An example is added about Cantor's set theory in the the section [Paradox Explanation]
https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
and thought it may be interested:

+---------------------+
| Paradox Explanation |
+---------------------+
......
......
The number of even number and the number of natural number are equal: Ans:
ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
are two different set of arithmetic systems. Thus, it is confusing to say
that the number of elements of an infinite set and its proper subset are
equal.

On 2024-03-09 11:42:22 +0000, wij said:

> An example is added about Cantor's set theory in the the section
> [Paradox Explanation]
> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> and thought it may be interested:
>
> +---------------------+
> | Paradox Explanation |
> +---------------------+
> .....
> .....
> The number of even number and the number of natural number are equal: Ans:
> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> are two different set of arithmetic systems. Thus, it is confusing to say
> that the number of elements of an infinite set and its proper subset are
> equal.

More generally, anything said about any infinite thing can be confusing.
Therefore one must be very careful when discussing infinity and inifinte
theings. As Cantor was.

--
Mikko

On 3/9/24 3:42 AM, wij wrote:
> An example is added about Cantor's set theory in the the section [Paradox Explanation]
> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> and thought it may be interested:
>
> +---------------------+
> | Paradox Explanation |
> +---------------------+
> .....
> .....
> The number of even number and the number of natural number are equal: Ans:
> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> are two different set of arithmetic systems. Thus, it is confusing to say
> that the number of elements of an infinite set and its proper subset are
> equal.
>
>

Basically, claiming systems are different is a good dodge for handling
things you can't handle.

The thing is you first need to define what it means for two infinite
sets to have the same number of elements. The normal ways is by
bijection, which shows that they DO have the same number of elements.

Without defining what you mean by equal number of elements, you can't
talk about the "size" of an infinite set.

On 03/09/2024 05:48 AM, Mikko wrote:
> On 2024-03-09 11:42:22 +0000, wij said:
>
>> An example is added about Cantor's set theory in the the section
>> [Paradox Explanation]
>> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
>>
>> and thought it may be interested:
>>
>> +---------------------+
>> | Paradox Explanation |
>> +---------------------+
>> .....
>> .....
>> The number of even number and the number of natural number are equal:
>> Ans:
>> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the
>> two sets
>> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1>
>> ). They
>> are two different set of arithmetic systems. Thus, it is confusing
>> to say
>> that the number of elements of an infinite set and its proper
>> subset are
>> equal.
>
> More generally, anything said about any infinite thing can be confusing.
> Therefore one must be very careful when discussing infinity and inifinte
> theings. As Cantor was.
>

One says, "the density..." or "the asymptotic density..." or
"the Schnirelmann density..." "... of the even integers in
the integers is one-half."

Fred Katz, wrote an MIT Ph.D., helping explain, OUTPACING,
a concept that proper supersets are larger and proper subsets
respectively smaller, than given sets. He wrote me once and
I really appreciated it.

Number theory, is what defines "density" and "asymptotic density".

Cantor's Set Theory, is a premier edifice in the foundation of
modern mathematics, establishing classes of cardinality, of sets,
that cardinally equivalent/equipollent sets have a Cartesian bijection
between them, an inequivalent don't, and relating those to the
finite cardinals, and the trans-finite cardinals, where the counting
numbers', is the least transfinite cardinal, named Aleph_0. There's
introduced a notation that looks like the exponent of real numbers,
"2 to the...", but upon cardinal numbers it means exactly and only
cardinal exponentiation, and results a prototype element 2^Aleph_0,
and the ordering that Aleph_0 < 2^Aleph_0.

The cardinal 2^Aleph_0 isn't _necessarily_ Aleph_1, about whether
there are or aren't transfinite cardinals between the least and
a prototypical greater least, whether Aleph_1 = 2^Aleph_0 is
called the Cantorian Continuum Hypothesis, CH, and whether each
Aleph_alpha+1 = 2^Aleph_alpha is called the Cantorian Generalized
Continuum Hypothesis, GCH. Cohen established that ZFC set theory,
the usual set theory, doesn't decide CH nor GCH either way,
using ordering theory.

The ordinals and transfinite ordinals are totally different
from the cardinals and transfinite cardinals, but share some
assignments, where transfinite ordinals are mostly used like
the usual inductive set, a set of ordinals making a prototype
for "next" for the cases of infinite induction.

So, cardinal ordering establishes a "size" relation among sets,
and, there are other relevant notions that establish a "size"
relation, among sets, particularly as of numbers, which have
their own other implicit relations, when it is said that the
sets "model" the numbers, while really a "model" of the numbers
is a model of all the relations of the numbers, not just
their names or "the definition", of the existence of a label
of a distinctness of a number.

That there is a big difference between "counting" as a sort
of act, and "numbering" as a sort of identity, is a thing,
that often doesn't matter, or is indeterminate, but that
they are two different things themselves, in what matters
about the notion of algebra's words, and geometry's numbers,
with regards to usual primary elements of theories of
mathematics: sets, words, numbers, and geometry's elements,
with regards to function theory open about words, and
topology open about numbers, and whether words define
geometry, or geometry defines words, or they are independent,
and vice-versa.

Anyways, yes you can say that "half of the integers are even",
and anybody who says that's not mathematical, is excluding
number theory from mathematics.

On Sat, 2024-03-09 at 09:43 -0800, Richard Damon wrote:
> On 3/9/24 3:42 AM, wij wrote:
> > An example is added about Cantor's set theory in the the section [Paradox Explanation]
> > https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
> > and thought it may be interested:
> >
> > +---------------------+
> > > Paradox Explanation |
> > +---------------------+
> > .....
> > .....
> > The number of even number and the number of natural number are equal: Ans:
> >      ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
> >      are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
> >      are two different set of arithmetic systems. Thus, it is confusing to say
> >      that the number of elements of an infinite set and its proper subset are
> >      equal.
> >
> >
>
> Basically, claiming systems are different is a good dodge for handling
> things you can't handle.
>
> The thing is you first need to define what it means for two infinite
> sets to have the same number of elements. The normal ways is by
> bijection, which shows that they DO have the same number of elements.
>
> Without defining what you mean by equal number of elements, you can't
> talk about the "size" of an infinite set.

I do mean 'cardinality'. 
Swapping to 'size' is a good dodge handling things you can't handle.

On 03/09/2024 06:05 PM, wij wrote:
> On Sat, 2024-03-09 at 09:43 -0800, Richard Damon wrote:
>> On 3/9/24 3:42 AM, wij wrote:
>>> An example is added about Cantor's set theory in the the section [Paradox Explanation]
>>> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
>>> and thought it may be interested:
>>>
>>> +---------------------+
>>>> Paradox Explanation |
>>> +---------------------+
>>> .....
>>> .....
>>> The number of even number and the number of natural number are equal: Ans:
>>> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
>>> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
>>> are two different set of arithmetic systems. Thus, it is confusing to say
>>> that the number of elements of an infinite set and its proper subset are
>>> equal.
>>>
>>>
>>
>> Basically, claiming systems are different is a good dodge for handling
>> things you can't handle.
>>
>> The thing is you first need to define what it means for two infinite
>> sets to have the same number of elements. The normal ways is by
>> bijection, which shows that they DO have the same number of elements.
>>
>> Without defining what you mean by equal number of elements, you can't
>> talk about the "size" of an infinite set.
>
> I do mean 'cardinality'.
> Swapping to 'size' is a good dodge handling things you can't handle.
>

No, it seems that you heard "size, is a cardinality",
but what it is, is, "cardinality, is a size".

It seems, what you heard, is,
"X, is a Y",
but what it is, is,
"X, is a kind, of Y".

There are others, ....

Or, you know, "Y, has a kind, of X".

According to number theory, half of the integers are even,
whereas all one (hundred percent) of the integers are integers.

It, seems what you heard is, "one",
but, what it is, is, "one hundred percent".

Of course, there are no paradoxes in logic,
and no paradoxes in numbers, nor
paradoxes in mathematics,
only missing and wrong.

Multiplicity theory introduces
the general notion of singularity theory,
introduces the general notion of "undefined",
as the usual idea of "overdefined: as undefined".

For example, why "not divide by zero",
when there can be, "not multiply by reciprocal of zero"?

I mean, you have to know, right.

In mathematics the usual theory of types,
is very loose both ways, while yet being
a strong theory of types.

It's fair to consider of your number modeling your
sets instead of the other way around, "set theory
in number theory", so that then the model of your
numbers in your set theory, "number theory in set
theory", seems most naturally primitive, meaning
primary, and that the fundamental relation, in
those or on those, "size", "cardinality", are same.

What that means is probably that your model looks
modular like integers already where you lay sets
on it, that automatically the definition, is,
"in set theory it's cardinality and in my number
theory it's density and they're same and that's size",
it makes sense, helping explain why number theorists
and set theorists don't "get" each other's idea while
still modeling the same things in each other's theories.

Sometimes called "talking past one another".

On 03/09/2024 06:05 PM, wij wrote:
> On Sat, 2024-03-09 at 09:43 -0800, Richard Damon wrote:
>> On 3/9/24 3:42 AM, wij wrote:
>>> An example is added about Cantor's set theory in the the section [Paradox Explanation]
>>> https://sourceforge.net/projects/cscall/files/MisFiles/logic_en.txt/download
>>> and thought it may be interested:
>>>
>>> +---------------------+
>>>> Paradox Explanation |
>>> +---------------------+
>>> .....
>>> .....
>>> The number of even number and the number of natural number are equal: Ans:
>>> ℕ=ℕ<0,+1> and ℕ<0,+2> are isomorphic but the "even numnber" in the two sets
>>> are semantically different (i.e. the 6 in ℕ<0,+2> is 3 in ℕ<0,+1> ). They
>>> are two different set of arithmetic systems. Thus, it is confusing to say
>>> that the number of elements of an infinite set and its proper subset are
>>> equal.
>>>
>>>
>>
>> Basically, claiming systems are different is a good dodge for handling
>> things you can't handle.
>>
>> The thing is you first need to define what it means for two infinite
>> sets to have the same number of elements. The normal ways is by
>> bijection, which shows that they DO have the same number of elements.
>>
>> Without defining what you mean by equal number of elements, you can't
>> talk about the "size" of an infinite set.
>
> I do mean 'cardinality'.
> Swapping to 'size' is a good dodge handling things you can't handle.
>

It sort of helps to explore the intuitive and
also unintuitive aspects of the infinite and
the trans-finite and closures and completions,
and incompleteness and undecideability,
with regards to the scale and order of things.

First you might want to forget all about it,
then think in terms of continuity and discreteness,
with regards to our limited means or finite means,
yet though thought arrives at infinite things as
consequences of that "finite things would be finite",
i.e., that "it's beyond finite means, to say that the
objects of mathematics aren't beyond finite means".

Then, mathematics is at least two things, together.
One of them is whole relation, and the other is part
relations, and there are relations of state, and
relations of change.

After it's granted "the objects of mathematics have
capacities beyond finite means", then there's a lot
expected from the linear curriculum, which provides
that all students as model instructees given model
instruction have a common language and school of
derivation, mathematical facts together, and largely,
as of for methods of finite means. It's great.

Then, the other usual notion is "enrichment", about
whatever's aside that improves these things, with
regards to something like Max Wertheimer's "Productive
Thinking", and these kinds of things.

Part of that is to help illustrate, that when elementary
objects are introduced into the linear curriculum,
that they reflect upon each other. Here this is basically
about points and lines, points discrete and lines continuous.
They have a dialectic together, points from lines and lines
from points, but it involves development beyond the
usual number-sense, to the coordination of number-sense,
word-sense, time-sense, and object-sense, what results a
"continuum-sense", not just the experience of it, but the
abstraction of it, and the terms of the dialectic of it.

Now, for the phenomenological, this basically introduces
that beyond our physical senses of the stimuli, result
real human-level senses of the object-sense, word-sense,
number-sense, time-sense, and a "continuum-sense",
that it sort of starts in development an object-sense,
then a time-sense, then a number-sense, then a word-sense,
the the abstraction, all the combinations of those, resulting
a fuller object-sense while in a continuum-sense.

Then, for usual theories and curricula where "all we can
relay is after the phenomenological because the model
instructee must be relayable in its own intermediary terms",
at least the mental model of the mental model accumulates
to develop in its own phenomenological terms.

Then, in the theory of the objects of numbers and sets,
modeling either either way in terms of the other gets
involved, it's not just their relations among each other,
it's their relations _as_ each other.

Then, quantity, after quality, results why it's for distinction
and disambiguation, that these sorts notions each have
their own absolute sorts notions, while as well, they each
have their relative notions.

It's a fuller dialectic.

So, these kinds of things.

Then, "mechanical thinking" basically involves "an object-sense,
a number-sense, a word-sense, a time-sense", vis-a-vis "sensor input",
state and change and models then of overall an object-sense,
state and change, what builds all those _among_ each other
and _as_ each other.

This then can usually result what's called "model theory",
a theory of models and a model of theories.