference. Otherwise it's maintained as valid, the
valid $1$ in its expression and as a constant and
as an invariant, and also here where direct establishment
via or under otherwise constant terms, sees that
"$\infty$" maintains as if it were either potentially,
or actually infinite, as about that in the terms, it's actual and constant.

This is where otherwise that only "$>$" in an ordering of
the numbers, not exhausting the ordering, where "$>$" is
the only evaluation and "$\infty$" is outside the space
(or otherwise "valid"), the only fact is that as the
ordering is unbounded, that for any value that is not
infinity (and none are), there exists an inverse
that is less-than the inverse of the value. That
it's also bounded below on 0.0 the additive identity
(and multiplicative annihilator, besides), establishes
zero as a critical point and also the only critical
point of multiplicative inverses of positive integers.

\[
\forall n \in N,
\exists m > n
\implies
\forall n \exists
\text {"m-many m such that 1/m is closer to zero than 1/n"}
\]

A central fixture in continuum analysis that in results
maintained directly by continuum analysis or space concerns,
that exhaustion is incomplete from enumeration but complete
from attenuation, here follows the simple establishment above, of that it's the only fact available to mutual
(and deductive) inference, the existence of quantity (in the value space).

The existence of "so many" larger, the "at least so many
larger" instead of "all the rest (after, in the ordering) larger" establishes results in symmetries,
under potential in usual potential spaces as for example
quadratic in area. The central property of locality, in
the value space, results in perfect equilibrium as usually
constant under terms.

This maintains that for the "actual infinity" in the
value space, extreme in the value space, that there
doesn't exist any other object in the value space,
because infinity is the only non-finite and non-zero
value, that "$\frac{1}{\infty} = 0$" and "$\lim_{n < \infty} \frac{1}{n} = 0$", that those equating the same zero in the value space
is the establishment of the value semantics of infinity, for and about $\text{measure} 1.0$. Thus, either stipulation of potential or actual infinity has the possibility of mutual inference, constant in and under terms.

\end{document}

[Ross A. Finlayson]

Sergio mentioned "computer program", proving systems:
here for example I should read this and see what it would think.

[markus...@gmail.com]

what

[mitchr...@gmail.com]

Mathematical continuity was Einstein's central continuum...
Eternal continuity is real. It is mathematical gravity.
God creates gravity.

Mitchell Raemsch

[Ross A. Finlayson]

In other words you describe an emergent property of monism.

If, massy bodies make for a fall gravity with occluding the
everpresent flux of the field gravity, or here the "gravific" field,
then, they're not always working all the time as would use energy,
with instead that usual thermokinetics are as well a natural property
of the inherent energy of the system. Also it's a convenient or direct
route to a unified field theory.

"The" continuum is in the sense of many the space-time continuum,
as all its contents, vis-a-vis the linear continuum, a numerical setting.

Here, G-d (politely) only needed make this field (or as it was an
emergent property of monism) and roll the die, time goes back
forever, space goes on forever, in a simple least complicated
model of a hologram of a space-time continuum.


Continuum analysis is real analysis, here the above is framed past
bounds, making just a usual formalism for the resulting establishment
of limit and so on.

Which is neat, direct, brief, and short.