On 1/12/2024 10:18 PM, Ross Finlayson wrote:
> On Friday, January 12, 2024
> at 3:46:43 PM UTC-8, Richard Damon wrote:

>> [...]
>
> Sorites and the Heap.

Make a claim which is true of each
and leap the Heap.

> There are no standard models of integers.

The minimal inductive set is
a standard standard model of integers.

Are you currently considering
non.standard standard models of integers?

> There are fragments,

A fragment isn't a model.

Granted, there are things other than
the minimal inductive set.
But, Shirley, that's not what you mean?

> there are extensions,

An extension isn't standard.

Granted, there are things other than
the minimal inductive set.
But, Shirley, that's not what you mean?

> the ordinary inductive set's
> a non-logical constant.

The only place I remember seeing "non-logical"
(which doesn't mean "illogical") used
is with axioms _in addition to_ the logical axioms.

We could call this sense of non-logical
metalogical, after.logical, which makes it less
likely to be mistaken for an admission of failure.

If you (RF) don't mean metalogical, then
I don't know what you mean by non-logical.

The unformalizable goal of stating
metalogical axioms is that everyone can say
"Yes, that is what everyone means by X".
That wouldn't be a _logical_ result, but
it would be a fact about what everyone means.

For the more obscure X,
there isn't really an "everyone".
There is the author, and
there is what they mean by X

However, natural numbers aren't obscure.

For _what we mean by_ the non.negative integers,
the minimal inductive set is a standard model.

> There are multiple models of integers.

Have we drifted away from (up.post)
> There are no standard models of integers.
?

What I think we mean, what I know I mean
by a model of the non.negative integers
is that it contain at least
each standard non.negative integer,

by a standard model of the non.negative integers
is that it contains no extensions.

If any model of the non.negative integers exists,
then a standard model exists, provably.

On 1/14/24 11:48 AM, WM wrote:
> Le 13/01/2024 à 00:46, Richard Damon a écrit :
>> On 1/12/24 9:05 AM, WM wrote:
>>> Le 12/01/2024 à 04:09, Richard Damon a écrit :
>>>
>>>> Which of those largest natural numbers can k not get to?
>>>
>>> If I could name it I had made it v isible.
>>>>
>>>> What is the boundry that can not be passed?
>>>
>>> That is the difficult point: There is no fixed threshold. Most can't
>>> comprehend it. Potential infinity!
>
>> So, you have two distinct sets with no boundry between them?
>
> The visible numbers are not a set but only a collection, because the
> membership is not fixed.
>>
>> What makes them different?
>
> This: You cannot use real numbers of the domain containing the ℵ
> smallest unit fractions and of the domain containing the ℵ largest
> natural numbers. It is obviously impossible to come closer to zero.
> There are always ℵ unit fractions between 0 and a number chosen by you.
> Same for ω, but, contrary to zero, ω is a vague end of the scale.

Really? You can't figure out how to use number like 1/10 or 12?

After all, those are part of the ℵ unit fractions between some unit
fraction and 0 and above some Natural Number.

There are no unit fractions smaller that ALL unit fractions or Natural
Numbers greater than ALL Natural Numbers, so the domain of numbers that
you can not use because of that condition is empty.

>
>> If there isn't a line that keeps the describable numbers out of your
>> dark numbers, then aren't all your dark numbers describable?
>
> Assume it. Then you can come close to zero such that nothing is between
> it and your chosen number. But that is a contradiction.

You can come as close as you want.

That doesn't means that you can reach the point that nothing is between.

That would require the existance of a Highest Natural Number.

That assumption is where the contradiction is, not that all Natural
Numbers/Unit Fractions are visible.

>>
>> If there is a line, then there must be a highest describable number,
>> so you can give it.
>
> That is the point hard to swallow. There is no line.

Then all your "dark" numbers are also visible, and thus not Dark.

Without a line, what keeps the visible numbers out of the set of "Dark"
numbers?

You are just hitting the flaw of naive set theory.

>
> Regards, WM
>
>

On 1/14/2024 11:43 AM, WM wrote:
> Le 12/01/2024 à 19:30, Jim Burns a écrit :
>> On 1/12/2024 9:01 AM, WM wrote:
>>> Le 12/01/2024 à 00:34, Jim Burns a écrit :

>>>> Since you (WM) have decided that
>>>> you are talking about _not.arithmetic_
>>>> this might be an especially apt time
>>>> for your students to remember that
>>>> a claim about _not.arithmetic_
>>>> even if it were _true_
>>>> doesn't contradict a claim about _arithmetic_
>>>
>>> That should not hinder an inquisitive student
>>> to learn that
>>> arithmetic does not cover the domain of
>>> the smallest ℵo unit fractions and of
>>> the largest ℵo natural numbers.
>>
>> Do your students learn that it doesn't from
>> their instructor, from you?
>
> They need not learn it,

Which is to say:
No,
you,
Wolfgang Mückenheim of Hochschule Augsburg,
do not tell your students that,
when you talk about
the smallest ℵ₀ unit fractions and
the largest ℵ₀ natural numbers,
you aren't talking about
arithmetic.

> because it is clear to everybody,
> perhaps after a short hint:
> You cannot use real numbers

Real numbers ℝ are the rationals ℚ and
enough points between non.empty splits of ℚ
that no function ℝ→ℝ which jumps is
continuous at each point in ℝ

Rationals ℚ are the integers ℤ and
enough points that each non.0 division in ℤ
has a solution in ℚ

Integers ℤ are the naturals ℕ and
enough points that each subtraction in ℕ
has a solution in ℤ

Naturals ℕ are each end of ordered ⟨0,…,k⟩
such that,
for each non.empty split F,H of ⟨0,…,k⟩
i‖i⁺¹ exists last‖first in F‖H,
and 0‖k exists first‖last in ⟨0,…,k⟩

i⁺¹ is non.0 non.doppelgänger non.final.

Also Known As arithmetic.

> You cannot use real numbers of
> the domain containing
> the ℵ smallest unit fractions and of
> the domain containing
> the ℵ largest natural numbers.

Each element of ℝ is
not the smallest unit fraction and
not the largest natural number.

> It is obviously impossible
> to come closer to zero.

No,
it is obvious in arithmetic
that each x ≠ 0 can come closer.
To x/2 for example.

> There are always ℵ unit fractions between
> 0 and a number [x] chosen by you.

For each x > 0
for each not-fitting-1-removed cardinality n
the cardinality of unit fractions ⊆ (0,x]
is not that cardinality.
|…,⅟mₓ⁺²,⅟mₓ⁺¹| ≥ |⅟mₓ⁺²,⅟mₓ⁺¹| > |⅟mₓ⁺¹|
|…,⅟mₓ⁺²,⅟mₓ⁺¹| ≥ |⅟mₓ⁺³,…,⅟mₓ⁺¹| > |⅟mₓ⁺²,⅟mₓ⁺¹|
|…,⅟mₓ⁺²,⅟mₓ⁺¹| ≥ |⅟mₓ⁺⁴,…,⅟mₓ⁺¹| > |⅟mₓ⁺³,…,⅟mₓ⁺¹|
|…,⅟mₓ⁺²,⅟mₓ⁺¹| ≥ |⅟mₓ⁺⁵,…,⅟mₓ⁺¹| > |⅟mₓ⁺⁴,…,⅟mₓ⁺¹|
|…,⅟mₓ⁺²,⅟mₓ⁺¹| ≥ |⅟mₓ⁺⁶,…,⅟mxₓ⁺¹| > |⅟mₓ⁺⁵,…,⅟mₓ⁺¹|
....

The cardinality of unit fractions ⊆ (0,x]
_is not_ any not.fitting.1.removed cardinality n

The cardinality of unit fractions ⊆ (0,x]
_is_ fitting.1.removed
|…,⅟mₓ⁺⁴,⅟mₓ⁺³,⅟mₓ⁺²,⅟mₓ⁺¹| =
|…,⅟mₓ⁺⁴,⅟mₓ⁺³,⅟mₓ⁺²| =
|…,⅟mₓ⁺⁴,⅟mₓ⁺³| =
|…,⅟mₓ⁺⁴| =
.... =
ℵ₀

> Same for ω, but,
> contrary to zero, ω is a vague end of the scale.

The cardinality of natural numbers ⊆ [x,∞)
_is not_ any not.fitting.1.removed cardinality n
|mₓ⁺¹,mₓ⁺²,…| ≥ |mₓ⁺¹,mₓ⁺²| > |mₓ⁺¹|
|mₓ⁺¹,mₓ⁺²,…| ≥ |mₓ⁺¹,…,mₓ⁺³| > |mₓ⁺¹,mₓ⁺²|
|mₓ⁺¹,mₓ⁺²,…| ≥ |mₓ⁺¹,…,mₓ⁺⁴| > |mₓ⁺¹,…,mₓ⁺³|
|mₓ⁺¹,mₓ⁺²,…| ≥ |mₓ⁺¹,…,mₓ⁺⁵| > |mₓ⁺¹,…,mₓ⁺⁴|
|mₓ⁺¹,mₓ⁺²,…| ≥ |mₓ⁺¹,…,mₓ⁺⁶| > |mₓ⁺¹,…,mₓ⁺⁵|
....

The cardinality of natural numbers ⊆ [x,∞)
_is_ fitting.1.removed
|mₓ⁺¹,mₓ⁺²,mₓ⁺³,mₓ⁺⁴,…| =
|mₓ⁺²,mₓ⁺³,mₓ⁺⁴,…| =
|mₓ⁺³,mₓ⁺⁴,…| =
|mₓ⁺⁴,…| =
.... =
ℵ₀ = |ω|

On Sunday, January 14, 2024 at 10:24:52 AM UTC-8, Jim Burns wrote:
> On 1/12/2024 10:18 PM, Ross Finlayson wrote:
> > On Friday, January 12, 2024
> > at 3:46:43 PM UTC-8, Richard Damon wrote:
> >> [...]
> >
> > Sorites and the Heap.
>
> Make a claim which is true of each
> and leap the Heap.
> > There are no standard models of integers.
> The minimal inductive set is
> a standard standard model of integers.
>
> Are you currently considering
> non.standard standard models of integers?
>
> > There are fragments,
>
> A fragment isn't a model.
>
> Granted, there are things other than
> the minimal inductive set.
> But, Shirley, that's not what you mean?
>
> > there are extensions,
>
> An extension isn't standard.
>
> Granted, there are things other than
> the minimal inductive set.
> But, Shirley, that's not what you mean?
> > the ordinary inductive set's
> > a non-logical constant.
> The only place I remember seeing "non-logical"
> (which doesn't mean "illogical") used
> is with axioms _in addition to_ the logical axioms.
>
> We could call this sense of non-logical
> metalogical, after.logical, which makes it less
> likely to be mistaken for an admission of failure.
>
> If you (RF) don't mean metalogical, then
> I don't know what you mean by non-logical.
>
>
> The unformalizable goal of stating
> metalogical axioms is that everyone can say
> "Yes, that is what everyone means by X".
> That wouldn't be a _logical_ result, but
> it would be a fact about what everyone means.
>
> For the more obscure X,
> there isn't really an "everyone".
> There is the author, and
> there is what they mean by X
>
> However, natural numbers aren't obscure.
>
>
> For _what we mean by_ the non.negative integers,
> the minimal inductive set is a standard model.
> > There are multiple models of integers.
> Have we drifted away from (up.post)
> > There are no standard models of integers.
> ?
>
> What I think we mean, what I know I mean
> by a model of the non.negative integers
> is that it contain at least
> each standard non.negative integer,
>
> by a standard model of the non.negative integers
> is that it contains no extensions.
>
> If any model of the non.negative integers exists,
> then a standard model exists, provably.

Hallo, James. I'm glad you're considering those notions in a usual way.

Here are some things you should be familiar with, if you talking
about models, structures that embody all relations, of integers,
here the natural or non-negative integers, for that model-theory
and proof-theory are the same thing, in terms of mathematical proofs,
insofar as that a model of all relation and structurally, is a proof,
and that any proof has a corresponding model, and vice-versa.

This is for constructivism, where the usual notion of intuitionism,
is as above the fundamental and elementary of the model,
its emergent and sublime, as what result the "fundamental theorems",
of the objects in relation, in the fundamentally more elementary,
the objects of the working milieu the theory.

I.e., model theory is the theory, it's the meta-theory and the theory.

So, about models of integers, then you'll be familiar with Peano and
Presburger, after you're familiar with set theory, where, set theory is
the most explored and well-known and developed elementary theory,
being as its a theory of objects with only one relation, "elt" or "element-of",
vis-a-vis sets and classes, which are sets, and elt and "contains" or membership,
the relation.

Here a brief aside defining predicates and relations makes for the usual
notion that a relation is a primary association pair-wise of any two things,
between two domains or sets of things, or a copy as self-domain, there's
that relations relate any two things, and a predicate relates a thing,
to one of true or false, which are values essentially stipulating membership
in the model, a model of satisfied predicates, where predicates are only
satisfied as "true".

So it's kind of said that Peano and Presburger make for a definition of a
model of integers, the integer numbers, the whole numbers, though as
that not always includes zero, the counting numbers which are the relations
of elements to counts or cardinals, the ordinals which are the relations of
elements to position or index, Peano assigns a sort of structure of sets,
in their forms being discernible and recognizable, as a given first and
then following, that the integers are infinitely-many. Then, for the fundamental
theorem of arithmetic, unique prime factorization, and the fundamental theorem
of algebra, the polynomials satisfying having integer roots express arithmetic,
those together make for a sort of usual notion of integers that suffice for
arithmetic, the _modular_.

Eventually then the idea is that a model of these integers exists, and it's related
to exactly one of the non-logical constants of the theory, here named ZF set
theory after Zermelo Fraenkel, its seven axioms one a schema that affect the
reflection that their relations as sets are satisfied. (Why at least one is a schema
is that it results the quantifications over these things have to be stacked or
what's the order of the well-formed formulas, but, they're otherwise axioms..)
These axioms first affect "expansion of comprehension", because it's just derived
that together that make structures in relation that establish the satisfaction of
the formation of pair-wise relations, sets in sets. Then, not all the axioms are
"expansion of comprehension". Two of the axioms introduce "non-logical" constants,
so called because they're just introduced as existing instead of that commonly
under relation their unique forms would exist, because they don't. So, "Zero" and
an "Infinity" are axiomatized, they're stipulated, not just as "expansion of comprehension",
but instead as "non-logical constants", so-called because they have to be axiomatized
unique. Furthermore there's an axiom of "restriction of comprehension", called well-foundedness,
that otherwise a usual quantification over sets could include themselves in their definition,
according to expansion of comprehension. (Quantifier comprehension. The quantifiers
posit the various existence and universality of relation, and freely.) Then, the axiom
of an "Infinity", also includes an aside that it's also well-founded, that not only is
ZF's Infinity non-logical, also, it's a restriction of comprehension.

So, in a theory without restriction of comprehension, ZF's Infinity is a fragment.
Yet, here the point is to get to why it comes around again, that what is a "standard"
model, is where "standard" essentially means "Archimedean", infinitely-many but
no infinitely-valued, while, as a set the "inductive set" that Infinity introduces has
also a property of "no-infinitely-descending-epsilon-chain", or "-elt-chain", where
elt is written epsilon, ZF's Infinity Axiom is there exists an "ordinary", i.e. well-founded,
model of sets _and_ ordinals _and_ integers, according to that both being a non-logical
constant that it exists and restriction of comprehension that it doesn't include itself,
where that otherwise free comprehension arrives anyways that it exists and does.

Now, next you should know about Skolem and Levy, and Skolem and Louwenheim,
and Levy, and perhaps Mostowski. These set theorists basically introduce that for
a given _uncountable_ model, that it has a _countable_ model, which is very surprising
to most people after they've proven to themselves that these ordinary infinite sets
can no way model an ordinary powerset of themself nor vice-versa. These are called
extensions, the greater for the lesser, of that exist only for the lesser for the greater.

Then also there's Cohen's Independence of the Continuum Hypothesis to consider,
which is framed more as of about ordinals. Skolem and Cohen are important set theorists.
Also Solomon Feferman helps introduce the extra-ordinary as in about quantifiers,
where Cohen's result is where sets run out for ordinals, into theories like "ubiquitous
ordinals", in model theory and set theory.

So, why "Infinity" is a non-logical constant in ZF is as above, and, why there's not
a standard model of integers, instead only fragments or extensions, is a concept
I arrived at some years ago. So, here then I'll research my writings on sci.logic and
sci.math about models of integers, and where I wrote quick proofs and results
why what I say is so, to help effect a reflection on model theory overall, as what's
called descriptive set theory after axiomatic set theory, what's called foundations.

Le 14/01/2024 à 19:42, Richard Damon a écrit :

> There are no unit fractions smaller that ALL unit fractions

But there are ℵ unit fraction smaller than all you can name.

>> Assume it. Then you can come close to zero such that nothing is between
>> it and your chosen number. But that is a contradiction.
>
> You can come as close as you want.

By naming an x you can come close as you want in the measure of distance
but not in the measure number of unit fractions between 0 and x.

∀eps > 0 ∀x ∈ (eps, 1]: NUF(x) = ℵo
>
> That doesn't means that you can reach the point that nothing is between.
>
> That would require the existance of a Highest Natural Number.

Nevertheless you can reach 0.
>
> That assumption is where the contradiction is, not that all Natural
> Numbers/Unit Fractions are visible.

That is the same. Fact is that you cannot name almost all numbers.

Regards, WM

Le 15/01/2024 à 01:23, Jim Burns a écrit :
> On 1/14/2024 11:43 AM, WM wrote:

>
>> You cannot use real numbers of
>> the domain containing
>> the ℵ smallest unit fractions and of
>> the domain containing
>> the ℵ largest natural numbers.
>
> Each element of ℝ is
> not the smallest unit fraction and
> not the largest natural number.

Each element of ℝ that can be named.
>
>> It is obviously impossible
>> to come closer to zero.
>
> No,
> it is obvious in arithmetic
> that each x ≠ 0 can come closer.
> To x/2 for example.

There is no boundary in the meaqsure of distance but there is a boundary
in the measure of remaining elements between 0 and the chosen eps > 0.

∀eps > 0, ∀x ∈ (eps, 1]: NUF(x) = ℵo

Regards, WM

On 1/15/24 3:15 AM, WM wrote:
> Le 14/01/2024 à 19:42, Richard Damon a écrit :
>
>> There are no unit fractions smaller that ALL unit fractions
>
> But there are ℵ unit fraction smaller than all you can name.

No, there are not. As I have shown, I can name any of them.

>
>>> Assume it. Then you can come close to zero such that nothing is
>>> between it and your chosen number. But that is a contradiction.
>>
>> You can come as close as you want.
>
> By naming an x you can come close as you want in the measure of distance
> but not in the measure number of unit fractions between 0 and x.

No, the number of unit fractions between 0 and x will always be Alpha_0,
the measure of Countable infinity, as there will ALWAYS be that many
unit fractions between 0 and x (unless x is 0, then the number is 0)

>
> ∀eps > 0 ∀x ∈ (eps, 1]: NUF(x) = ℵo

Right. There are always ℵo unit fractions below a positive number, so no
number exists where NUF(x) == 1

That wasn't for all "visible" numbers, that was for all "Numbers" (from
one of the standard sets of numbers, like Rational or Reals)

>>
>> That doesn't means that you can reach the point that nothing is between.
>>
>> That would require the existance of a Highest Natural Number.
>
> Nevertheless you can reach 0.

Only by leaving the set of unit fractions.

>>
>> That assumption is where the contradiction is, not that all Natural
>> Numbers/Unit Fractions are visible.
>
> That is the same. Fact is that you cannot name almost all numbers.

Except that you can.

Show a set that I can not name a member of it.

Your logic is just insufficient to handle unbounded sets.

>
> Regards, WM

On 1/15/2024 3:24 AM, WM wrote:
> Le 15/01/2024 à 01:23, Jim Burns a écrit :
>> On 1/14/2024 11:43 AM, WM wrote:

>>> It is obviously impossible
>>> to come closer to zero.
>>
>> No,
>> it is obvious in arithmetic
>> that each x ≠ 0 can come closer.
>> To x/2  for example.
>
> There is no boundary in
> the measure of distance
> but
> there is a boundary in
> the measure of remaining elements
> between 0 and the chosen eps > 0.
>
> ∀eps > 0, ∀x ∈ (eps, 1]: NUF(x) = ℵo

In arithmetic,
∀eps > 0
∀k ∈ ℕ
∃mᵉᵖˢ ∈ ℕ:
eps > ⅟mᵉᵖˢ > ⅟(mᵉᵖˢ+k+1) > 0
k < NUF(eps)

In arithmetic,
∀eps > 0
∀k ∈ ℕ
k < NUF(eps)
NUF(eps) ≮ NUF(eps)
NUF(eps) ∉ ℕ

In arithmetic,
∀eps > 0
NUF(eps) ∉ ℕ

In arithmetic,
¬∃mᵂᴹ ∈ ℕ:
∀eps > 0
∀k ∈ ℕ
eps > ⅟mᵂᴹ > ⅟(mᵂᴹ+k+1) > 0

> there is a boundary in
> the measure of remaining elements
> between 0 and the chosen eps > 0.

In arithmetic,
there is a boundary in
the cardinality of remaining elements
between 0 and the chosen eps > 0.
but
it's not a positive boundary,
the boundary is 0

| Assume otherwise.
| Assume 0 < β/2 < β < 2β
| for remaining.element.card.boundary β
| | β < 2β
| NUF(2β) ∉ ℕ
| ∀k ∈ ℕ
| ∃mₖ ∈ ℕ:
| 2β > ⅟mₖ > ⅟(mₖ+k+1) > 0
| [1]
| k < NUF(2β)
| | β/2 < β
| NUF(β/2) ∈ ℕ
| ¬∀k ∈ ℕ
| ∃mₖ ∈ ℕ:
| β/2 > ⅟mₖ > ⅟(mₖ+k+1) > 0
| | ∃k ∈ ℕ:
| ¬(k < NUF(β/2))
| | ∃k ∈ ℕ:
| ∀m′ ∈ ℕ
| ¬(β/2 > ⅟m′ > ⅟(m′+k+1) > 0)
| β/2 ≤ ⅟m′ ∨ ¬(⅟m′ > ⅟(m′+k+1) > 0)
| ⅟m′ > ⅟(m′+k+1)) > 0
| β/2 ≤ ⅟m′
| | ∀m′ ∈ ℕ
| β/2 ≤ ⅟m′
| [2]
| | From [1]
| ∀k ∈ ℕ
| ∃mₖ ∈ ℕ:
| 2β > ⅟mₖ
| β/2 > ⅟(4mₖ)
| | However, from [2]
| in the case of m′ = 4mₖ
| β/2 ≤ ⅟(4mₖ)
| Contradiction.

Therefore,
in arithmetic,
the remaining.element.measure.boundary β
isn't positive.

In arithmetic,
∀eps > 0:
NUF(eps) ∉ ℕ

On Monday, January 15, 2024 at 3:53:29 AM UTC-8, Mild Shock wrote:
> You are a moron. Take this proof:
>
> p(a) v p(b) |- p(a) v p(b)
>
> Whats the "corresponding" model?
>
> I wonder whether your ass already envies your mouth,
> for the amount of shit that comes out of it.
> Ross Finlayson schrieb am Montag, 15. Januar 2024 um 06:52:36 UTC+1:
> > and that any proof has a corresponding model, and vice-versa.

You mean "TND: the domain of the universe of binary propositions?"

It's pretty simple that anything in proof theory has a model in model theory.

It's pretty simple that constructions of inferential relations are first-class,
objects in a model theory of a theory, the theory of model theory, its model.

To relate "first-class" and "first-order", is about simple representations of
logical constructions as logical constructions.

Heh, you wonder.

Seven deadly sins: ....

Le 15/01/2024 à 13:48, Richard Damon a écrit :
> On 1/15/24 3:15 AM, WM wrote:
>> Le 14/01/2024 à 19:42, Richard Damon a écrit :
>>
>>> There are no unit fractions smaller that ALL unit fractions
>>
>> But there are ℵ unit fraction smaller than all you can name.
>
> No, there are not. As I have shown, I can name any of them.

Then show it. Name a unit fraction that has not ℵ smaller ones.
>
> No, the number of unit fractions between 0 and x will always be Alpha_0,
> the measure of Countable infinity, as there will ALWAYS be that many
> unit fractions between 0 and x (unless x is 0, then the number is 0)

The term is aleph_0, not alpha_0.
The mathematics is this: if x is less than every positive number, then x
is less than (0, oo). That is impossible for positive x, let alone for
ℵo unit fractions.

>> ∀eps > 0 ∀x ∈ (eps, 1]: NUF(x) = ℵo
>
> Right. There are always ℵo unit fractions below a positive number, so no
> number exists where NUF(x) == 1

The statement says not below "a positive number" but below "every positive
number", but that means below (0, oo) and is nonsense.
>
> Your logic is just insufficient to handle unbounded sets.

My logic is based on mathematics. Your claims are not logic but dogmas of
matheology in contradiction with mathematics.

Regards, WM

Le 15/01/2024 à 20:57, Jim Burns a écrit :
> In arithmetic,
> ∀eps > 0:
> NUF(eps) ∉ ℕ

In arithmetic

∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.

Regards, WM

On 1/16/24 6:16 AM, WM wrote:
> Le 15/01/2024 à 20:57, Jim Burns a écrit :
>
>> In arithmetic,
>> ∀eps > 0:
>> NUF(eps) ∉ ℕ
>
> In arithmetic
>
> ∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.
>
> Regards, WM
>

Except that the statement does't hold if, say, y was x/2.

You also need to define the domain of x and y. If not specified it will
be presumed something like The Reals.

Your problem is you have the order of operations wrong in your logic,
the Qualifier is run first, so when we look at the inequality, we have
am x we can use.

On 1/16/24 6:12 AM, WM wrote:
> Le 15/01/2024 à 13:48, Richard Damon a écrit :
>> On 1/15/24 3:15 AM, WM wrote:
>>> Le 14/01/2024 à 19:42, Richard Damon a écrit :
>>>
>>>> There are no unit fractions smaller that ALL unit fractions
>>>
>>> But there are ℵ unit fraction smaller than all you can name.
>>
>> No, there are not. As I have shown, I can name any of them.
>
> Then show it. Name a unit fraction that has not ℵ smaller ones.

So, you don't unddrstand English and just working with Strawman.

I never claimed there was a unit fraction that doesn't have aleph_0
smaller ones.

>>
>> No, the number of unit fractions between 0 and x will always be
>> Alpha_0, the measure of Countable infinity, as there will ALWAYS be
>> that many unit fractions between 0 and x (unless x is 0, then the
>> number is 0)
>
> The term is aleph_0, not alpha_0.
> The mathematics is this: if x is less than every positive number, then x
> is less than (0, oo). That is impossible for positive x, let alone for
> ℵo unit fractions.

So, you just proved that there is no

>
>>> ∀eps > 0 ∀x ∈ (eps, 1]: NUF(x) = ℵo
>>
>> Right. There are always ℵo unit fractions below a positive number, so
>> no number exists where NUF(x) == 1
>
> The statement says not below "a positive number" but below "every
> positive number", but that means below (0, oo) and is nonsense.

So, you agree that NUF(x) = ℵo for all positive x and is never a finite
value.

Since for all eps > 0 NUF(x) == ℵo, there is no eps for which NUF(x) is
smaller than that, not even a "dark" one.

>>
>> Your logic is just insufficient to handle unbounded sets.
>
> My logic is based on mathematics. Your claims are not logic but dogmas
> of matheology in contradiction with mathematics.

Nope. YOUR claims are based on the dogma of WMism, not actual logic or
mathematics.

You can't get to actual fundamental statements that show your claims, so
you are just shown to be a dogmatist.

>
> Regards, WM
>
>

On 1/16/2024 6:16 AM, WM wrote:
> Le 15/01/2024 à 20:57, Jim Burns a écrit :

>> In arithmetic,
>> ∀eps > 0:
>> NUF(eps) ∉ ℕ
>
> In arithmetic
> ∀x ∈ (0, 1]: y < x ==> y =< 0,
> i.e., y is not positive.

In arithmetic,
∀eps > 0: eps > 0
i.e., eps is positive.

There is an implicit ∀y

I don't know which you intend:

(i)
(∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
true

if y is a lower bound of (0,1]
then y ≤ glb(0,1] = 0

0 ∉ (0,1]

(ii)
∀x ∈ (0,1]: (y < x ⇒ y ≤ 0)
false

∀y: ∀x ∈ (0,1]: (y < x ⇒ y ≤ 0)
if and only if
¬∃y: ∃x ∈ (0,1]: (y < x ∧ 0 < y)
if and only if
¬∃x ∈ (0,1]: ∃y ∈ (0,x)
false
Instead,
∃x ∈ (0,1]: ∃y ∈ (0,x)

x ∈ ⋃{ (eps,1] | eps ∈ (0,1] }
if and only if
x ∈ (0,1]

∀eps > 0: NUF(eps) ∉ ℕ
true

>
> Regards, WM
>

Richard Damon schrieb:
> On 1/16/24 6:16 AM, WM wrote:
>> Le 15/01/2024 à 20:57, Jim Burns a écrit :
>>
>>> In arithmetic,
>>> ∀eps > 0:
>>> NUF(eps) ∉ ℕ
>>
>> In arithmetic
>>
>> ∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.
>>
>> Regards, WM
>>
>
> Except that the statement does't hold if, say, y was x/2.
>

That's right. But what WM means is the following:

for every y which is smaller than every x ∈ (0,1] : y <= 0.

One of his problems is that he doesn't understand quantifiers and their
proper application. Example: The correct statement about unit fractions
1/n (where n∈ℕ)

∀ x ∈ (0,1] ∃^ℵo 1/n : 1/n < x

he misunderstands in a way that is equivalent to the quantifier shift:

∃^ℵo 1/n ∀ x ∈ (0,1] : 1/n < x .

Then he concludes:

there are ℵo unit fractions left from zero.

Discussing with WM you should always keep in mind: WM doesn't write
about mathematics but his private system of ideas which are based on
his inability to understand infinite sets and set theory.

Dieter Heidorn

On 1/15/2024 12:52 AM, Ross Finlayson wrote:
> On Sunday, January 14, 2024
> at 10:24:52 AM UTC-8, Jim Burns wrote:

>> [...]
>
> Here are some things
> you should be familiar with,
> if you talking about models,
> structures that embody all relations,
> of integers,
> here the natural or non-negative integers,
> for that model-theory and proof-theory are
> the same thing,
> in terms of mathematical proofs,
> insofar as that
> a model of all relation and structurally,
> is a proof,
> and that
> any proof has a corresponding model,
> and vice-versa.

As I understand it,
the semantic (model.theory) and
the syntactic (proof.theory) points of view
are two sides of the same coin.

There are some very nice proofs showing
how the two are related, but
they aren't exactly the same thing.

For a certain theory (syntax),
there might or might not exist
a structure which it describes (semantics).
A structure which a theory describes
is a model of the theory.

Provably,
if no contradiction follows from a theory,
then a model of it exists.

The proof constructs (shows exists) a model from
the objects of the language of the theory.
That's why I note that, even if a theory has a model,
it still might not have the model you're thinking of.

Every structure that satisfies a theory is
a model of the theory.

Perhaps significantly different structures
satisfy the same theory (are models).

Provably,
if each model (semantics) which satisfies a theory
satisfies an additional formula,
then a proof (syntax) exists of that formula
starting from the theory (syntax).

----
Because there are formally undecidable formulas
in the natural numbers,
if any model exists,
then more than one model exists.

Also,
if any inductive set exists,
then a unique minimal inductive set exists,
subset to each inductive set, and
containing only each countable.to number.
By any other word, a standard model.

tl,;dr
The natural numbers either
are complete nonsense, top to bottom, or
have a standard model and a non.standard model.

On Tuesday, January 16, 2024 at 1:31:50 PM UTC-8, Jim Burns wrote:
> On 1/15/2024 12:52 AM, Ross Finlayson wrote:
> > On Sunday, January 14, 2024
> > at 10:24:52 AM UTC-8, Jim Burns wrote:
> >> [...]
> >
> > Here are some things
> > you should be familiar with,
> > if you talking about models,
> > structures that embody all relations,
> > of integers,
> > here the natural or non-negative integers,
> > for that model-theory and proof-theory are
> > the same thing,
> > in terms of mathematical proofs,
> > insofar as that
> > a model of all relation and structurally,
> > is a proof,
> > and that
> > any proof has a corresponding model,
> > and vice-versa.
> As I understand it,
> the semantic (model.theory) and
> the syntactic (proof.theory) points of view
> are two sides of the same coin.
>
> There are some very nice proofs showing
> how the two are related, but
> they aren't exactly the same thing.
>
> For a certain theory (syntax),
> there might or might not exist
> a structure which it describes (semantics).
> A structure which a theory describes
> is a model of the theory.
>
> Provably,
> if no contradiction follows from a theory,
> then a model of it exists.
>
> The proof constructs (shows exists) a model from
> the objects of the language of the theory.
> That's why I note that, even if a theory has a model,
> it still might not have the model you're thinking of.
>
> Every structure that satisfies a theory is
> a model of the theory.
>
> Perhaps significantly different structures
> satisfy the same theory (are models).
>
> Provably,
> if each model (semantics) which satisfies a theory
> satisfies an additional formula,
> then a proof (syntax) exists of that formula
> starting from the theory (syntax).
>
> ----
> Because there are formally undecidable formulas
> in the natural numbers,
> if any model exists,
> then more than one model exists.
>
> Also,
> if any inductive set exists,
> then a unique minimal inductive set exists,
> subset to each inductive set, and
> containing only each countable.to number.
> By any other word, a standard model.
>
> tl,;dr
> The natural numbers either
> are complete nonsense, top to bottom, or
> have a standard model and a non.standard model.

New RECUSITY lines...

RECUSITY:
FFAST

PRIDE

it's personal....

If you can read this,
look behind you.

I don't have eyes in the back of my head.

If wi-fi means wired fidelity,
I'm pretty sure it's cheating.

It's RECUSITY,
wear it out.

That model theory and proof theory are equi-interpretable,
and often thoroughly intermixed, pretty much separates
the men from the toys.

On Tuesday, January 16, 2024 at 2:01:02 PM UTC-8, Ross Finlayson wrote:
> On Tuesday, January 16, 2024 at 1:31:50 PM UTC-8, Jim Burns wrote:
> > On 1/15/2024 12:52 AM, Ross Finlayson wrote:
> > > On Sunday, January 14, 2024
> > > at 10:24:52 AM UTC-8, Jim Burns wrote:
> > >> [...]
> > >
> > > Here are some things
> > > you should be familiar with,
> > > if you talking about models,
> > > structures that embody all relations,
> > > of integers,
> > > here the natural or non-negative integers,
> > > for that model-theory and proof-theory are
> > > the same thing,
> > > in terms of mathematical proofs,
> > > insofar as that
> > > a model of all relation and structurally,
> > > is a proof,
> > > and that
> > > any proof has a corresponding model,
> > > and vice-versa.
> > As I understand it,
> > the semantic (model.theory) and
> > the syntactic (proof.theory) points of view
> > are two sides of the same coin.
> >
> > There are some very nice proofs showing
> > how the two are related, but
> > they aren't exactly the same thing.
> >
> > For a certain theory (syntax),
> > there might or might not exist
> > a structure which it describes (semantics).
> > A structure which a theory describes
> > is a model of the theory.
> >
> > Provably,
> > if no contradiction follows from a theory,
> > then a model of it exists.
> >
> > The proof constructs (shows exists) a model from
> > the objects of the language of the theory.
> > That's why I note that, even if a theory has a model,
> > it still might not have the model you're thinking of.
> >
> > Every structure that satisfies a theory is
> > a model of the theory.
> >
> > Perhaps significantly different structures
> > satisfy the same theory (are models).
> >
> > Provably,
> > if each model (semantics) which satisfies a theory
> > satisfies an additional formula,
> > then a proof (syntax) exists of that formula
> > starting from the theory (syntax).
> >
> > ----
> > Because there are formally undecidable formulas
> > in the natural numbers,
> > if any model exists,
> > then more than one model exists.
> >
> > Also,
> > if any inductive set exists,
> > then a unique minimal inductive set exists,
> > subset to each inductive set, and
> > containing only each countable.to number.
> > By any other word, a standard model.
> >
> > tl,;dr
> > The natural numbers either
> > are complete nonsense, top to bottom, or
> > have a standard model and a non.standard model.
> New RECUSITY lines...
>
>
> RECUSITY:
> FFAST
>
>
>
> PRIDE
>
>
> it's personal....
>
>
>
>
> If you can read this,
> look behind you.
>
> I don't have eyes in the back of my head.
>
>
>
>
> If wi-fi means wired fidelity,
> I'm pretty sure it's cheating.
>
>
>
> It's RECUSITY,
> wear it out.
>
>
>
> That model theory and proof theory are equi-interpretable,
> and often thoroughly intermixed, pretty much separates
> the men from the toys.

Always a favorite:

I CAN READ

writing on the wall

Now with more THE.

James, your cogent response deserves a more thorough
and appreciative reply, so I hope that you'll find some
further in this thread, and I'll make some too.

Thanks for saving us all.

On Tuesday, January 16, 2024 at 2:29:07 PM UTC-8, Ross Finlayson wrote:
> On Tuesday, January 16, 2024 at 2:01:02 PM UTC-8, Ross Finlayson wrote:
> > On Tuesday, January 16, 2024 at 1:31:50 PM UTC-8, Jim Burns wrote:
> > > On 1/15/2024 12:52 AM, Ross Finlayson wrote:
> > > > On Sunday, January 14, 2024
> > > > at 10:24:52 AM UTC-8, Jim Burns wrote:
> > > >> [...]
> > > >
> > > > Here are some things
> > > > you should be familiar with,
> > > > if you talking about models,
> > > > structures that embody all relations,
> > > > of integers,
> > > > here the natural or non-negative integers,
> > > > for that model-theory and proof-theory are
> > > > the same thing,
> > > > in terms of mathematical proofs,
> > > > insofar as that
> > > > a model of all relation and structurally,
> > > > is a proof,
> > > > and that
> > > > any proof has a corresponding model,
> > > > and vice-versa.
> > > As I understand it,
> > > the semantic (model.theory) and
> > > the syntactic (proof.theory) points of view
> > > are two sides of the same coin.
> > >
> > > There are some very nice proofs showing
> > > how the two are related, but
> > > they aren't exactly the same thing.
> > >
> > > For a certain theory (syntax),
> > > there might or might not exist
> > > a structure which it describes (semantics).
> > > A structure which a theory describes
> > > is a model of the theory.
> > >
> > > Provably,
> > > if no contradiction follows from a theory,
> > > then a model of it exists.
> > >
> > > The proof constructs (shows exists) a model from
> > > the objects of the language of the theory.
> > > That's why I note that, even if a theory has a model,
> > > it still might not have the model you're thinking of.
> > >
> > > Every structure that satisfies a theory is
> > > a model of the theory.
> > >
> > > Perhaps significantly different structures
> > > satisfy the same theory (are models).
> > >
> > > Provably,
> > > if each model (semantics) which satisfies a theory
> > > satisfies an additional formula,
> > > then a proof (syntax) exists of that formula
> > > starting from the theory (syntax).
> > >
> > > ----
> > > Because there are formally undecidable formulas
> > > in the natural numbers,
> > > if any model exists,
> > > then more than one model exists.
> > >
> > > Also,
> > > if any inductive set exists,
> > > then a unique minimal inductive set exists,
> > > subset to each inductive set, and
> > > containing only each countable.to number.
> > > By any other word, a standard model.
> > >
> > > tl,;dr
> > > The natural numbers either
> > > are complete nonsense, top to bottom, or
> > > have a standard model and a non.standard model.
> > New RECUSITY lines...
> >
> >
> > RECUSITY:
> > FFAST
> >
> >
> >
> > PRIDE
> >
> >
> > it's personal....
> >
> >
> >
> >
> > If you can read this,
> > look behind you.
> >
> > I don't have eyes in the back of my head.
> >
> >
> >
> >
> > If wi-fi means wired fidelity,
> > I'm pretty sure it's cheating.
> >
> >
> >
> > It's RECUSITY,
> > wear it out.
> >
> >
> >
> > That model theory and proof theory are equi-interpretable,
> > and often thoroughly intermixed, pretty much separates
> > the men from the toys.
> Always a favorite:
>
>
> I CAN READ
>
> writing on the wall
>
>
> Now with more THE.
>
>
>
> James, your cogent response deserves a more thorough
> and appreciative reply, so I hope that you'll find some
> further in this thread, and I'll make some too.
>
>
> Thanks for saving us all.

Calvin on Calvinism

Somebody had to keep it running

when nobody else knew how it did

Geil. Sehr geil.

In proof theory, it's provable, that in classical logic, one must
always assert Empty, Infinity, and well-foundedness for the
inference of the existence of any set, for otherwise inference
may arrive at Russell's set via quantification.

So, in classical logic, ZF's axioms aren't independent.
Of course, these days we know that that's "quasi-modal" logic.
Yet, it's not the point here that "classical" logic isn't monotonic.

Yet, the idea that "there's isn't a standard model of the integers",
and, also, "there are multiple non-standard models of integers",
in set theory, and as well in number theory, has a lot going on
with "there's an infinite integer in some infinitudes of integers".

Largely, an unbounded fragment, is the model of usual infinite induction.
That's not even necessarily an infinity, just an unbounded exercise
of induction.

Then, a true infinity, makes for integers, about the number-theoretic
properties of its elements, and what one has. This gets into things
like "primes at infinity", "double primes at infinity", "triple primes at
infinity", the last one of those being 2, 3, 5, and "quadruple primes at
infinity", the first one of those being at infinity, and on.

"Mathematics: truer than initially thought"

Now the above already has a lot why "there aren't standard models of
integers and there are bounded and unbounded fragments and there
are non-standard extensions", there's plenty more to it, not that that's
not plenty, to it.

"Hey, classical quasi-modal logic isn't all bad. You might get lucky."

Le 16/01/2024 à 13:11, Richard Damon a écrit :
> On 1/16/24 6:16 AM, WM wrote:
>> Le 15/01/2024 à 20:57, Jim Burns a écrit :
>>
>>> In arithmetic,
>>> ∀eps > 0:
>>> NUF(eps) ∉ ℕ
>>
>> In arithmetic
>>
>> ∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.
>>

> Except that the statement does't hold if, say, y was x/2.

The statement concerns all real numbers.
>
> You also need to define the domain of x and y. If not specified it will
> be presumed something like The Reals.
>
> Your problem is you have the order of operations wrong in your logic,
> the Qualifier is run first, so when we look at the inequality, we have
> am x we can use.

The "less than" relation is independent of quantifier-exchange. If for
every x ∈ (0, 1] there a smaller y, then there is an y smaller than
every x ∈ (0, 1] and hence smaller than (0, 1]. Of course for ℵ
elements y this is much clearer.

Regards, WM

Le 16/01/2024 à 21:19, Dieter Heidorn a écrit :
> The correct statement about unit fractions
> 1/n (where n∈ℕ)
>
> ∀ x ∈ (0,1] ∃^ℵo 1/n : 1/n < x
>
> he misunderstands in a way that is equivalent to the quantifier shift:
>
> ∃^ℵo 1/n ∀ x ∈ (0,1] : 1/n < x .
>
> Then he concludes:
>
> there are ℵo unit fractions left from zero.

Would be required to make your statement correct.

For the less-than relation there is no quantifier magic.
NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.

Regards, WM

Le 16/01/2024 à 18:11, Jim Burns a écrit :

> (i)
> (∀x ∈ (0,1]: y < x) ⇒ y ≤ 0
> true

Yes, that is correct.

Regards, WM

On 1/17/24 6:49 AM, WM wrote:
> Le 16/01/2024 à 21:19, Dieter Heidorn a écrit :
>> The correct statement about unit fractions
>> 1/n (where n∈ℕ)
>>
>>     ∀ x ∈ (0,1]  ∃^ℵo 1/n : 1/n < x
>>
>> he misunderstands in a way that is equivalent to the quantifier shift:
>>
>>     ∃^ℵo 1/n  ∀ x ∈ (0,1] : 1/n < x .
>>
>> Then he concludes:
>>
>>     there are ℵo unit fractions left from zero.
>
> Would be required to make your statement correct.
>
> For the less-than relation there is no quantifier magic.
> NUF(x) cannot grow anywhere from 0 to ℵo without passing finite values.
>
> Regards, WM
>

Why not?

And what says those "values" that it happens at have to be in the set of
Unit Fractions/rational/real numbers?

If the "value" where NUF(x) == 1 isn't in the domain of Unit
factions/Rational Numbers/ Real Numbers, then there is no need for your
"dark" numbers, they are just visible numbers in some other system.

On 1/17/24 6:43 AM, WM wrote:
> Le 16/01/2024 à 13:11, Richard Damon a écrit :
>> On 1/16/24 6:16 AM, WM wrote:
>>> Le 15/01/2024 à 20:57, Jim Burns a écrit :
>>>
>>>> In arithmetic,
>>>> ∀eps > 0:
>>>> NUF(eps) ∉ ℕ
>>>
>>> In arithmetic
>>>
>>> ∀x ∈ (0, 1]: y < x ==> y =< 0, i.e., y is not positive.
>>>
>
>> Except that the statement does't hold if, say, y was x/2.
>
> The statement concerns all real numbers.
>>
>> You also need to define the domain of x and y. If not specified it
>> will be presumed something like The Reals.
>>
>> Your problem is you have the order of operations wrong in your logic,
>> the Qualifier is run first, so when we look at the inequality, we have
>> am x we can use.
>
> The "less than" relation is independent of quantifier-exchange. If for
> every x ∈ (0, 1] there a smaller  y, then there is an y smaller than
> every x ∈ (0, 1] and hence smaller than (0, 1]. Of course for ℵ elements
> y this is much clearer.
>
> Regards, WM
>
>
>

Right, for every x ∈ (0, 1] there exist a y such that 0 < y < x.

That doesn't mean that y < (0, 1], because that is a category error. x
and y were elements of the set and don't have a numeric relations to the
set (sets aren't numbers, but sets of numbers)

This is just expressing the unbounded nature of this set of numbers,
there is NO "lowest bound" of the set in the set.

On 1/17/24 6:53 AM, WM wrote:
> Le 16/01/2024 à 18:11, Jim Burns a écrit :
>
>> (i)
>> (∀x ∈ (0,1]: y < x)  ⇒  y ≤ 0
>> true
>
> Yes, that is correct.
>
> Regards, WM
>
>
>
>

Except we show it isn't for y = x/2