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Re: Question on Hilbert's Hotel. PLO

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olcott

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Jan 27, 2023, 2:43:32 PM1/27/23
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On 12/19/2022 4:04 AM, WM wrote:
> If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
> Is it possible to add a natural number to the set ℕ?
>
> Regards, WM

You don't actually mean "add" as in arithmetic you actually mean append
as in string concatenation.

https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

Hilbert's Hotel is incoherent in that all of the infinite number of
rooms are occupied and a vacant room can be made for one more guest.

--
Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer

Jeffrey Rubard

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Jan 27, 2023, 5:16:22 PM1/27/23
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"Do you suppose Edmund Husserl stayed at this 'Hilbert's Hotel'?"

WM

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Jan 30, 2023, 10:45:16 AM1/30/23
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_ Olcott schrieb am Freitag, 27. Januar 2023 um 20:43:32 UTC+1:
> On 12/19/2022 4:04 AM, WM wrote:
> > If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
> > Is it possible to add a natural number to the set ℕ?
> >
> You don't actually mean "add" as in arithmetic you actually mean append
> as in string concatenation.
>
> https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
>
> Hilbert's Hotel is incoherent in that all of the infinite number of
> rooms are occupied and a vacant room can be made for one more guest.
>
If the set ℕ exists, then it is impossible to create a natural number that is not already in the set ℕ (i.e., a room number of Hilbert's hotel). If the set ℕ does not exist, then Hilberts hotel does not exist either.

Regards, WM

olcott

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Jan 30, 2023, 12:30:14 PM1/30/23
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Yes you and I are in agreement that the Hilbert's hotel thought
experiment is incorrect.

Dan Christensen

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Jan 30, 2023, 12:38:20 PM1/30/23
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On Friday, January 27, 2023 at 2:43:32 PM UTC-5, _ Olcott wrote:
> On 12/19/2022 4:04 AM, WM wrote:
> > If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
> > Is it possible to add a natural number to the set ℕ?
> >
> > Regards, WM
>
> You don't actually mean "add" as in arithmetic you actually mean append
> as in string concatenation.
>
> https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
>
> Hilbert's Hotel is incoherent in that all of the infinite number of
> rooms are occupied and a vacant room can be made for one more guest.
>

This bizarre scenario can be avoided by first getting a handle on FINITE sets. Then an infinite set is just one that is not finite. See my blog posting on this topic: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
Visit my Math Blog at http://www.dcproof.wordpress.com

olcott

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Jan 30, 2023, 1:04:31 PM1/30/23
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None-the-less if every room of a countably infinite set of rooms is
filled, then each room is already associated with an element of N as
its room number, leaving zero room numbers available for occupancy.

Hilbert simply got this wrong.

olcott

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Jan 30, 2023, 1:05:59 PM1/30/23
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Credit goes to WM for involving room numbers into this analysis.

Dan Christensen

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Jan 30, 2023, 2:18:28 PM1/30/23
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I disagree. I think Hilbert's Hotel was simply a thought experiment to justify that, on an infinite set X, there exists a bijection from X to a proper subset of X (from Dedekind). It succeeds at this much.

In my development, I start with a thought experiment to develop the notion of a Dedekind FINITE set. It avoids a lot of the supernatural silliness of Hilbert's Hotel.

Mostowski Collapse

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Jan 30, 2023, 2:22:07 PM1/30/23
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Hell No! A thought experiment. Please stick to Historical Facts,
and not some Mathematical Fiction.

Similarly I recommend retracting the induction axiom schema
from your Peano Axioms. It cannot be that we theorems like:

n + m = m + n

In the light of dark numbers. This is surely humbug predicting
the future of ones calculation this, way, that

is impossibru!

Dan Christensen

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Jan 30, 2023, 2:36:23 PM1/30/23
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On Monday, January 30, 2023 at 2:22:07 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Montag, 30. Januar 2023 um 20:18:28 UTC+1:
> > On Monday, January 30, 2023 at 1:04:31 PM UTC-5, _ Olcott wrote:
> > > On 1/30/2023 11:38 AM, Dan Christensen wrote:
> > > > On Friday, January 27, 2023 at 2:43:32 PM UTC-5, _ Olcott wrote:
> > > >> On 12/19/2022 4:04 AM, WM wrote:
> > > >>> If the complete set ℕ of natural numbers exists, then it can be used to populate Hilbert's Hotel.
> > > >>> Is it possible to add a natural number to the set ℕ?
> > > >>>
> > > >>> Regards, WM
> > > >>
> > > >> You don't actually mean "add" as in arithmetic you actually mean append
> > > >> as in string concatenation.
> > > >>
> > > >> https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel
> > > >>
> > > >> Hilbert's Hotel is incoherent in that all of the infinite number of
> > > >> rooms are occupied and a vacant room can be made for one more guest.
> > > >>
> > > >
> > > > This bizarre scenario can be avoided by first getting a handle on FINITE sets. Then an infinite set is just one that is not finite. See my blog posting on this topic: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> > > >
> > > > Dan
> > > >
> > > > Download my DC Proof 2.0 freeware at http://www.dcproof.com
> > > > Visit my Math Blog at http://www.dcproof.wordpress.com
> > > >
> >
> > > None-the-less if every room of a countably infinite set of rooms is
> > > filled, then each room is already associated with an element of N as
> > > its room number, leaving zero room numbers available for occupancy.
> > >
> > > Hilbert simply got this wrong.
> > I disagree. I think Hilbert's Hotel was simply a thought experiment to justify that, on an infinite set X, there exists a bijection from X to a proper subset of X (from Dedekind). It succeeds at this much.
> >
> > In my development, I start with a thought experiment to develop the notion of a Dedekind FINITE set. It avoids a lot of the supernatural silliness of Hilbert's Hotel.

> Hell No! A thought experiment. Please stick to Historical Facts,
> and not some Mathematical Fiction.
>

Maybe you didn't know, but there are actually no infinite hotels. Maybe there would if those unicorns of yours had existed, Mr. Collapse. HA, HA, HA!

Dan

olcott

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Jan 30, 2023, 3:16:52 PM1/30/23
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It is the case that when a hotel has room numbers that form a bijection
to the elements of N and each of these rooms are full that there are no
empty rooms that can be filled with new guests.

The problem with comprehending this is that a hotel with an infinite
number of rooms it itself an incoherent concept, as Dan suggested.

Mostowski Collapse

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Jan 30, 2023, 4:17:42 PM1/30/23
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Well you termed it though experiment. Is this
some kind of stupidity contest of yours?

It exists in someones imagination, doesn't it?
Similar like your Dedekind infinite sets,

they also do not exist only virtually.

Jeffrey Rubard

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Jan 30, 2023, 4:20:30 PM1/30/23
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Did Gerhard Gentzen at least stay there, then?

Dan Christensen

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Jan 30, 2023, 4:37:14 PM1/30/23
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On Monday, January 30, 2023 at 4:17:42 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Montag, 30. Januar 2023 um 20:36:23 UTC+1:
> > Maybe you didn't know, but there are actually no infinite hotels.
> > Maybe there would if those unicorns of yours had existed, Mr. Collapse. HA, HA, HA!

> Well you termed it though experiment. Is this
> some kind of stupidity contest of yours?
>

You seem to be in first place, Mr. Collapse.

> It exists in someones imagination, doesn't it?
> Similar like your Dedekind infinite sets,
>
> they also do not exist only virtually.

Is this suppose to convince us of the existence of your unicorns and infinite hotels? It isn't working.

If you have this much trouble with the definition of an infinite set, you should first try to define a FINITE set, preferably a non-numeric definition. It is quite a bit easier. See: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/

Mostowski Collapse

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Jan 30, 2023, 4:39:27 PM1/30/23
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Whats your point?

Mostowski Collapse

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Jan 30, 2023, 4:51:10 PM1/30/23
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You have never heard of Hilberts country?
It consists of all villages with finitely many houses.

* * * * * * ....

The country is fully inhabitated, now
some new person arrives from

the neightbour country. How could he
find a house in the country. Very easy,

one person from village n moves to
village n+1 and so on.

LoL

Dan Christensen

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Jan 30, 2023, 5:07:34 PM1/30/23
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On Monday, January 30, 2023 at 4:51:10 PM UTC-5, Mostowski Collapse wrote:

> Mostowski Collapse schrieb am Montag, 30. Januar 2023 um 22:39:27 UTC+1:
> > Whats your point?
> > Dan Christensen schrieb am Montag, 30. Januar 2023 um 22:37:14 UTC+1:
> > > On Monday, January 30, 2023 at 4:17:42 PM UTC-5, Mostowski Collapse wrote:
> > >
> > > > Dan Christensen schrieb am Montag, 30. Januar 2023 um 20:36:23 UTC+1:
> > > > > Maybe you didn't know, but there are actually no infinite hotels.
> > > > > Maybe there would if those unicorns of yours had existed, Mr. Collapse. HA, HA, HA!
> > > > Well you termed it though experiment. Is this
> > > > some kind of stupidity contest of yours?
> > > >
> > > You seem to be in first place, Mr. Collapse.
> > > > It exists in someones imagination, doesn't it?
> > > > Similar like your Dedekind infinite sets,
> > > >
> > > > they also do not exist only virtually.
> > > Is this suppose to convince us of the existence of your unicorns and infinite hotels? It isn't working.
> > >
> > > If you have this much trouble with the definition of an infinite set, you should first try to define a FINITE set, preferably a non-numeric definition. It is quite a bit easier. See: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/

> You have never heard of Hilberts country?

No.

> It consists of all villages with finitely many houses.
>
> * * * * * * ....
>
> The country is fully inhabitated, now
> some new person arrives from
>
> the neightbour country. How could he
> find a house in the country. Very easy,
>
> one person from village n moves to
> village n+1 and so on.
>

The point here is to develop a non-numeric definition of a finite set. A set X is said to be finite iff ______________________________. (Fill in the blank)

How about it, Mr. Collapse? Eventually, we want to obtain the definition of a Dedekind finite set.

Mostowski Collapse

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Jan 30, 2023, 7:38:19 PM1/30/23
to
Looking at their cardinality, how many finite sets are there?
In Hilbert Country for every cardinality n,
there is a village with n houses:

* * ... * *
1 ... n

Now the country has a village for every n. A new person arrives
in the country. One person from village n moves to village n+1,
and so on, and the person that arrive can move into village 1.

Mostowski Collapse

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Jan 30, 2023, 7:42:27 PM1/30/23
to
Or symbolicaly:

ω = U { n | FiniteA(s)=n } = U { n | FiniteB(s)=n }

You talk about finite sets, yet you talk about infinitely many finite sets.

Dan Christensen

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Jan 30, 2023, 9:48:32 PM1/30/23
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On Monday, January 30, 2023 at 7:38:19 PM UTC-5, Mostowski Collapse wrote:

> Dan Christensen schrieb am Montag, 30. Januar 2023 um 23:07:34 UTC+1:

> > The point here is to develop a non-numeric definition of a finite set. A set X is said to be finite iff ______________________________. (Fill in the blank)
> >
> > How about it, Mr. Collapse? Eventually, we want to obtain the definition of a Dedekind finite set.
> > See: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> >

> Looking at their cardinality, how many finite sets are there?
> In Hilbert Country for every cardinality n,
> there is a village with n houses:
>
> * * ... * *
> 1 ... n
>
> Now the country has a village for every n. A new person arrives
> in the country. One person from village n moves to village n+1,
> and so on, and the person that arrive can move into village 1.

Still no definition of a finite set. Hint: You don't need the machinery of cardinalities or numbers, just functions on the set in question. It's really quite simple. Try again.

Mostowski Collapse

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Jan 31, 2023, 2:52:13 AM1/31/23
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I guess one cannot prove in DC Proof:

ω = { |s| | FiniteA(s) } = U { |s| | FiniteB(s) }

It takes some more axioms to make the above happen,
than only those that are present in DC Proof concerning sets.

Already one cannot prove in DC Proof:

EXIST(s):Set(s)

LoL

Dan Christensen schrieb:

Dan Christensen

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Jan 31, 2023, 11:05:01 AM1/31/23
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On Tuesday, January 31, 2023 at 2:52:13 AM UTC-5, Mostowski Collapse wrote:

>
> Dan Christensen schrieb:
> > On Monday, January 30, 2023 at 7:38:19 PM UTC-5, Mostowski Collapse wrote:
> >
> >> Dan Christensen schrieb am Montag, 30. Januar 2023 um 23:07:34 UTC+1:
> >
> >>> The point here is to develop a non-numeric definition of a finite set. A set X is said to be finite iff ______________________________. (Fill in the blank)
> >>>
> >>> How about it, Mr. Collapse? Eventually, we want to obtain the definition of a Dedekind finite set.
> >>> See: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> >>>
> >
> >> Looking at their cardinality, how many finite sets are there?
> >> In Hilbert Country for every cardinality n,
> >> there is a village with n houses:
> >>
> >> * * ... * *
> >> 1 ... n
> >>
> >> Now the country has a village for every n. A new person arrives
> >> in the country. One person from village n moves to village n+1,
> >> and so on, and the person that arrive can move into village 1.
> >
> > Still no definition of a finite set. Hint: You don't need the machinery of cardinalities or numbers, just functions on the set in question. It's really quite simple. Try again.
> >

Still waiting, Mr. Collapse.

> > See: https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
> >

> I guess one cannot prove in DC Proof:
>
> ω = { |s| | FiniteA(s) } = U { |s| | FiniteB(s) }
>
> It takes some more axioms to make the above happen,
> than only those that are present in DC Proof concerning sets.
>
> Already one cannot prove in DC Proof:
>
> EXIST(s):Set(s)
>

True, DC Proof does not have built into it an assumption of the existence of a non-empty universe. Such a notion is not unheard of, even in philosophy textbooks. It avoids silliness like ALL(x):P(x) => EXIST(x):P(x). It allows rules of logic that are simpler and more like those implicitly used in most math textbooks.

Anyway, it seems you cannot even provide a workable, non-numerical definition of a finite set. What seems to be the problem, Mr. Collapse?

Jim Burns

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Jan 31, 2023, 1:20:38 PM1/31/23
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On 1/31/2023 11:04 AM, Dan Christensen wrote:

> a workable, non-numerical definition of
> a finite set.

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

This is my favorite:
|
| 3. (Paul Stäckel) S can be given a total ordering
| which is well-ordered both forwards and backwards.
| That is, every non-empty subset of S has both
| a least and a greatest element in the subset.

My expression of Stäckel's definition
has evolved into
|
| There is an order of S such that
|
| Each element of S is before or after
| each of the other elements of S
|
| There is a first in S and a last in S.
|
| For each split of S,
| There is a last-before and a first-after.

In my opinion, that definition seems to
more clearly express the concept of
being able to get there from here
than other definitions do.

"Because we can get there from here"
seems to often be the answer to
"Why are we only talking about finites?"
"Definables?" "Visibles?" "Accesssibles?"
"Individually-usables?" and so on.


Dan Christensen

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Jan 31, 2023, 2:58:31 PM1/31/23
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On Tuesday, January 31, 2023 at 1:20:38 PM UTC-5, Jim Burns wrote:
> On 1/31/2023 11:04 AM, Dan Christensen wrote:
>
> > a workable, non-numerical definition of
> > a finite set.
> https://en.wikipedia.org/wiki/Finite_set
>
> This is my favorite:
> |
> | 3. (Paul Stäckel) S can be given a total ordering
> | which is well-ordered both forwards and backwards.
> | That is, every non-empty subset of S has both
> | a least and a greatest element in the subset.
>

You can do without any kind of ordering of the set. Simpler and based on Dedekind: A set X is finite iff every injective function f: X --> X is surjective.

Dan

Jim Burns

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Jan 31, 2023, 4:16:43 PM1/31/23
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On 1/31/2023 2:58 PM, Dan Christensen wrote:
> On Tuesday, January 31, 2023
> at 1:20:38 PM UTC-5, Jim Burns wrote:

>> https://en.wikipedia.org/wiki/Finite_set
>>
>> This is my favorite:
>> |
>> | 3. (Paul Stäckel) S can be given a total ordering
>> | which is well-ordered both forwards and backwards.
>> | That is, every non-empty subset of S has both
>> | a least and a greatest element in the subset.
>
> You can do without any kind of ordering of the set.
> Simpler and based on Dedekind:
> A set X is finite iff
> every injective function f: X --> X is surjective.

Yes, I agree that Dedekind-finiteness is cleaner
than the splits and last-befores and first-afters
that I have been using. It is Dedeking-finiteness
that I have been thinking of when I have been
mentioning Bob-conservation.

However,
we have an instance here in which I prefer
less cleanliness. My preference here, with
Bob and end segments and unit fractions,
is to show why things are the way they are.

Dedekind-finiteness is a beautiful jewel,
lovingly polished. But I would like to protray
a rock getting dug out of a hole in the ground,
and processed, and processed, and processed,
until we have this jewel.

Different purposes, different results.


Jeffrey Rubard

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Feb 7, 2023, 4:41:35 PM2/7/23
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"I think you guys need to work on it."

Dan Christensen

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Feb 7, 2023, 5:08:25 PM2/7/23
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That is what I try to do at my blog posting:

I begin:

"Starting at any house in this finite village, you set out to visit some or all of the houses there. You keep walking non-stop until you return to your starting point, visiting no house more than once.

"Intuitively, it seems obvious that you could not avoid returning to your starting point under these conditions. You could, of course, return to your starting point without first going to every other house in the village. If you kept going, however, you would eventually run out of different places to go, and you would have to return to your starting point. Going anywhere else at that point would be going there more than once. (As you might imagine, in an infinite village, it might be possible to keep walking from house to house, never stopping and never returning to your starting point.)"

Using some clever diagrams, I then go about justifying a formalization of this scenario to eventually obtain this definition of a finite set:

ALL(s):[Set(s) => [Finite(s)
<=> ALL(x0):ALL(f):[x0 in s
& ALL(a):[a in s => f(a) in s]
& ALL(a):ALL(b):[a in s & b in s => [f(a)=f(b) => a=b]]
=> EXIST(xn):[xn e s & f(xn)=x0]]]]

Polishing this up a bit, as you put it, I then show that this is equivalent to the notion of Dedekind finiteness:

ALL(s):[Set(s) => [Finite(s)
<=> ALL(f):[ALL(a):[a e s => f(a) e s]
& ALL(a):ALL(b):[a e s & b e s => [f(a)=f(b) => a=b]]
=> ALL(a):[a e s => EXIST(xn):[xn e s & f(xn)=a]]]

https://dcproof.wordpress.com/2014/09/17/infinity-the-story-so-far/
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