XPost: sci.math

On 7/29/2024 9:31 PM, Ross Finlayson wrote:

On 07/29/2024 02:12 PM, Jim Burns wrote:

On 7/29/2024 3:44 PM, Ross Finlayson wrote:

On 07/29/2024 05:32 AM, Jim Burns wrote:

On 7/28/2024 7:42 PM, Ross Finlayson wrote:

about ubiquitous ordinals

What are ubiquitous ordinal?

Well, you know that ORD, is, the order type of ordinals,
and so it's an ordinal, of all the ordinals.

Is a ubiquitous ordinal a finite ordinal?
I would appreciate a "yes" or a "no" in your response.

The ubiquitous ordinals are, for example,
a theory where the primary elements are ordinals,
for ordering theory, and numbering theory,
which may be more fundamental, than set theory,
with regards to a theory of one relation.

Apparently,
what you mean by ubiquitous ordinals are ordinals,
without further qualification.

Ordinals can be represented as sets, and are,
most often by the von Neumann scheme, λ = [0,λ)ᴼʳᵈ

Apparently,
ubiquitous ordinals are
what are represented by the [0,λ)ᴼʳᵈ

Ordinals are well.ordered.
The [0,λ)ᴼʳᵈ are well.ordered.
That is entirely non.accidental.
There isn't much reason to choose between
the von Neumann ordinal.representations and
the raw, unfiltered "ubiquitous" ordinals.

The one advantage which
representations have over the ubiquitous(?) is that
they are are objects in a theory of sets which
we have great confidence isn't contradictory,
and that extends our great confidence to
the non.contradictoriness of theorems about ordinals.

Unless we are considering their existence,
which is to say, their non.contradictoriness,
ordinals.of.unspecified.origin are well.ordered,
and that's an end to their description.

It's like the universe of set theory,

Do you and I mean the same by "universe of set theory"?

I am most familiar with theories of
well.founded sets without urelements.

In the von Neumann hierarchy of hereditary well.founded sets
V[0] = {}
V[β+1] = 𝒫(V[β])
V[γ] = ⋃[β<γ] V[β]

V[ω] is the universe of hereditarily finite sets.

For the first inaccessible ordinal κ
V[κ] is a model of ZF+Choice.

For first inaccessible ordinal κ
[0,κ) holds an uncountable ordinal and
is closed under cardinal arithmetic.

There is no universe in ZFC, don't be saying otherwise.

I will continue not.saying that
there is a universe _IN_ ZFC.

There are universes _OF_ ZFC which are also called domains.
V[κ] is one of the domains of ZFC.

None of the sets described by the theory is
the universal set, holding all sets IN the domain.


Not precisely to your point, but interesting:
Some of the sets IN ZFC satisfy all its axioms,
which make them also domains OF ZFC

Compare that to the way in which
each end.segment of ℕ is also a model of ℕ

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XPost: sci.math

On 7/30/2024 4:56 PM, Ross Finlayson wrote:

On 07/30/2024 11:18 AM, Jim Burns wrote:

[...]

The idea that there's one theory for all this theory,
has that otherwise there isn't
and you're not talking about any of them.

If I remember correctly, your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

However,
it is because we are hypocriticalᴿꟳ (in your sense?)
that such discussions produce results.
"Conclusions", if you like.

We make finite.length.statements which
we know are true in infinitely.many senses.

We can know they are so because
we have narrowed our attention to
those for which they are true without exception.
Stated once, finitely, for infinitely.many.

Non.hypocrisyᴿꟳ (sincerityᴿꟳ?) throws that away.

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XPost: sci.math

On 7/31/2024 8:30 PM, Ross Finlayson wrote:

On 07/31/2024 01:21 PM, Jim Burns wrote:

If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

However,
it is because we are hypocriticalᴿꟳ (in your sense?)
that such discussions produce results.
"Conclusions", if you like.

We make finite.length.statements which
we know are true in infinitely.many senses.

We can know they are so because
we have narrowed our attention to
those for which they are true without exception.
Stated once, finitely, for infinitely.many.

Non.hypocrisyᴿꟳ (sincerityᴿꟳ?) throws that away.

You're talking about a field,
I'm talking about foundations.

I doubt that
you and I are calling the same thing a field:
a set with addition, multiplication, identities, inverses
such that
a+(b+c)=(a+b)+c a+b=b+a a+0=a a+(-a)=0
a⋅(b⋅c)=(a⋅b)⋅c a⋅b=b⋅a a⋅1=a a≠0 ⇒ a⋅⅟a=1 a⋅(b+c)=(a⋅b)+(a⋅c)
?

fieldᴿꟳ == domainⁿᵒᵗᐧᴿꟳ ?

The counterpart of a variable is its domainⁿᵒᵗᐧᴿꟳ
== those to which the variable possibly refers.

From what I can see,
both fieldsᴿꟳ and foundationsᴿꟳ are domainsⁿᵒᵗᐧᴿꟳ

I'm guessing that the distinction between
fieldsᴿꟳ and foundationsᴿꟳ is the distinction between
retail mathematics and wholesale mathematics,
issues of the day and grand unification.

In given circumstances, there may well be
excellent reasons to do retail mathematics or
to do wholesale mathematics.
I am skeptical about there ever being
logical reasons to choose one over the other.

You're talking about a field,
I'm talking about foundations.
... Of which there is one and a universe of it.

If a theory has any model of infinite cardinality,
it has models of each infinite cardinality.

That's a general result.
The empty theory (with no extralogical axioms)
has models of each infinite cardinality.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

About triangles and right triangles,
and classes and sets in an ordinary theory
like ZFC with classes, now your theory has
classes that aren't sets.

Somewhere, in axioms or definitions,
there are statements we know are true
as long as they are referring to a right triangle.

I fully expect that
things other than right triangles exist.
Those other things' existence doesn't change
the truth of those statements
as long as they are referring to a right triangle.

That isn't a particularly difficult insight.
We know they're true because
we know what a right triangle is. Duh.

I think that I find myself repeating
that not.particularly.difficult insight
because it _sounds like_
teeny, tiny finite beings <waves at camera> are
somehow engaging in some sort of infinite activity.

We are not engaged in any sort of
infinite activity.
Making finitely.many finite.length statements
is not an infinite activity.
Yes, they are true _about_ infinitely.many, but
we do not "true" the statements infinitely.often
as though we're laying infinitely.many bricks.

Yeah, my mathematical conscience demands that
hypocrisy is bad.

Bad why?

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XPost: sci.math

On 8/1/2024 3:28 PM, Ross Finlayson wrote:

On 08/01/2024 04:23 AM, Jim Burns wrote:

On 7/31/2024 8:30 PM, Ross Finlayson wrote:

On 07/31/2024 01:21 PM, Jim Burns wrote:

If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

Yeah, my mathematical conscience demands that
hypocrisy is bad.

Bad why?

"Wrong", ....

It is wrong to treat claims about right triangles
as though they are claims about more than right triangles.

Definition usually expands,

The hypocrisyᴿꟳ of NOT expanding
the definition of right triangle ABC
to encompass triangles without right angles
leaves it NOT wrong that
a segment CH from right angle C
perpendicular to and meeting side AB at H
makes two more triangles ACH BCH,
which are both similar to ABC
which, as similar triangles,
have corresponding sides in the same ratio
so that
A͞H/A͞C = A͞C/A͞B
H͞B/B͞C = B͞C/A͞B
(A͞H+H͞B)⋅A͞B = A͞C² +B͞C²
and
A͞B² = A͞C² + B͞C² is NOT wrong.

hypocrisy is bad.

If it is, then it isn't for making things wrong,
which is something hypocrisyᴿꟳ
(not.talking about outside the domain)
doesn't do.

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XPost: sci.math

On 8/1/2024 8:52 PM, Ross Finlayson wrote:

On 08/01/2024 05:36 PM, Jim Burns wrote:

On 8/1/2024 3:28 PM, Ross Finlayson wrote:

On 08/01/2024 04:23 AM, Jim Burns wrote:

On 7/31/2024 8:30 PM, Ross Finlayson wrote:

On 07/31/2024 01:21 PM, Jim Burns wrote:

If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,
that we _should not_ for example, not.talk about
_all_ triangles when we discuss whether
the square of its longest side equals
the sum of the squares of the two remaining sides.

Yeah, my mathematical conscience demands that
hypocrisy is bad.

Bad why?

"Wrong", ....

It is wrong to treat claims about right triangles
as though they are claims about more than right triangles.

Definition usually expands,

The hypocrisyᴿꟳ of NOT expanding
  the definition of right triangle ABC
  to encompass triangles without right angles
leaves it NOT wrong that
a segment CH from right angle C
  perpendicular to and meeting side AB at H
makes two more triangles ACH BCH,
  which are both similar to ABC
  which, as similar triangles,
  have corresponding sides in the same ratio
so that
A͞H/A͞C = A͞C/A͞B
H͞B/B͞C = B͞C/A͞B
(A͞H+H͞B)⋅A͞B = A͞C² +B͞C²
and
  A͞B² = A͞C² + B͞C²  is NOT wrong.

hypocrisy is bad.

If it is, then it isn't for making things wrong,
  which is something hypocrisyᴿꟳ
  (not.talking about outside the domain)
doesn't do.

There is no "outside" the universe.

Anything else, there is.

There very often are things outside
the things which we are discussing.

It is perhaps surprising how important it is that,
when we are discussing certain things,
we are not discussing other things, outside.

It is, it seems to me, a key part of a technique for
exploring the infinite by finite means.

That key part seems to me to be
what you are calling hypocrisyᴿꟳ

It strikes me as _at least_ unwise to discard
such a powerful and reliable tool.

It sort of seems the straw-man of you to say
I'm disputing Pythagoras
when all I did was point out that
Russell was more-or-less lying to you.

I picked Pythagoras as a concrete example of
my best guess at what you (RF) mean by hypocrisyᴿꟳ

If my best guess is wide of the mark,
perhaps you should tell me what you DO mean.

That sounds like you're delivering a value.judgment:
that we _should not_ not.talk about
what's outside the domain of discussion,

Yeah, my mathematical conscience demands that
hypocrisy is bad.

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XPost: sci.math

On 8/1/2024 8:52 PM, Ross Finlayson wrote:

On 08/01/2024 05:36 PM, Jim Burns wrote:

On 8/1/2024 3:28 PM, Ross Finlayson wrote:

On 08/01/2024 04:23 AM, Jim Burns wrote:

On 7/31/2024 8:30 PM, Ross Finlayson wrote:

On 07/31/2024 01:21 PM, Jim Burns wrote:

If I remember correctly,  your (RF's) name for
not.talking about
what's outside the domain of discussion
is hypocrisyᴿꟳ.

Yeah, my mathematical conscience demands that
hypocrisy is bad.

Bad why?

"Wrong", ....
Definition usually expands,

It sort of seems the straw-man of you
to say I'm disputing Pythagoras
when all I did was point out that
Russell was more-or-less lying to you.

This is the heart of the matter:

Expanding the definition of right triangle
does not dispute Pythagoras.

Expanding the definition of right triangle
moves us to a second conversation.
Pythagoras is in the first conversation.
The second has no effect there.

First conversation.
⎛ Consider △ABC and AC⟂BC
⎜
⎜ H in AB: CH⟂AH and CH⟂BH
⎜
⎜ △ACH ∝ △ABC
⎜ A͞H/A͞C = A͞C/A͞B
⎜
⎜ △CBH ∝ △ABC
⎜ H͞B/B͞C = B͞C/A͞B
⎜
⎜ (A͞H+H͞B)⋅A͞B = A͞C² + B͞C²
⎝ A͞B² = A͞C² + B͞C²

Second conversation.
⎛ Consider △ABC ...
⎜ Comrades!
⎜ Throw off the shackles of the past!
⎜ AC⟂BC or AC̸⟂BC
⎜ You have been lied to.
⎜ For A͞B = A͞C = B͞C
⎜ A͞B² ≠ A͞C² + B͞C²
⎝ Freedom!

I am exaggerating for effect.
If you have called the rest of us comrades, etc,
I have missed it.

Nor do I think that you dispute Pythagoras.
I hope that you do not.
I want you to come over to our side and
look at the shape of your argument once more.

Are non.standard right triangles
an objection or a change of topic?

Is non.standard analysis
an objection or a change of topic?

Would you like to know why
your objections to a vast swath of of mathematics
attract such a tiny bit of attention?
No one else perceives what you're doing
to be raising an objection.

In my own case, it took me years
to guess that's what you intend.

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XPost: sci.math

On 8/2/2024 3:55 PM, Ross Finlayson wrote:

On 08/02/2024 03:39 AM, FromTheRafters wrote:

[...]

What I ask,
if that you surpass,
the inductive impasse,
of the infinite super-task.

I don't see what infinite super.task or
what inductive impasse you are asking about.

I see transfinite induction being
a consequence of ordinals being
well.ordered.

I see cisfinite induction being
a consequence of finite ordinals being
well.ordered and bounding only 2.ended sets.

----
In transfinite induction,
{ξ ∈ 𝕆: ¬P(ξ)} holds
no first ordinal β:
¬P(β) ∧ ¬∃α<β: ¬P(α)

⎛ Because ordinals are well.ordered,
⎝ the set is no.first only.if {ξ ∈ 𝕆: ¬P(ξ)} = {}.

¬∃β ∈ 𝕆:( ¬P(β) ∧ ¬∃α<β: ¬P(α) )
⟹
¬∃ξ ∈ 𝕆: ¬P(ξ)

∀β ∈ 𝕆:( P(β) ⇐ ∀α<β: P(α) )
⟹
∀ξ ∈ 𝕆: P(ξ)

----
⎛ Because a finite.ordinal bounds only
⎜ 2.ended non.{}.sets
⎜ finite.ordinal β bounds only
⎜ non.{} [0,α)ᴼ with a second.end
⎝ [0,α)ᴼ = [0,α-1]ᴼ for 0 < α ≤ finite β

In cisfinite induction,
{ξ ∈ 𝕆ᶠⁱⁿ: ¬P(ξ)} doesn't hold
0 or ordinal β+1: ¬P(β+1) ∧ P(β)

...which implies
{ξ ∈ 𝕆ᶠⁱⁿ: ¬P(γ)} holds
no first ordinal β:
¬P(β) ∧ ¬∃α<β: ¬P(α)
and
no.first implies
{ξ ∈ 𝕆ᶠⁱⁿ: ¬P(ξ)} = {}.

P(0) ∧ ¬∃β ∈ 𝕆ᶠⁱⁿ: ¬P(β+1) ∧ P(β)
⟹
¬∃β ∈ 𝕆ᶠⁱⁿ:( ¬P(β) ∧ ¬∃α<β: ¬P(α) )
⟹
¬∃ξ ∈ 𝕆: ¬P(ξ)

P(0) ∧ ∀β ∈ 𝕆ᶠⁱⁿ: P(β) ⇒ P(β+1)
⟹
∀β ∈ 𝕆ᶠⁱⁿ:( P(β) ⇐ ∀α<β: P(α) )
⟹
∀ξ ∈ 𝕆ᶠⁱⁿ: P(ξ)

----

Then what *is* restricted comprehension?

Usually it's just
the antonym of expansion of comprehension.

I am more familiar with unrestricted comprehension
being the antonym of restricted comprehension.

Unrestricted comprehension grants that
{x:P(x)} exists because
description P(x) of its elements exists.

Restricted comprehension grants that
{x∈A:P(x)} exists because
description P(x) and set A exist.

The existence of set A might have been granted
because of Restricted.Comprehension or Infinity or
Power.Set or Union or Replacement or Pairing,
but A would be logically prior to {x∈A:P(x)}
by some route.

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XPost: sci.math

On 8/3/2024 9:08 PM, Ross Finlayson wrote:

On 08/03/2024 12:08 PM, Jim Burns wrote:

On 8/2/2024 3:55 PM, Ross Finlayson wrote:

On 08/02/2024 03:39 AM, FromTheRafters wrote:

Then what *is* restricted comprehension?

Usually it's just the antonym of
expansion of comprehension.
What I ask,
if that you surpass,
the inductive impasse,
of the infinite super-task.

I am more familiar with unrestricted comprehension
being the antonym of restricted comprehension.

Unrestricted comprehension grants that
{x:P(x)} exists because
description P(x) of its elements exists.

Restricted comprehension grants that
{x∈A:P(x)} exists because
description P(x) and set A exist.

The existence of set A might have been granted
because of Restricted.Comprehension or Infinity or
Power.Set or Union or Replacement or Pairing,
but A would be logically prior to {x∈A:P(x)}
by some route.

Geometry, axiomatic geometry or Euclid's,
is a classical theory, and it's constructive,
there's only expansion of comprehension,

I know what comprehension, restricted.comprehension,
and unrestricted.comprehension are by having seen
set axioms which were called Comprehension,
Restricted.Comprehension, and Unrestricted.Comprehension.

What does 'comprehension' mean where there are no sets?

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XPost: sci.math

On 8/3/2024 11:51 PM, Ross Finlayson wrote:

On 08/03/2024 08:45 PM, Jim Burns wrote:

On 8/3/2024 9:08 PM, Ross Finlayson wrote:

On 08/03/2024 12:08 PM, Jim Burns wrote:

On 8/2/2024 3:55 PM, Ross Finlayson wrote:

On 08/02/2024 03:39 AM, FromTheRafters wrote:

Then what *is* restricted comprehension?

Usually it's just the antonym of
expansion of comprehension.
What I ask,
if that you surpass,
the inductive impasse,
of the infinite super-task.

I am more familiar with unrestricted comprehension
being the antonym of restricted comprehension.

Unrestricted comprehension grants that
{x:P(x)} exists because
description P(x) of its elements exists.

Restricted comprehension grants that
{x∈A:P(x)} exists because
description P(x) and set A exist.

The existence of set A might have been granted
because of Restricted.Comprehension or Infinity or
Power.Set or Union or Replacement or Pairing,
but A would be logically prior to {x∈A:P(x)}
by some route.

Geometry, axiomatic geometry or Euclid's,
is a classical theory, and it's constructive,
there's only expansion of comprehension,

I know what comprehension, restricted.comprehension,
and unrestricted.comprehension are by having seen
set axioms which were called Comprehension,
Restricted.Comprehension, and Unrestricted.Comprehension.

What does 'comprehension' mean where there are no sets?

What can you think it means.

Your rhetoric suggests that
_you_ don't have something in mind for the term
_you_ introduced,
and you'd like someone else to provide something
to have in mind. Please prove me wrong.

What does 'comprehension' mean where there are no sets?

Specifically,
what does 'expansion of comprehension' mean
in the context of
"geometry, axiomatic geometry or Euclid's"?

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XPost: sci.math

On 8/4/2024 10:36 AM, Ross Finlayson wrote:

On 08/03/2024 10:25 PM, Jim Burns wrote:

On 8/3/2024 11:51 PM, Ross Finlayson wrote:

On 08/03/2024 08:45 PM, Jim Burns wrote:

What does 'comprehension' mean where there are no sets?

What can you think it means.

Your rhetoric suggests that
_you_ don't have something in mind for the term
_you_ introduced,
and you'd like someone else to provide something
to have in mind. Please prove me wrong.

What does 'comprehension' mean where there are no sets?

Specifically,
what does 'expansion of comprehension' mean
in the context of
"geometry, axiomatic geometry or Euclid's"?

No, "what can you think", it means.

Usually it just means "construction".

Okay. Then you did answer my question.

"Comprehension", "construction" and "what can you think"
each seem to me very different from the other two.

I will let you carry on doing what it is you are doing.

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XPost: sci.math

On 8/4/2024 1:44 PM, Ross Finlayson wrote:

On 08/04/2024 09:48 AM, Jim Burns wrote:

On 8/4/2024 10:36 AM, Ross Finlayson wrote:

On 08/03/2024 10:25 PM, Jim Burns wrote:

On 8/3/2024 11:51 PM, Ross Finlayson wrote:

On 08/03/2024 08:45 PM, Jim Burns wrote:

What does 'comprehension' mean where there are no sets?

What can you think it means.

Your rhetoric suggests that
_you_ don't have something in mind for the term
_you_ introduced,
and you'd like someone else to provide something
to have in mind. Please prove me wrong.
What does 'comprehension' mean where there are no sets?
Specifically,
what does 'expansion of comprehension' mean
in the context of
"geometry, axiomatic geometry or Euclid's"?

No, "what can you think", it means.
Usually it just means "construction".

Okay. Then you did answer my question.
"Comprehension", "construction" and "what can you think"
each seem to me very different from the other two.
I will let you carry on doing what it is you are doing.

I don't need your help nor permission, thanks.

I didn't offer help or permission.

You are using English words in a manner which
I don't follow, and,
not to brag, but
I've been speaking English for a really, really long time.
Thus, I'm going to stop trying to follow your use.
My comment was intended to be read as <wave bye bye>.

And it's rather presumptious of you to not
make what is equi-interpretable to be equi-interpretable.

Jaded, say, biased, willfully ignorant, hypocritical, ...,
"wrong".

It is not wrong, when talking about certain things,
to not.be.talking about other things.
It is not wrong, it is not possible to do otherwise.

There is no pair of integers in the ratio of √2

To say that
there are pairs of reals in the ratio of √2
does not argue against the first claim,
because they are claims about different things.

To pretend that it argues against the first claim
is wrong.
https://en.wikipedia.org/wiki/Straw_man

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XPost: sci.math

On 8/4/2024 2:38 PM, Ross Finlayson wrote:

On 08/04/2024 11:26 AM, Jim Burns wrote:

On 8/4/2024 1:44 PM, Ross Finlayson wrote:

And it's rather presumptious of you to not
make what is equi-interpretable to be equi-interpretable.

Jaded, say, biased, willfully ignorant, hypocritical, ...,
"wrong".

It is not wrong, when talking about certain things,
to not.be.talking about other things.
It is not wrong, it is not possible to do otherwise.

There is no pair of integers in the ratio of √2

To say that
there are pairs of reals in the ratio of √2
does not argue against the first claim,
because they are claims about different things.

To pretend that it argues against the first claim
is wrong.
https://en.wikipedia.org/wiki/Straw_man

Numbers

https://en.wikipedia.org/wiki/Number

It is not wrong, when talking about certain numbers,
to not.be.talking about other numbers.
It is not wrong, it is not possible to do otherwise.

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