On 10/25/2024 12:44 PM, Jim Burns wrote:
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_
>
>
Do you yet recall that these properties:
extent density completeness measure,
would establish that ran(f) that being ran(EF)
is a continuous domain?
Then that completeness is as simply trivial
that it's defined that the least-upper-bound
of the set is an element of the set, that
for f(...m) that f(m+1) is this?
Then that measure is according to any suitable
sigma algebra for example most directly length?
Of course then there's still that the putative
function (between discrete domain and continuous
range) would have to fall out of all the arguments
otherwise for un-countability as un-contradicted,
lo, it is so.
Then, about the "anti" and "only", and there being
this way that this ultimately tenuous continuum
limit (I'm glad at least we've arrived at that
being a word, "continuum-limit"), is just so that
in a descriptive theory where there's numbers,
functions, all in sets, then the sets that model
these numbers and these functions, and the functions
the relations among the numbers and numbers the
fixed-points among the relations, that in the logical
setting of "terms, predicates or propositions, and _relations_"
and that a theory of logical objects may be all
_relations_, then that this function, where a function
is a particular concrete relation, makes for that
its range is a "continuous domain" itself (the unit
interval, built in that it's bounded above and below
of course), then with regards to others like the
complete ordered field, they only connect to that
with functions as via relations, then that the "bridge"
as it were as over these "ponts", is a natural mathematical
object, and of course un-contradicted.
So, it exists and doesn't not exist.
Then, regardless that you can always crib out the
complete ordered field and plain usual ordinary
descriptive set theory and the whole bit, the
"anti-diagonal" bit, then there's also this most
simple sort of infinitary continuum-limit where
it's established that thusly there _must_ exist
at least this example non-Cartesian function,
"only-diagonal", not Cartesian, both.
Still shows both, ....
How many different unit fractions are lessorequal than all unit fractions?
Re: How many different unit fractions are lessorequal than all unit fractions?