1. Scrooge McDuck's bankrupt

Scrooge Mc Duck earns 1000 $ daily and spends only 1 $ per day. As a
cartoon-figure he will live forever and his wealth will increase without
bound. But according to set theory he will get bankrupt if he spends the
dollars in the same order as he receives them. Only if he always spends
them in another order, for instance every day the second dollar
received, he will get rich. These different results prove set theory to
be useless for all practical purposes.

The above story is only the story of Tristram Shandy in simplified
terms, which has been narrated by Fraenkel, one of the fathers of ZF set
theory.

"Well known is the story of Tristram Shandy who undertakes to write his
biography, in fact so pedantically, that the description of each day
takes him a full year. Of course he will never get ready if continuing
that way. But if he lived infinitely long (for instance a 'countable
infinity' of years [...]), then his biography would get 'ready',
because, expressed more precisely, every day of his life, how late ever,
finally would get its description because the year scheduled for this
work would some time appear in his life." [A. Fraenkel: "Einleitung in
die Mengenlehre", 3rd ed., Springer, Berlin (1928) p. 24] "If he is
mortal he can never terminate; but did he live forever then no part of
his biography would remain unwritten, for to each day of his life a year
devoted to that day's description would correspond." [A.A. Fraenkel, A.
Levy: "Abstract set theory", 4th ed., North Holland, Amsterdam (1976) p.
30]

2. Failed enumeration of the fractions

All natural numbers are said to be enough to index all positive
fractions. This can be disproved when the natural numbers are taken from
the first column of the matrix of all positive fractions

1/1, 1/2, 1/3, 1/4, ...
2/1, 2/2, 2/3, 2/4, ...
3/1, 3/2, 3/3, 3/4, ...
4/1, 4/2, 4/3, 4/4, ...
... .

To cover the whole matrix by the integer fractions amounts to the idea
that the letters X in

XOOO...
XOOO...
XOOO...
XOOO...
...

can be redistributed to cover all positions by exchanging them with the
letters O. (X and O must be exchanged because where an index has left,
there is no index remaining.) But where should the O remain if not
within the matrix at positions not covered by X?

3. Violation of translation invariance

Translation invariance is fundamental to every scientific theory. With n
m ∈ ℕ and q ∈ {ℚ ∩ (0, 1]} there is precisely the same number of
rational points n + q in (n, n+1] as of rational points m + q in (m,
m+1] . However, half of all positive rational numbers of Cantor's
enumeration
1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
5/1, ...
are of the form 0 + q and lie in the first unit interval between 0 and
1. There are less rational points in (1, 2] but more than in (2, 3] and
so on.

4. Violation of inclusion monotony

Every endsegment E(n) = {n, n+1, n+2, ...} of natural numbers has an
infinite intersection with all other infinite endsegments.
∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
Set theory however comes to the conclusion that there are only infinite
endsegments and that their intersection is empty. This violates the
inclusion monotony of the endegments according to which, as long as only
non-empty endsegments are concerned, their intersection is non-empty.

5. Actual infinity implies a smallest unit fraction

All unit fractions 1/n have finite distances from each other
∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
Therefore the function Number of Unit Fractions between 0 and x, NUF(x),
cannot be infinite for all x > 0. The claim of set theory
∀x ∈ (0, 1]: NUF(x) = ℵo
is wrong. If every positive point has ℵo unit fractions at its left-hand
side, then there is no positive point with less than ℵo unit fractions
at its left-hand side, then all positive points have ℵo unit fractions
at their left-hand side, then the interval (0, 1] has ℵo unit fractions
at its left-hand side, then ℵo unit fractions are negative.
Contradiction.

6. There are more path than nodes in the infinite Binary Tree

Since each of n paths in the complete infinite Binary Tree contains at
least one node differing from all other paths, there are not less nodes
than paths possible. Everything else would amount to having more houses
than bricks.

7. The diagonal does not define a number

An endless digit sequence without finite definition of the digits cannot
define a real number. After every known digit almost all digits will
follow.

Regards, WM

On Thursday, January 4, 2024 at 1:59:34 AM UTC-8, WM wrote:
> 1. Scrooge McDuck's bankrupt
>
> Scrooge Mc Duck earns 1000 $ daily and spends only 1 $ per day. As a
> cartoon-figure he will live forever and his wealth will increase without
> bound. But according to set theory he will get bankrupt if he spends the
> dollars in the same order as he receives them. Only if he always spends
> them in another order, for instance every day the second dollar
> received, he will get rich. These different results prove set theory to
> be useless for all practical purposes.
>
> The above story is only the story of Tristram Shandy in simplified
> terms, which has been narrated by Fraenkel, one of the fathers of ZF set
> theory.
>
> "Well known is the story of Tristram Shandy who undertakes to write his
> biography, in fact so pedantically, that the description of each day
> takes him a full year. Of course he will never get ready if continuing
> that way. But if he lived infinitely long (for instance a 'countable
> infinity' of years [...]), then his biography would get 'ready',
> because, expressed more precisely, every day of his life, how late ever,
> finally would get its description because the year scheduled for this
> work would some time appear in his life." [A. Fraenkel: "Einleitung in
> die Mengenlehre", 3rd ed., Springer, Berlin (1928) p. 24] "If he is
> mortal he can never terminate; but did he live forever then no part of
> his biography would remain unwritten, for to each day of his life a year
> devoted to that day's description would correspond." [A.A. Fraenkel, A.
> Levy: "Abstract set theory", 4th ed., North Holland, Amsterdam (1976) p.
> 30]
>
> 2. Failed enumeration of the fractions
>
> All natural numbers are said to be enough to index all positive
> fractions. This can be disproved when the natural numbers are taken from
> the first column of the matrix of all positive fractions
>
> 1/1, 1/2, 1/3, 1/4, ...
> 2/1, 2/2, 2/3, 2/4, ...
> 3/1, 3/2, 3/3, 3/4, ...
> 4/1, 4/2, 4/3, 4/4, ...
> .. .
>
> To cover the whole matrix by the integer fractions amounts to the idea
> that the letters X in
>
> XOOO...
> XOOO...
> XOOO...
> XOOO...
> ..
>
> can be redistributed to cover all positions by exchanging them with the
> letters O. (X and O must be exchanged because where an index has left,
> there is no index remaining.) But where should the O remain if not
> within the matrix at positions not covered by X?
>
> 3. Violation of translation invariance
>
> Translation invariance is fundamental to every scientific theory. With n
> m ∈ ℕ and q ∈ {ℚ ∩ (0, 1]} there is precisely the same number of
> rational points n + q in (n, n+1] as of rational points m + q in (m,
> m+1] . However, half of all positive rational numbers of Cantor's
> enumeration
> 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1, 1/5, 2/4, 3/3, 4/2,
> 5/1, ...
> are of the form 0 + q and lie in the first unit interval between 0 and
> 1. There are less rational points in (1, 2] but more than in (2, 3] and
> so on.
>
> 4. Violation of inclusion monotony
>
> Every endsegment E(n) = {n, n+1, n+2, ...} of natural numbers has an
> infinite intersection with all other infinite endsegments.
> ∀k ∈ ℕ_def: ∩{E(1), E(2), ..., E(k)} = E(k) /\ |E(k)| = ℵ₀ .
> Set theory however comes to the conclusion that there are only infinite
> endsegments and that their intersection is empty. This violates the
> inclusion monotony of the endegments according to which, as long as only
> non-empty endsegments are concerned, their intersection is non-empty.
>
> 5. Actual infinity implies a smallest unit fraction
>
> All unit fractions 1/n have finite distances from each other
> ∀n ∈ ℕ: 1/n - 1/(n+1) = d_n > 0.
> Therefore the function Number of Unit Fractions between 0 and x, NUF(x),
> cannot be infinite for all x > 0. The claim of set theory
> ∀x ∈ (0, 1]: NUF(x) = ℵo
> is wrong. If every positive point has ℵo unit fractions at its left-hand
> side, then there is no positive point with less than ℵo unit fractions
> at its left-hand side, then all positive points have ℵo unit fractions
> at their left-hand side, then the interval (0, 1] has ℵo unit fractions
> at its left-hand side, then ℵo unit fractions are negative.
> Contradiction.
>
> 6. There are more path than nodes in the infinite Binary Tree
>
> Since each of n paths in the complete infinite Binary Tree contains at
> least one node differing from all other paths, there are not less nodes
> than paths possible. Everything else would amount to having more houses
> than bricks.
>
> 7. The diagonal does not define a number
>
> An endless digit sequence without finite definition of the digits cannot
> define a real number. After every known digit almost all digits will
> follow.
>
> Regards, WM

I replied to this in the other identical thread now you're spamming in this one.

Your problems include set theory has problems, but you're not helping.

Really if you want to address set theory's problems then you'll review my slates,
about uncountability and logical paradox, about continuous domains and foundations.

So, "Mister McDuck", or, "Mister Magoo", as it were, first you should read "Hodges'
Hopeless: an editor recalls some papers", about usual mistakes in studying set theory,
then strike all those examples from yours, then what you're left with is to read mine.

....
way it is

On 1/4/24 10:59, WM wrote:
> [snip]

what point are you trying to make? infinity is strange

immibis schrieb am Donnerstag, 4. Januar 2024 um 22:53:35 UTC+1:

> what point are you trying to make? infinity is strange

But it is based on logic. This logic is violated in the seven points I
made.

Regards, WM

On 1/5/24 5:22 AM, WM wrote:
> immibis schrieb am Donnerstag, 4. Januar 2024 um 22:53:35 UTC+1:
>
>> what point are you trying to make? infinity is strange
>
> But it is based on logic. This logic is violated in the seven points I
> made.
>
> Regards, WM

Which are base on logic that doesn't handle infinity.

Yes, it is well know that infinite sets break some of the seemingly
obvious properties that hold for finite sets.

YOU just don't seem to understand and accept that fact, and keep on
making the ERROR of asuming it must.

On 1/4/2024 4:59 AM, WM wrote:

> 7. The diagonal does not define a number
>
> An endless digit sequence without
> finite definition of the digits
> cannot define a real number.
> After every known digit
> almost all digits will follow.

After every known and unknown digit,
almost all digits follow.

For each digit in the sequence,
the cardinal of digits.before is ⁺¹.able,
the cardinal of digits.after is
larger than each ⁺¹.able cardinal,
and thus is not.⁺¹.able.

It is sufficient that
an endless digit sequence without
finite definition of the digits
exist.

There is at most one real number
which is permitted by each
finite initial sub.sequence of digits.
That one real number is the one which
the endless digit sequence represents.

| Assume otherwise.
| Assume two points at a distance d > 0
| are permitted by each
| finite initial sub.sequence of digits.
| | However,
| there is an initial n.digit sub.sequence which
| permits only points apart by 10-n < d
| Those two points cannot both be permitted
| by that n.digit sub.sequence or
| by each finite initial sub.sequence.
| Contradiction.

Therefore,
there is at most one real number
which is permitted by each
finite initial sub.sequence of digits.

Also,
there is at least one real number
which is permitted by each
finite initial sub.sequence of digits.

That follows from the requirement that
functions which jump
are discontinuous at some point.

For each endless digit sequence,
each digit preceded by ⁺¹.ably.many digits,
there is no more and no less than
one point.

For each point,
there is no less than one and
no more than one
(no more than two for trailing 0's and 9's)
endless digit sequence,
each digit preceded by ⁺¹.ably.many digits,

The representation of real points by digits
is obviously cousin to
the representation of rational points by digits.

However, the cousins are not the same.
The points _exist_
The digit.sequences _exist_
We know that by augmenting descriptive claims
with only not.first.false claims.
That isn't a calculation in the sense that
a rational point is calculated.

Nevertheless, we know they exist, uncalculated,
because we know that
a finite sequence of _claims_
if it has no first.false _claim_
has no false _claim_

On Friday, January 5, 2024 at 10:32:19 AM UTC-8, Jim Burns wrote:
> On 1/4/2024 4:59 AM, WM wrote:
>
> > 7. The diagonal does not define a number
> >
> > An endless digit sequence without
> > finite definition of the digits
> > cannot define a real number.
> > After every known digit
> > almost all digits will follow.
> After every known and unknown digit,
> almost all digits follow.
>
> For each digit in the sequence,
> the cardinal of digits.before is ⁺¹.able,
> the cardinal of digits.after is
> larger than each ⁺¹.able cardinal,
> and thus is not.⁺¹.able.
>
>
> It is sufficient that
> an endless digit sequence without
> finite definition of the digits
> exist.
>
> There is at most one real number
> which is permitted by each
> finite initial sub.sequence of digits.
> That one real number is the one which
> the endless digit sequence represents.
>
> | Assume otherwise.
> | Assume two points at a distance d > 0
> | are permitted by each
> | finite initial sub.sequence of digits.
> |
> | However,
> | there is an initial n.digit sub.sequence which
> | permits only points apart by 10-n < d
> | Those two points cannot both be permitted
> | by that n.digit sub.sequence or
> | by each finite initial sub.sequence.
> | Contradiction.
>
> Therefore,
> there is at most one real number
> which is permitted by each
> finite initial sub.sequence of digits.
>
> Also,
> there is at least one real number
> which is permitted by each
> finite initial sub.sequence of digits.
>
> That follows from the requirement that
> functions which jump
> are discontinuous at some point.
>
> For each endless digit sequence,
> each digit preceded by ⁺¹.ably.many digits,
> there is no more and no less than
> one point.
>
> For each point,
> there is no less than one and
> no more than one
> (no more than two for trailing 0's and 9's)
> endless digit sequence,
> each digit preceded by ⁺¹.ably.many digits,
>
> The representation of real points by digits
> is obviously cousin to
> the representation of rational points by digits.
>
> However, the cousins are not the same.
> The points _exist_
> The digit.sequences _exist_
> We know that by augmenting descriptive claims
> with only not.first.false claims.
> That isn't a calculation in the sense that
> a rational point is calculated.
>
> Nevertheless, we know they exist, uncalculated,
> because we know that
> a finite sequence of _claims_
> if it has no first.false _claim_
> has no false _claim_

If it's sufficient to establish a model of arithmetic then
by the GIT's you'll agree it's at best incomplete.

.... That it's false to say it's, the "true", claim.
Only scientific and not falsified.

Platonism then sort of demands "there are true
numbers, so work it up".

"There is no 'but', only 'yet', ...."

....

On 1/5/2024 5:58 PM, Ross Finlayson wrote:
> On Friday, January 5, 2024
> at 10:32:19 AM UTC-8, Jim Burns wrote:
>> On 1/4/2024 4:59 AM, WM wrote:

>>> An endless digit sequence without
>>> finite definition of the digits
>>> cannot define a real number.
>>> After every known digit
>>> almost all digits will follow.

>> The representation of real points by digits
>> is obviously cousin to
>> the representation of rational points by digits.
>>
>> However, the cousins are not the same.
>> The points _exist_
>> The digit.sequences _exist_
>> We know that by augmenting descriptive claims
>> with only not.first.false claims.
>> That isn't a calculation in the sense that
>> a rational point is calculated.
>>
>> Nevertheless, we know they exist, uncalculated,
>> because we know that
>> a finite sequence of _claims_
>> if it has no first.false _claim_
>> has no false _claim_
>
> If it's sufficient
> to establish a model of arithmetic
> then by the GIT's
> you'll agree it's at best incomplete.

I'm not sure we're on the same page
with regard to incompleteness.

Incomplete == some things we can't know.
Were you (RF) seriously contemplating
not (some things we can't know)?

Gödel didn't surprise with the result itself.
Yes, some things we can't know. Whatever.
Gödel surprised with _proving_ it.
| On Formally Undecidable Propositions of
| Principia Mathematica and Related Systems
| _Formally_ undecidable.

----
If a theory has a model,
then it's consistent.

If a theory can express
recursive definitions
non.recursively
(re.stated without the defined term)
(which is something arithmetic can do),
then
the theory can _quote_
recursive definitions of
what it is to be a formula and
what it is to be a proof.

By "quote", I mean: represent
an object of language (formula, proof) by
an object of study (number, set)
(Gödel.numbers et al)

If a theory can quote
what it is to be a formula and
what it is to be a proof,
then the theory can express G(x) such that
G("H(x)") ⟹ proof of H("H(x)") not.exists.
¬G("H(x)") ⟹ proof of contradiction exists.

If a theory can express G("H(x)")
then the theory can express G("G(x)")
G("G(x)") ⟹ proof of G("G(x)") not.exists.
¬G("G(x)") ⟹ proof of contradiction exists.

If a theory has a certain
very.attainable level of expressiveness,
then choose one: incomplete or inconsistent.

But there is a model. It's consistent.
So, it's incomplete.

> ... That it's false to say
> it's, the "true", claim.
> Only scientific and not falsified.
>
> Platonism then sort of demands
> "there are true numbers, so work it up".
>
> "There is no 'but', only 'yet', ...."

We know that
each theory is true of
whatever that theory is true of.

Some theories are true of nothing.
Contradictions can be proved in those.

Some theories are true of something,
of what we intend or of something unintended.
Contradictions cannot be proved in those.

Some theories, numbers, for example,
if they can prove they're consistent,
would prove that they _aren't_ consistent.

There is no "yet" to not.proving consistency.
We know there is no such proof --
or, if (sadly) there is, it's meaningless,
about nothing.

However,
there are better options than
a theory proving itself consistent.

Gödel's work does not deny that
some other theory can prove
our theory consistent.

For example,
ZFC is a theorem of
ZFC+inaccessible.cardinal.exists

Some theories are so simple that,
even though we know there is no proof,
we don't take seriously the possibility
that they have no model.

For example,
the set.theory fragment,
-- {} exists
-- x∪{y} exists
-- same.element.sets x and y are equal
proves arithmetic and Gödel.incompleteness

On Saturday, January 6, 2024 at 11:29:28 AM UTC-8, Jim Burns wrote:
> On 1/5/2024 5:58 PM, Ross Finlayson wrote:
> > On Friday, January 5, 2024
> > at 10:32:19 AM UTC-8, Jim Burns wrote:
> >> On 1/4/2024 4:59 AM, WM wrote:
>
> >>> An endless digit sequence without
> >>> finite definition of the digits
> >>> cannot define a real number.
> >>> After every known digit
> >>> almost all digits will follow.
> >> The representation of real points by digits
> >> is obviously cousin to
> >> the representation of rational points by digits.
> >>
> >> However, the cousins are not the same.
> >> The points _exist_
> >> The digit.sequences _exist_
> >> We know that by augmenting descriptive claims
> >> with only not.first.false claims.
> >> That isn't a calculation in the sense that
> >> a rational point is calculated.
> >>
> >> Nevertheless, we know they exist, uncalculated,
> >> because we know that
> >> a finite sequence of _claims_
> >> if it has no first.false _claim_
> >> has no false _claim_
> >
> > If it's sufficient
> > to establish a model of arithmetic
> > then by the GIT's
> > you'll agree it's at best incomplete.
> I'm not sure we're on the same page
> with regard to incompleteness.
>
> Incomplete == some things we can't know.
> Were you (RF) seriously contemplating
> not (some things we can't know)?
>
> Gödel didn't surprise with the result itself.
> Yes, some things we can't know. Whatever.
> Gödel surprised with _proving_ it.
> | On Formally Undecidable Propositions of
> | Principia Mathematica and Related Systems
> |
> _Formally_ undecidable.
>
> ----
> If a theory has a model,
> then it's consistent.
>
> If a theory can express
> recursive definitions
> non.recursively
> (re.stated without the defined term)
> (which is something arithmetic can do),
> then
> the theory can _quote_
> recursive definitions of
> what it is to be a formula and
> what it is to be a proof.
>
> By "quote", I mean: represent
> an object of language (formula, proof) by
> an object of study (number, set)
> (Gödel.numbers et al)
>
> If a theory can quote
> what it is to be a formula and
> what it is to be a proof,
> then the theory can express G(x) such that
> G("H(x)") ⟹ proof of H("H(x)") not.exists.
> ¬G("H(x)") ⟹ proof of contradiction exists.
>
> If a theory can express G("H(x)")
> then the theory can express G("G(x)")
> G("G(x)") ⟹ proof of G("G(x)") not.exists.
> ¬G("G(x)") ⟹ proof of contradiction exists.
>
> If a theory has a certain
> very.attainable level of expressiveness,
> then choose one: incomplete or inconsistent.
>
> But there is a model. It's consistent.
> So, it's incomplete.
> > ... That it's false to say
> > it's, the "true", claim.
> > Only scientific and not falsified.
> >
> > Platonism then sort of demands
> > "there are true numbers, so work it up".
> >
> > "There is no 'but', only 'yet', ...."
> We know that
> each theory is true of
> whatever that theory is true of.
>
> Some theories are true of nothing.
> Contradictions can be proved in those.
>
> Some theories are true of something,
> of what we intend or of something unintended.
> Contradictions cannot be proved in those.
>
> Some theories, numbers, for example,
> if they can prove they're consistent,
> would prove that they _aren't_ consistent.
>
> There is no "yet" to not.proving consistency.
> We know there is no such proof --
> or, if (sadly) there is, it's meaningless,
> about nothing.
>
> However,
> there are better options than
> a theory proving itself consistent.
>
> Gödel's work does not deny that
> some other theory can prove
> our theory consistent.
>
> For example,
> ZFC is a theorem of
> ZFC+inaccessible.cardinal.exists
>
> Some theories are so simple that,
> even though we know there is no proof,
> we don't take seriously the possibility
> that they have no model.
>
> For example,
> the set.theory fragment,
> -- {} exists
> -- x∪{y} exists
> -- same.element.sets x and y are equal
> proves arithmetic and Gödel.incompleteness

It's not so much "Goedel's Incompleteness Theorems knows unknowables",
as, "Goedel's Incompleteness Theorems show there are more truths than
what are already theorems". There's, "the extra-ordinary".

When you say model, that's a structure, it's structuralism, and for constructivists.

When you say "some theories are true of nothing, contradictions can be proved in them",
that's really a great statement for Ex Falso Nihilum contra Ex Falso Quodlibet,
and it's true and it's good.

So, consider the theories of "cardinals in ordinals", and, "ordinals in cardinals", as
two theories about one theory, with different primary objects, elementarily..
This is already framed as "ordering theory" vis-a-vis "set theory". (Then,
the goal is that it's one theory, or that the one theory has structure the
objects either way.)

So, Goedel's Incompleteness Theorems, have their sorts of beginnings and ends.

I.e., a plain sequence of Geodel functions still carries on out to infinity..

As a complement to Goedel's, don't forget to include Cohen's forcing,
he tops it off with an ordinal its own order type. ("Extra-ordinary.")

It's not so much so that there are large cardinals, which aren't cardinals nor sets,
in set theory. It's usually then about class/set distinction, but, it results that
it's better to have the extra-ordinary theory up front then find set theory in that.

Then, this is related to my slates of uncountability and logical paradox,
about DesCartes to Cohen, a sort of paleo-classical super-modern Platonism.

"In my 10,000's posts to sci.math, sci.logic, sci.physics, sci.physics.relativity, ...."

Good luck dear sir and thanks for your reply, what you can find is that "A Theory"
and the "Null Axiom Theory" really represent A Theory.

....

Le 05/01/2024 à 16:04, Richard Damon a écrit :
> On 1/5/24 5:22 AM, WM wrote:
>> immibis schrieb am Donnerstag, 4. Januar 2024 um 22:53:35 UTC+1:
>>
>>> what point are you trying to make? infinity is strange
>>
>> But it is based on logic. This logic is violated in the seven points I
>> made.
>
> Which are base on logic that doesn't handle infinity.

This logic is indispensable.
>
> Yes, it is well know that infinite sets break some of the seemingly
> obvious properties that hold for finite sets.

But it has not yet been recognized that ZF breaks indispensable laws of
logic.
>
> YOU just don't seem to understand and accept that fact, and keep on
> making the ERROR of asuming it must.

That is not an error. It shows that ZF could never acquire any relevance
for reality where the basic laws of logic are referenced. It shows that ZF
is only a religion to be believed by its proponents and poor captured
students.

Regards, WM

Le 05/01/2024 à 19:32, Jim Burns a écrit :
> On 1/4/2024 4:59 AM, WM wrote:
>
>> 7. The diagonal does not define a number
>>
>> An endless digit sequence without
>> finite definition of the digits
>> cannot define a real number.
>> After every known digit
>> almost all digits will follow.

> It is sufficient that
> an endless digit sequence without
> finite definition of the digits
> exist.

No. Even and endless digit sequence does not describe an irrational
number. The irrational number is the limit only. Compare 0.999... which
does not contain 1 but only has the limit 1.
>
> There is at most one real number
> which is permitted by each
> finite initial sub.sequence of digits.

There are infinitely many. Therefore it is impossible to decide which real
number is described.
>
> Therefore,
> there is at most one real number
> which is permitted by each
> finite initial sub.sequence of digits.

By each defined digit sequence infinitely many numbers are permitted.
And the dark digits cannot describe any real number.

Regards, WM

On 1/8/24 5:40 AM, WM wrote:
> Le 05/01/2024 à 16:04, Richard Damon a écrit :
>> On 1/5/24 5:22 AM, WM wrote:
>>> immibis schrieb am Donnerstag, 4. Januar 2024 um 22:53:35 UTC+1:
>>>
>>>> what point are you trying to make? infinity is strange
>>>
>>> But it is based on logic. This logic is violated in the seven points
>>> I made.
>>
>> Which are base on logic that doesn't handle infinity.
>
> This logic is indispensable.

Maybe for your small mind.

>>
>> Yes, it is well know that infinite sets break some of the seemingly
>> obvious properties that hold for finite sets.
>
> But it has not yet been recognized that ZF breaks indispensable laws of
> logic.

No, it shows that some supposed laws of logic can't handle unbounded sets.

>>
>> YOU just don't seem to understand and accept that fact, and keep on
>> making the ERROR of asuming it must.
>
> That is not an error. It shows that ZF could never acquire any relevance
> for reality where the basic laws of logic are referenced. It shows that
> ZF is only a religion to be believed by its proponents and poor captured
> students.
>
> Regards, WM
>
>

In other words, you don't understand what it says so you ignore it.

On 1/8/2024 7:18 AM, WM wrote:
> Le 05/01/2024 à 19:32, Jim Burns a écrit :

>>> An endless digit sequence without
>>> finite definition of the digits
>>> cannot define a real number.
>>> After every known digit
>>> almost all digits will follow.

>> There is at most one real number
>> which is permitted by each
>> finite initial sub.sequence of digits.
>
> There are infinitely many.

Assume there are two permitted points.
Each power 10⁻ⁿ is larger than their distance,
a positive distance d > 0 apart.

β ≥ d > 0 is the least upper bound of
distances which each 10⁻ⁿ is larger than.

10β > β is not
a distance which each 10⁻ⁿ is larger than.
Thus b exists: 10⁻ᵇ < 10β

β/10 < β is
a distance which each 10⁻ⁿ is larger than.
Thus, in particular,
β/10 < 10⁻ᵇ⁻²

However,
(10⁻ᵇ)/100 < (10β)/100
10⁻ᵇ⁻² < β/10
Contradiction.

Therefore,
there _aren't_ two points which
are permitted by each
finite initial sub.sequence of
the endless digit sequence.

> Therefore it is impossible to decide which
> real number is described.

Whether or not we describe these points,
these points _exist_
no more than one point to each
endless digit sequence.

On Monday, January 8, 2024 at 1:19:18 PM UTC-8, Fritz Feldhase wrote:
> On Monday, January 8, 2024 at 1:19:02 PM UTC+1, WM wrote:
> > Le 05/01/2024 à 19:32, Jim Burns a écrit :
> > >
> > > It is sufficient that an endless digit sequence [...] exists.
> > >
> > No. Even and endless digit sequence does not describe an irrational number.
> It does, you silly idiot.
>
> See: https://www.dpmms.cam.ac.uk/~wtg10/decimals.html

Where's Simon Stevin when you need him?

Le 08/01/2024 à 19:46, Jim Burns a écrit :
> On 1/8/2024 7:18 AM, WM wrote:

>
>> Therefore it is impossible to decide which
>> real number is described.
>
> Whether or not we describe these points,
> these points _exist_
> no more than one point to each
> endless digit sequence.

Dark numbers, dark points and dark parts of infinite sequences exist. But
we cannot distinguish and use them. And we can prove that there are not
more irrational numbers than rational numbers. Just today I showed some
proofs to my students. One is this: Between ***every*** pair of irrational
numbers there is a rational number. Another one, which was very well
received, is the game Conquer the Binay Tree:

You start with one cent, buy a path in the Binary Tree and get one cent
for every covered node. Then you buy another path and get one cent for
every node not yet covered by the first path. You will never earn less
than one cent, because every path is distinct by at least one node from
every other path. Therefore you will not get bankrupt. But if there were
more paths than nodes, you would get bankrupt. Hence Cantor is defeated.

Regards, WM

On Tuesday, January 9, 2024 at 9:41:06 AM UTC-8, WM wrote:
> Le 08/01/2024 à 19:46, Jim Burns a écrit :
> > On 1/8/2024 7:18 AM, WM wrote:
>
> >
> >> Therefore it is impossible to decide which
> >> real number is described.
> >
> > Whether or not we describe these points,
> > these points _exist_
> > no more than one point to each
> > endless digit sequence.
> Dark numbers, dark points and dark parts of infinite sequences exist. But
> we cannot distinguish and use them. And we can prove that there are not
> more irrational numbers than rational numbers. Just today I showed some
> proofs to my students. One is this: Between ***every*** pair of irrational
> numbers there is a rational number. Another one, which was very well
> received, is the game Conquer the Binay Tree:
>
> You start with one cent, buy a path in the Binary Tree and get one cent
> for every covered node. Then you buy another path and get one cent for
> every node not yet covered by the first path. You will never earn less
> than one cent, because every path is distinct by at least one node from
> every other path. Therefore you will not get bankrupt. But if there were
> more paths than nodes, you would get bankrupt. Hence Cantor is defeated.
>
> Regards, WM

That's better than usual, and terms that aren't so mangled,
where sometimes "mangle" means flatten like a steam press
and other times means "spindle" as the over and under in definition,
so there is that rationals are HUGE like Friedman's,
and, there is a constructible tree like Hausdorff,
then Cantor is not "defeated", rather, refined in apologetics.

There is Cantor space, and, there is square Cantor space.

It's all the 0's and 1's, ....

.....

On 1/9/2024 12:40 PM, WM wrote:
> Le 08/01/2024 à 19:46, Jim Burns a écrit :
>> On 1/8/2024 7:18 AM, WM wrote:

>>>>> An endless digit sequence without
>>>>> finite definition of the digits
>>>>> cannot define a real number.
>>>>> After every known digit
>>>>> almost all digits will follow.
>>
>>>> There is at most one real number
>>>> which is permitted by each
>>>> finite initial sub.sequence of digits.
>>>
>>> There are infinitely many.

>>> Therefore it is impossible to decide
>>> which real number is described.
>>
>> Whether or not we describe these points,
>> these points _exist_
>> no more than one point to each
>> endless digit sequence.
>
> Dark numbers, dark points
> and dark parts of infinite sequences
> exist.

Elsewhere, you have told us that
darkᵂᴹ numbers and their ilk
are never one step away from
visibleᵂᴹ numbers and their ilk.

I'll clarify.
There is at most one point
which is permitted by each
finite initial sub.sequence of
visibleᵂᴹ digits.

Because,
if there are two permitted points,
there is a positive least upper bound β of
distances < each visibleᵂᴹ 10⁻ⁿ
β/10 < β < 10β
and visibleᵂᴹ 10⁻ᵇ < 10β
and visibleᵂᴹ 10⁻ᵇ⁻² > β/10
but also
10⁻ᵇ/100 < 10β/100
10⁻ᵇ⁻² < β/10
Contradiction.
Thus,
not two.

> Just today I showed some proofs to my students.

If you are claiming your students for
some form of peer.review, then
you are accepting pre-calculus students as
_peers_

I won't object if you choose to do that.
I just want to make sure you understand that
they are who you are accepting as peers.

On Tuesday, January 9, 2024 at 11:34:19 AM UTC-8, Jim Burns wrote:
> On 1/9/2024 12:40 PM, WM wrote:
> > Le 08/01/2024 à 19:46, Jim Burns a écrit :
> >> On 1/8/2024 7:18 AM, WM wrote:
> >>>>> An endless digit sequence without
> >>>>> finite definition of the digits
> >>>>> cannot define a real number.
> >>>>> After every known digit
> >>>>> almost all digits will follow.
> >>
> >>>> There is at most one real number
> >>>> which is permitted by each
> >>>> finite initial sub.sequence of digits.
> >>>
> >>> There are infinitely many.
> >>> Therefore it is impossible to decide
> >>> which real number is described.
> >>
> >> Whether or not we describe these points,
> >> these points _exist_
> >> no more than one point to each
> >> endless digit sequence.
> >
> > Dark numbers, dark points
> > and dark parts of infinite sequences
> > exist.
> Elsewhere, you have told us that
> darkᵂᴹ numbers and their ilk
> are never one step away from
> visibleᵂᴹ numbers and their ilk.
>
> I'll clarify.
> There is at most one point
> which is permitted by each
> finite initial sub.sequence of
> visibleᵂᴹ digits.
>
> Because,
> if there are two permitted points,
> there is a positive least upper bound β of
> distances < each visibleᵂᴹ 10⁻ⁿ
> β/10 < β < 10β
> and visibleᵂᴹ 10⁻ᵇ < 10β
> and visibleᵂᴹ 10⁻ᵇ⁻² > β/10
> but also
> 10⁻ᵇ/100 < 10β/100
> 10⁻ᵇ⁻² < β/10
> Contradiction.
> Thus,
> not two.
> > Just today I showed some proofs to my students.
> If you are claiming your students for
> some form of peer.review, then
> you are accepting pre-calculus students as
> _peers_
>
> I won't object if you choose to do that.
> I just want to make sure you understand that
> they are who you are accepting as peers.

Everybody knows the entire point of pre-calculus
is to make it clear that the usual notion of an Aristotelean
continuum, which is about the most usual sort of intuition
about the continuum after constant motion, and the
course-of-passage as through points in a line, is gently
shushing that down for students who do have such
an intuitive notion of analytical character, and then
explaining for all that the usual laws of arithmetic and
delta-epsilonics, together, make for defining infinite limit.

That is all.

--
Today we call it line-reals and don't use quasi-modal logic
And it's standard, too

On 1/9/2024 6:29 PM, Ross Finlayson wrote:
> On Tuesday, January 9, 2024
> at 11:34:19 AM UTC-8, Jim Burns wrote:
>> On 1/9/2024 12:40 PM, WM wrote:

>>> Just today I showed some proofs to my students.
>>
>> If you are claiming your students for
>> some form of peer.review, then
>> you are accepting pre-calculus students as
>> _peers_

> Everybody knows
> the entire point of pre-calculus is
> to make it clear that the usual notion of
> an Aristotelean continuum, [...]

I could have said that better.

In that post,
the only thing I meant by "pre-calculus"
was "having not yet been a calculus student".
I wasn't thinking of "being a student in
the course just before calculus".

Calculus and WM's dark.number game
make a poor fit together.
I expect WM to agree with that much,
even if he and I assign blame for
the poor fit differently.

I haven't actually heard from anyone that
WM's students are not.yet.calculus.
It only seems to me as though they must be.

Sorry for any confusion.

One sentence:

> Everybody knows
> the entire point of pre-calculus is
> to make it clear that
> the usual notion of
> an Aristotelean continuum,
> which is about
> the most usual sort of intuition about
> the continuum after constant motion,
> and
> the course-of-passage as through
> points in a line,
> is
> gently shushing that down for
> students who do have
> such an intuitive notion of
> analytical character,
> and then
> explaining for all
> that the usual laws of
> arithmetic and delta-epsilonics,
> together,
> make for defining infinite limit.

Gently shushing down notions which are
not the preferred notions
is the purpose for which we hold classes.

If it can be done,
I think that
giving the reasons that
some other notion is preferred
can be an excellent strategy for
effective gentle.shushing.down.

However,
in some instances,
we are, today, looking at preferences
resulting from resolution of a long controversy
between brilliant people
encyclopedically educated in their field.

Should we ask students,
upon their first contact with that field,
to reproduce
the best of that field?

If that is a reasonable ask,
that would be wonderful to see.

However,
it might well not.be a reasonable ask.

It might well be necessary for
students to _trust_ that
reasons exist for preferring
one set of notions over
another,
to trust, at least,
until they acquire their own
brilliance and
encyclopedic knowledge.

My own preference is
to share the beauty in
these deep notions,
with all and sundry,
with those with only
a momentary interest.

However,
I have to recognize that
I can't always get what I want.

Le 09/01/2024 à 20:34, Jim Burns a écrit :
> On 1/9/2024 12:40 PM, WM wrote:

>> Just today I showed some proofs to my students.
>
> If you are claiming your students for
> some form of peer.review, then
> you are accepting pre-calculus students as
> _peers_

Nonsense. There are 8th semester informatics and engineering studens. They
know mathematics very well (that which Cantor is not needed for, according
to Feferman) but they do not use matheology. Try to show how the player
gets bankrupt in the game "We conquer the Binary Tree".

Regards, WM

Le 10/01/2024 à 06:49, Jim Burns a écrit :

> Calculus and WM's dark.number game
> make a poor fit together.

Of course. Calculus uses potential infinity only. Dark numbers can exist
only in actual infinity, the complement of potential infinity.
Feferman's book In the light of logic answers the question: Is Cantor
necessary for the maths of the real world? with a resounding no.

Regards, WM

On 1/10/24 2:14 PM, WM wrote:
> Le 10/01/2024 à 06:49, Jim Burns a écrit :
>
>> Calculus and WM's dark.number game
>> make a poor fit together.
>
> Of course. Calculus uses potential infinity only. Dark numbers can exist
> only in actual infinity, the complement of potential infinity.
> Feferman's book In the light of logic answers the question: Is Cantor
> necessary for the maths of the real world? with a resounding no.
>
> Regards, WM

But you claim them to be part of the sets that are only "Potential
Infinity", or are you retracting that claim. (All Natural Numbers are
only by your definition "Potentially Infinite")

All members of the Natural Numbers are Finite, so none of them are
"Actually Infinite".

The set of them has a SIZE that is infinite, but none of the members of
it ever are.

On Wednesday, January 10, 2024 at 4:49:39 PM UTC-8, Richard Damon wrote:
> On 1/10/24 2:14 PM, WM wrote:
> > Le 10/01/2024 à 06:49, Jim Burns a écrit :
> >
> >> Calculus and WM's dark.number game
> >> make a poor fit together.
> >
> > Of course. Calculus uses potential infinity only. Dark numbers can exist
> > only in actual infinity, the complement of potential infinity.
> > Feferman's book In the light of logic answers the question: Is Cantor
> > necessary for the maths of the real world? with a resounding no.
> >
> > Regards, WM
> But you claim them to be part of the sets that are only "Potential
> Infinity", or are you retracting that claim. (All Natural Numbers are
> only by your definition "Potentially Infinite")
>
> All members of the Natural Numbers are Finite, so none of them are
> "Actually Infinite".
>
> The set of them has a SIZE that is infinite, but none of the members of
> it ever are.

The order type of ordinals is an ordinal.

Finite ordinals as the classes of each those not containing themselves,
according to simple quantification or "the set of all sets that don't
contain themselves", contains itself.

Such are simple reasons why the infinite integers have an infinite integer.

Yes I know that ZF two axioms of restriction of comprehension or "don't look"
include one that is "there's an infinity that isn't thusly various".

So, you might find it interesting that in more naive theories,
it's those one-line proofs as above that do give that there are
true infinites courtesy the unbounded its quantification.

Then about MW's "I invoke the Absolute!", it's he's drunk
then also that potential and actual infinities do have differences
in real mathematics, one recalls "A Sober Mind Speaks".

So, RD's "there is only one theory and it ignores all my problems"
and MWs "there are at least two theories and they don't agree",
seems for a sort of "yet somehow, there's a theory".

.... With laws, plural, of large numbers.

If you want a gentle introduction to set theory,
try reading Quine's Set Theory then after it introduces
the class-set distinction and x not in x and x not equals x,
and why neither of those is first-order in ZF, quit.

.... in a theory, ....

On 1/10/24 8:10 PM, Ross Finlayson wrote:
> On Wednesday, January 10, 2024 at 4:49:39 PM UTC-8, Richard Damon wrote:
>> On 1/10/24 2:14 PM, WM wrote:
>>> Le 10/01/2024 à 06:49, Jim Burns a écrit :
>>>
>>>> Calculus and WM's dark.number game
>>>> make a poor fit together.
>>>
>>> Of course. Calculus uses potential infinity only. Dark numbers can exist
>>> only in actual infinity, the complement of potential infinity.
>>> Feferman's book In the light of logic answers the question: Is Cantor
>>> necessary for the maths of the real world? with a resounding no.
>>>
>>> Regards, WM
>> But you claim them to be part of the sets that are only "Potential
>> Infinity", or are you retracting that claim. (All Natural Numbers are
>> only by your definition "Potentially Infinite")
>>
>> All members of the Natural Numbers are Finite, so none of them are
>> "Actually Infinite".
>>
>> The set of them has a SIZE that is infinite, but none of the members of
>> it ever are.
>
> The order type of ordinals is an ordinal.
>
> Finite ordinals as the classes of each those not containing themselves,
> according to simple quantification or "the set of all sets that don't
> contain themselves", contains itself.
>
> Such are simple reasons why the infinite integers have an infinite integer.
>
> Yes I know that ZF two axioms of restriction of comprehension or "don't look"
> include one that is "there's an infinity that isn't thusly various".
>
> So, you might find it interesting that in more naive theories,
> it's those one-line proofs as above that do give that there are
> true infinites courtesy the unbounded its quantification.
>
>
> Then about MW's "I invoke the Absolute!", it's he's drunk
> then also that potential and actual infinities do have differences
> in real mathematics, one recalls "A Sober Mind Speaks".
>
> So, RD's "there is only one theory and it ignores all my problems"
> and MWs "there are at least two theories and they don't agree",
> seems for a sort of "yet somehow, there's a theory".
>
> ... With laws, plural, of large numbers.
>
> If you want a gentle introduction to set theory,
> try reading Quine's Set Theory then after it introduces
> the class-set distinction and x not in x and x not equals x,
> and why neither of those is first-order in ZF, quit.
>
>
> ... in a theory, ....
>

I understand the multitude of different ways to define "Numbers", but
their are a core set of fundamental numbers with basically agreed on
definitions, and reasonable people don't try calling something that
doesn't match the accepted definition as being one of those sets.

"Natural Numbers" is such a set, and while there may be several ways of
expressing its "generation", they all are essentially, you start with a
base number, called Zero, and then you have an operation, called
something like Successor, and the set of the Natural Numbers is the set
of things you get by applying that operation an unlimited number of times.

Thus, I admit to other number theories, they just are not the "standard"
systems of Natural Numbers, Integers, Rational Numbers, or Real Numbers.
(The Reals have a couple of different generational methods, but they
result in the same set of numbers with the same basic properties).

If you want to talk about other systems, feel free, just don't be
"deceptive" (or lying) and try to use the name for one of the standard sets.

And most of the "alternative" systems alternative have accepted names
for them, so that should be used.

On Wednesday, January 10, 2024 at 5:32:25 PM UTC-8, Richard Damon wrote:
> On 1/10/24 8:10 PM, Ross Finlayson wrote:
> > On Wednesday, January 10, 2024 at 4:49:39 PM UTC-8, Richard Damon wrote:
> >> On 1/10/24 2:14 PM, WM wrote:
> >>> Le 10/01/2024 à 06:49, Jim Burns a écrit :
> >>>
> >>>> Calculus and WM's dark.number game
> >>>> make a poor fit together.
> >>>
> >>> Of course. Calculus uses potential infinity only. Dark numbers can exist
> >>> only in actual infinity, the complement of potential infinity.
> >>> Feferman's book In the light of logic answers the question: Is Cantor
> >>> necessary for the maths of the real world? with a resounding no.
> >>>
> >>> Regards, WM
> >> But you claim them to be part of the sets that are only "Potential
> >> Infinity", or are you retracting that claim. (All Natural Numbers are
> >> only by your definition "Potentially Infinite")
> >>
> >> All members of the Natural Numbers are Finite, so none of them are
> >> "Actually Infinite".
> >>
> >> The set of them has a SIZE that is infinite, but none of the members of
> >> it ever are.
> >
> > The order type of ordinals is an ordinal.
> >
> > Finite ordinals as the classes of each those not containing themselves,
> > according to simple quantification or "the set of all sets that don't
> > contain themselves", contains itself.
> >
> > Such are simple reasons why the infinite integers have an infinite integer.
> >
> > Yes I know that ZF two axioms of restriction of comprehension or "don't look"
> > include one that is "there's an infinity that isn't thusly various".
> >
> > So, you might find it interesting that in more naive theories,
> > it's those one-line proofs as above that do give that there are
> > true infinites courtesy the unbounded its quantification.
> >
> >
> > Then about MW's "I invoke the Absolute!", it's he's drunk
> > then also that potential and actual infinities do have differences
> > in real mathematics, one recalls "A Sober Mind Speaks".
> >
> > So, RD's "there is only one theory and it ignores all my problems"
> > and MWs "there are at least two theories and they don't agree",
> > seems for a sort of "yet somehow, there's a theory".
> >
> > ... With laws, plural, of large numbers.
> >
> > If you want a gentle introduction to set theory,
> > try reading Quine's Set Theory then after it introduces
> > the class-set distinction and x not in x and x not equals x,
> > and why neither of those is first-order in ZF, quit.
> >
> >
> > ... in a theory, ....
> >
> I understand the multitude of different ways to define "Numbers", but
> their are a core set of fundamental numbers with basically agreed on
> definitions, and reasonable people don't try calling something that
> doesn't match the accepted definition as being one of those sets.
>
> "Natural Numbers" is such a set, and while there may be several ways of
> expressing its "generation", they all are essentially, you start with a
> base number, called Zero, and then you have an operation, called
> something like Successor, and the set of the Natural Numbers is the set
> of things you get by applying that operation an unlimited number of times..
>
> Thus, I admit to other number theories, they just are not the "standard"
> systems of Natural Numbers, Integers, Rational Numbers, or Real Numbers.
> (The Reals have a couple of different generational methods, but they
> result in the same set of numbers with the same basic properties).
>
> If you want to talk about other systems, feel free, just don't be
> "deceptive" (or lying) and try to use the name for one of the standard sets.
>
> And most of the "alternative" systems alternative have accepted names
> for them, so that should be used.

Well, yes that's one of the maxims of language, that it's not abused.

That's one of the greatest claims against MW by old Virgil, again.
He doesn't know his words, which for example our professorial
James B. distinguishes notationally.

Then, in logic, and not just "an introduction to" but "the theory of"
and furthermore "foundations of", then there's addressed ambiguity,
inherent in definition, like quantifier disambiguation for Heap and Sorites,
like impredicativity for Heap and Sorites, like the sublime for Heap and Sorites,
the extra-ordinary for Heap and Sorites, and so on.

At the same time, or yet, ignorance either way is an abuse thereof, of language.
It's an insult to it.

Things like conflation for Heap and Sorites, invalid metaphor for Heap and Sorites,
false supposition for Heap and Sorites, these aren't proper conflation, metaphor,
and supposition, which are framed modally in contingents, simile, and fusion,
theories thereof, and therein.

So, I agree with what you wrote, but also sort of demand you acknowledge,
that the extra-ordinary, exists from lesser of its independent fundamental
principles, so rather precedes the otherwise usual formalism of "a regularity".

The regularity or well-foundedness understood as being made first order
in ZF set theory, and as noted above it puts membership the relation instead
of element-of the relation, out of first-order and so strictly intensional,
along though with equality or the fundamentally intensional.

Thus, class/set distinction, is a good example of when definitions go bad
from because axioms went, "wrong". The "principle axiom distinctions",
sort of mold theory following throughout.

That said, a dialectic in the Hegelian sense, thesis antithesis synthesis,
shows for "descriptive dynamics", that such as these vagaries or the vague,
are tussled out according to their definition and refinement and respect of language.

What's a thesis, ....