Re: Incompleteness of Cantor's enumeration of the rational numbers (expressions)

On 11/12/2024 07:07 PM, Moebius wrote:

Am 13.11.2024 um 03:44 schrieb Chris M. Thomasson:
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[...] different ways to represent zero?
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0 = 3-4+1 = 4-4 = 2*0 = 0
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ect.

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Right. "3-4+1", "4-4", "2*0", "0", are just different NAMEs (math:
TERMs) to denote (refer to) _one and the same_ number, namely zero
(usualy, denoted with "0").
>
Hint: Suoerman is Clark Kent.

In positive numbers it's "additive partitions", for addition.
Then, those would usually be "expressions", of "terms".
The "numeral" as representing the "number", with regards
to that being an equivalence class of values in numbers
for example the real-valued, yes usually is "positional
notation" and as according to "a radix" and "the radix",
that it has a radix the "decimal point" and is in a
radix "the base".
The "transitive" equalities the expressions what "associate",
are each terms and terms and terms again and
expressions and expressions and expressions again.
So, the additive partitions and multiplicative partitions,
how many of those is a combinatorial expression, as is
the class of those that exist.
Are numerals and operators and relaters each expressions?
In a sense they represent the classes of all they fulfill.
So, much like logic can be "terms, predicates, relations",
or "terms, propositions", and some would have those be
"constants, terms", here it's most generally "relations".
So, more than "ways to write zero",
is "ways to write, where zero".
Then, I don't know a better word than "relater" for
the symbol of relates, for equality or inequality or
otherwise composition according to associativity and
transitivity, where as well, operators relate.  A
usual symbol for relates is "~", or "R" when introductory.
How about
Int f dx = df/dx
f = d^2 f /dx^2
f = 0
f = e^x
   or
f = EF(n)
There are some other functions their own anti-derivative.
Then there are functions like sin/cos, that oscillate,
sin/-cos/-sin/cos, each own _fourth_ derivative.
Anyways, it's funny that EF(n) is its own anti-derivative,
when you wouldn't usually figure that a discrete function
was even integrable.
There are functions whose derivative is their inverse.
f(x) = x^phi / phi^{-phi}  ?
https://math.stackexchange.com/questions/735695/function-whose-inverse-is-also-its-derivative
Notice they're looking symmetrical about the identity.
That is to say, for a wide class of functions and their
inverses in [0, oo) symmetrical about x = y.
There's only one real number 1/phi = phi - 1, it's phi.