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ordinala

Ordinala is a term used in speculative mathematics and in constructed-language literature to denote a hypothetical, layered extension of ordinal numbers. It is not an established mathematical object, and its precise definition varies among authors. In many treatments, ordinala serves as a thought experiment to explore how ordinal hierarchies could be organized beyond classical ordinals and how such a system might interact with ordinal operations and proofs.

In a representative scheme, the collection of ordinala is built in layers O0, O1, O2, and so

Operations on ordinala are defined to extend, but not uniquely determine, standard ordinal arithmetic. Different authors

Applications of ordinala are mostly theoretical and pedagogical. It appears in thought experiments about the foundations

on,
indexed
by
ordinals.
The
base
layer
O0
consists
of
the
natural
numbers.
Each
successor
layer
adds
new
elements
that
encode
the
existence
of
a
later
stage,
while
limit
layers
introduce
limit
elements
that
stand
for
the
collective
content
of
all
earlier
layers.
The
union
of
all
layers
forms
the
full
ordinala.
This
layered
approach
aims
to
preserve
a
well-ordered
structure
while
permitting
a
richer
landscape
of
“stages”
than
standard
ordinals
alone.
propose
varying
rules
for
addition,
multiplication,
and
limit-combinations
within
and
across
layers.
Because
of
this
lack
of
a
single
canonical
definition,
properties
of
ordinala
can
differ
between
formulations.
of
mathematics,
in
discussions
of
hierarchy
and
stage
theory,
and
in
some
works
of
fiction
or
philosophy
that
explore
alternative
bases
for
number
systems.
See
also
ordinal,
well-ordering,
transfinite
induction,
and
ordinal
arithmetic.