Re: How many different unit fractions are lessorequal than all unit fractions? (infinitary)

On 10/23/2024 1:38 PM, Ross Finlayson wrote:

On 10/23/2024 08:05 AM, Jim Burns wrote:

Put d and anything else behind your back.
Swap swap swap, pull them out.
I pick both.
One of those picked is in [0,1]ᴿ\f(ℕ)
f(ℕ) ≠ [0,1]ᴿ
If you think I'm wrong, say why.

Ah, the main thing I want you to notice,
is, that the usual definition of "function",
in the usual descriptive milieu of functions
according to ZF set theory, is: a sub-set
of the Cartesian product of left-hand-side
and right-hand-side, or "Cartesian functions".

Please consider me to be noticing that.

So, then this "only diagonal", has that
this function EF the n/d n->d d-> oo
is NOT a Cartesian function:
it simply does NOT have
the same definition of function as
being a Cartesian function, so
all that results from Cantor-Schroeder-Bernstein
about the transitivity of functions,
that's built upon Cartesian functions:
does not hold.

Your non.Cartesian functions are NOT
Cartesian functions. Okay.
Other than that,
what ARE your non.Cartesian functions?
I hope that we can at least say that
non.Cartesian functions
are terms in a first.order language.
But maybe they aren't.
⎛ <formula> ::=
⎜   <connective-formula> |
⎜   <quantifier-formula> |
⎜   <predicate-formula>
⎜
⎜ <predicate-formula> ::=
⎜   <1.ary-predicate> "(" <1.ary-argument> ")" |
⎜   <2.ary-predicate> "(" <2.ary-argument> ")" |
⎜   <3.ary-predicate> "(" <3.ary-argument> ")" |
⎜   ...
⎜
⎜ <1.ary-argument> ::= <term>
⎜
⎜ <k+1.ary-argument> ::= <k.ary-argument> "," <term>
⎜
⎜ <term> ::=
⎜   <variable> |
⎜   <constant> |
⎜   <function-term>
⎜
⎜ <function-term> ::=
⎜   <1.ary-function> "(" <1.ary-argument> ")" |
⎜   <2.ary-function> "(" <2.ary-argument> ")" |
⎜   <3.ary-function> "(" <3.ary-argument> ")" |
⎝   ...
The semantics of first.order language L
can be expressed by augmenting L with
a constant.name for each object in the domain of discourse,
perhaps with uncountably.many names.
L⁺⁺ is augmented.L
For k.ary function f
for each k.ary sequence d₁,…,dₖ of domain.constants,
there exists a unique range.constant r
such that r = f(d₁,…,dₖ)
I'm calling that a Fregean function.
What you call a Cartesian function looks to me to be
a Fregean function represented by a set.
My best guess at what you mean by EF
does not describe a Fregean function.
If, by 'non.Cartesian function',
you mean 'non.Fregean function', then
I have no idea at all what you mean.

[...] that the most direct mapping between
discrete domain and continuous range is
this totally simple continuum limit of n/d
for natural integers as only d is not finite
and furthermore
is constant monotone strictly increasing
with a bounded range in [a,b], an infinite domain.

The continuum limit is not the continuum.
I know:
it sounds like it should be, but it isn't.
The continuum limit is
the spacing of a lattice approaching 0.
If we are _already_ working in the continuum,
the lattice points _in the limit_
are sufficient to
uniquely determine a _continuous_ function.
For many purposes,
uniquely determining a continuous function
is sufficient for that purpose.
But that isn't the continuum.
In a continuum,
each split has a point at the split,
either one which ends the foresplit
or one which begins the hindsplit
_which is different_