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The Misuse of Bijection when Comparing Infinite Sets



There is no case where a bijection between infinite sets can be relied upon to compare their cardinalities. Their cardinalities can in some cases be compared using asymptotic density, but they can always be compared when expressed in terms of the infinite number PHI. Listing (enumerating) an infinite set does not measure its cardinality. The count of real numbers in the open interval (0, 1) is a choice, not some pre-defined value.
Misuse of Bijection
Leslie Green CEng MIEE 1 of 11 v1.00: May 2022
The Misuse of Bijection when
Comparing Infinite Sets
Abstract ............................................................................................ 1
Introduction ...................................................................................... 1
The Natural Numbers ....................................................................... 2
1:1 Correspondence ........................................................................ 3
Partition ............................................................................................ 3
Density ............................................................................................. 4
Cauchy’s Contribution ...................................................................... 5
Bijection over a Partial Domain ........................................................ 5
Bijection with the Rationals .............................................................. 6
Bijection via Enumeration ................................................................. 7
Geometric Bijections ........................................................................ 8
an infinite number ..................................................................... 9
Conclusions ................................................................................... 11
Version History ............................................................................... 11
There is no case where a bijection between infinite sets can be relied upon
to compare their cardinalities. Their cardinalities can in some cases be
compared using asymptotic density, but they can always be compared
when expressed in terms of the infinite number .
Listing (enumerating) an infinite set does not measure its cardinality.
The count of real numbers in the open interval (0, 1) is a choice, not some
pre-defined value.
Any graduate mathematician has a personal investment in the
mathematics of infinity as created by Galileo, Dedekind, Cantor, and
others. If this has become an article of Faith for you then there is no point
in continuing with this paper. One has to accept as a known fact that not
everything we were taught is true.
It therefore remains to find out which
teachings are robust, and which need to be revised or (rarely) even
Historically, the idea of infinity and infinitesimals
was found to be in
conflict with the teachings of the Church. Such heretical teachings were of
course dealt with harshly. Fortunately it is now expected that we should
robustly challenge theories, ideas, and opinions. I therefore make no
apology for challenging ideas which you may feel are certainly true.
In order to represent the ‘orthodox position’ I shall frequently refer to
The Book of Proof” by Richard Hammack; 3rd edition, 2018. There are two
reasons for choosing this particular text: Firstly, it presents a very clear
and well defined exposition of the subject matter. And secondly, Professor
Hammack has generously made the PDF text freely available so it is only a
click away.
Infinitesimal: How a Dangerous mathematical theory shaped the modern world. Amir
Alexander, 1988, (2015) Oneworld.
Misuse of Bijection
Leslie Green CEng MIEE 2 of 11 v1.00: May 2022
The Natural Numbers
The positive integers, also known as the natural numbers, can be written
as: + = = { 1, 2, 3, 4, 5, 6, 7, 8, 9, … }
The three dots, an ellipsis, mean carry on in the same way indefinitely
(forever). If we put vertical bars either side of a set, |{ 1, 2, 3 }| = 3, this
notation is used to count the number of elements in the set. For a set with
“infinitely many” elements, whatever that means, we have a difficultly
with the wording “number of elements” since infinity, as universally
defined, means “beyond any number”.
The new word cardinality is used to express the size without using the
word number. Sets with the same cardinality are therefore said to be of
equal size in some unspecified sense, but ideally not equinumerous,
because again we try to avoid using the root of the word number in this
infinite context.
We are pretty happy to say something like: |+| = ||
as the equals sign is comparing two like-quantities, even though they are
both ‘infinite’.
|{ 1, 2, 3, 4, 5, 6, 7, 8, 9 }| = 9 is fine
|{ 1, 2, 3, 4, 5, 6, 7, 8, 9, … }| = is less than ideal.
The left hand side (LHS) may be thought of as some sort of number,
whereas is not-a-number, but some sort of concept. Why is it
reasonable for me to claim that the LHS is some sort of number? Consider
a finite set of natural numbers: |{ 1, 2, 3, 4, 5, 6, 7, 8, 9, … , H }|, where H
is some HUGE value such as 1E1234567890. H can exceed any other value
we could have chosen, and yet it is always true that:
|{ 1, 2, 3, 4, 5, 6, 7, 8, 9, … , H }| = H
Notice that H appears within the set, and also is the cardinality of the set.
We are then told that
|| = 0 which is equivalent to
|{ 1, 2, 3, 4, 5, 6, 7, 8, 9, … }| = 0
which, in order to be true, would also have to be written as:
|{ 1, 2, 3, 4, 5, 6, 7, 8, 9, … , 0 }| = 0
But 0 , read this as aleph-null (or aleph-nought) is not a natural number,
it is a transfinite cardinal number, and therefore does not belong in the set
of natural numbers.
Let’s not worry too much about that technicality for now.
Book of Proof, Definition 14.3, page 275
Misuse of Bijection
Leslie Green CEng MIEE 3 of 11 v1.00: May 2022
1:1 Correspondence
Every element in set A corresponds to exactly one element in set B, and
this element in set B is not also ‘shared’ by another element in set A.
Every element in set B corresponds to exactly one element in set A, and
this element in set A is not also ‘shared’ by another element in set B.
In the same way that a function f (x) has only one value, say y, for each
value of x, 1:1 correspondence means that for each element in A you can
find a unique corresponding value in B, and vice versa. If 1:1
correspondence genuinely exists between two sets, they certainly have
the same size, even if this size is numerically too large to count.
Notice that the two sets need not be of the same type of object. In the
example above we are viewing the correspondence between natural
numbers and arrow symbols. This disparity of type can make it difficult to
see what is going on in sufficient detail when comparing infinite sets.
A partition of a set, S is defined
as any collection of disjoint proper
subsets of S, the union of which equals S.
Here we partition the natural numbers in an obvious way:
= { odd natural numbers } { even natural numbers }
This next equivalent set-builder approach will be useful shortly.
= { n : n 1 (mod 2) } { n : n 0 (mod 2) }
We are told that: || = |{ n : n 1 (mod 2) }|
and || = |{ n : n 0 (mod 2) }|
so that 0 = 0 + 0
and that this works because this is how transfinite cardinals are summed.
But why? You have to understand the sequence of operations here.
Cantor ‘proved’ that the even numbers could be put into 1:1
correspondence with the natural numbers, meaning they have the same
cardinality. This being the case, we have to be able to sum any number of
aleph-nulls and still get a single aleph null. If you like, the transfinite rules
are fitted to the (supposed) summation properties (incorrectly) observed
for infinite sets.
We can then get this supposed but improbable equality:
|| = |{ n : n 1 (mod H) }| for H = 1E123456789
In other words we can take the set of natural numbers, we can discard all
but one in every 1E123456789 values, and the new set has not changed its
size in any way. Of course the set is still infinite, but surely it is smaller in
some quantifiable sense? Mathematicians claim it is not smaller in any
way. Needless to say, I find that claim suspicious at best!
See also Book of Proof, Definition 11.5, page 216.
Misuse of Bijection
Leslie Green CEng MIEE 4 of 11 v1.00: May 2022
There are ‘infinitely many’ natural numbers. There are ‘infinitely many’
natural numbers which are perfect squares. Surely there is a sense in
which there are less perfect squares than natural numbers? This idea was
considered in the text of Hardy & Wright’s An Introduction to the Theory
of Numbers, 1938 (5th ed. 1979, , 2004) using the idea of “almost all” in
an asymptotic sense.
Suppose we wish to count the number of perfect squares up to some
arbitrary natural value H². There are H perfect squares, out of natural
numbers. The proportion of perfect squares up to is H/H² = 1/H. As H
heads off towards infinity, 1/H tends to zero. Even though there are
infinitely many perfect squares, almost no natural numbers are perfect
squares (in this asymptotic sense).
This idea of almost all or almost none is a bit vague. We should prefer a
more quantitative approach. What proportion of all natural numbers are
evenly divisible by 3? We can’t evaluate /3 as anything other than .
Likewise, we can’t evaluate 0/3 as anything other than 0. Those
symbols are not ordinary numbers, and they don’t behave ‘nicely’
(properly; in a number-like manner).
Again we need to consider a finite set up to say 3H. H of these are evenly
divisible by 3. The proportion that are divisible by 3 is H/(3H) = 1/3. Note
that the H divides out (whilst still finite). If you plotted a graph on a log
scale, the proportion would still be 1/3 even as you increased H in 1000-
decade intervals.
It is therefore very reasonable to say that half of all natural numbers are
odd, and a half are even, but you must prefix the statement by mentioning
that you are talking in an “asymptotic sense”. Unfortunately this powerful
notion of density is not mentioned in any high-school texts (and in fact in
no texts that I can find, other than a brief footnote in the Princeton
Companion to Mathematics, 2008, page 170, III.1.1, Countable and
Uncountable Sets ).
If two infinite sets contain the same type of numbers, eg natural numbers,
but their densities differ by 1000 orders of magnitude, eg one set contains
numbers even divisible by 1E1000 and the other set contains numbers not
evenly divisible by 1E1000, it is both illogical and ludicrous to claim that
there is any sense in which these two sets have equal sizes. If such a claim
is made, the method used to justify the claim is itself questionable.
The use of the asymptotic density concept means that in a partition of an
infinite set, the sum of the densities of the subsets correctly equals the
whole (1.000). The idea that you can put a strict subset into 1:1
correspondence with the initial set is untenable.
Misuse of Bijection
Leslie Green CEng MIEE 5 of 11 v1.00: May 2022
Cauchy’s Contribution
The great French Mathematician Cauchy, a giant in his field, published
some excellent ideas about infinitesimals and infinities. These ideas were
expressed in his 1821 book Cours d’analyse. In addition to the ground-
breaking content, perhaps what is most remarkable about this text is that
it was only translated into English in 2009 (by Bradley & Sandifer).
In section 2.1 Cauchy considers an infinitesimally small’ quantity , and
proves that although both and ² tend to zero, it is always true that
² << . Likewise for their reciprocals, the squared value is always much
larger. There is no difficulty found in dealing with powers of infinitesimally
small or infinitely large values. In order to compare infinite (or
infinitesimal) values Cauchy simply takes their ratio (Theorem I), just like
we do when considering densities.
Another relevant contribution by Cauchy was the understanding of the
meaning of the value of an infinite series. In order to understand what it
means to sum an infinite series, given that it is not possible to continue
summing terms forever, Cauchy presented the idea of the convergence of
the sequence of partial sums. This understanding meant that the value of
a (convergent) infinite series was put on a firm analytical footing. Maybe
something similar can help us with bijections between infinite sets?
Bijection over a Partial Domain
The evidence seen so far casts doubt over the use of a bijection to
compare infinite sets. We can then see if the idea of a partial domain
helps us to understand what such a bijection between infinite sets should
What has been done consistently in the past is to consider two infinite
sets, apparently both using the same domain limit, namely limited only by
the set of natural numbers. But is that what is actually happening?
Let’s consider two sets over the partial natural domain up to some
ridiculously HUGE limiting natural value 2H.
A = { n : (H ) n 2H } |A| = 2H
B = { n : (H ) n 2H, n 0 (mod 2) } |B| = H
As H heads off towards infinity there is no bijection possible between the
sets A and B. They clearly have different (and trivially calculable)
cardinalities. There is no point at which 2H ‘becomes infinite’ and sorts
out the problem. The supposed bijection between A and B is a fallacy
caused by careless thinking. In order to obtain the supposed 1:1
correspondence between elements in the sets, the domain limit of B has
to always be twice as large as the domain limit of A.
It is possible for somebody to claim: “BUT twice infinity is still infinity, so
the domain limits are the same size AT infinity.”
Nice try (if you are deliberately trying to perpetuate a fraud) but
remember that n , H , . If you claim to have reached infinity,
you are wrong.
The partial domain method is seen to be similar to the idea of asymptotic
density, and gives equal results. Thus all the modular set comparisons
work ‘correctly’ in accordance with our in-built intuition of how numbers
should work. Unfortunately there are other claimed bijections between
sets of dissimilar object types where the partial domain method cannot be
applied. We therefore have more work to do.
Misuse of Bijection
Leslie Green CEng MIEE 6 of 11 v1.00: May 2022
Bijection with the Rationals
This (supposed) bijection is again due to Cantor. The claim is that the set
of natural numbers can be put into 1:1 correspondence with the set of
Below we plot each positive rational at the point (x, y), each having the z-
value x/y. Here the value of H is not very large, and the individual (x, y)
points are not explicitly shown.
The claim is that by using
this geometrically-
diagonal matching
procedure, we will
eventually match all the
elements in the set of
rational numbers,
provided we continue
for ‘long enough’. The
nature of the rationals
map may not be
immediately obvious. If
so, a more complete
explanation is only a click
A = { n : (H ) n H } |A| = H
B = { x/y : (x , y , H ), x H, y H } |B| = H²
On the Density of Rational Numbers, (2018), Green. L.O. (v1.10, 2021)
The set A is a list of natural numbers from 1 to H inclusive. These are
required to match, in a 1:1 fashion, an array of H H rationals.
The array is not as big as H2 in practice because we are trying to create a
bijection. We only want the first member of each equivalence class for the
rationals. To be clear, by way of an example, we only want the ½, and not
the equivalent values: 2/4, 3/6, 4/8, …
We could add the extra requirement to B that x and y are coprime
[ gcd(x, y) = 1 ], to eliminate unwanted rationals in the equivalence
classes, but then |B| becomes difficult to count analytically. We should, of
course, also eliminate all but one of the cases where x = y.
Suppose we start with H = 10, even though this is not a HUGE value. We
have 10 elements in A to match with 100 elements in B. So we increase
set A to give us the required 100 elements before we realise that it is then
necessary to increase H² to 10,000. Rather than making things better,
increasing H is making things worse, as a lower percentage of B is being
This supposed 1:1 correspondence requires us to believe that:
O(H) = O(H²) … which is un-mathematical nonsense.
Thinking that increasing H towards infinity somehow makes the matching
work itself out (in some hard to understand way) is extremely careless. A
and B have different domains, but we are setting H as the maximum
available natural number at any particular stage in the matching process.
It is unreasonable to arbitrarily make the limit H larger in set A than set B.
Misuse of Bijection
Leslie Green CEng MIEE 7 of 11 v1.00: May 2022
Bijection via Enumeration
For unlike-sets it is considered convenient to create an ‘infinite list’ of the
elements of the set to be matched to the naturals.
In order to create a
bijection from such a list it is essential that each element in the list is
unique (not repeated). Having created such a list, the enumerated set is
declared to have the same cardinality as the naturals. Such a set is called
countably infinite, enumerable, or denumerable, according to the
preferences of the author.
Everyone (but not me) seems to agree that the reals are not countably
infinite since it was ‘proved’ by Cantor that such a list (enumeration) is not
The reals are called “uncountable” because they (apparently)
cannot be enumerated, and have been ‘proven’ to be so.
It is well worth
looking at this so-called proof (which is merely a correct republication of
Cantor’s spurious argument). If you look at Cantor’s argument, and feel
that it is deliberately written to confuse the issue, you may start to realise
why Cantor was labelled as a “scientific charlatan” and a “corrupter of
In Cantor’s diagram (republished in the Book of Proof) the decimal values
are placed in some randomised order. Why? Certainly it makes it more
difficult to see the careless reasoning involved. The idea is to create a new
number whose nth digit is different to any previously used sequence of n
digits, such that the new number is not present in the list. If the new
number is not in the list, then the list is incomplete, and no bijection is
Actually, this idea of incomplete listing was actually used constructively to
make a complete enumeration of the reals.
Book of Proof, Theorem 14.3, page 276.
Book of Proof, Theorem 14.2, page 272.
Book of Proof, diagram at the bottom of page 271.
Consider this table of binary numbers. All 8 of the available 3-digit
binary numbers are listed. None are missing. You cannot find an
extra sequence that has never been listed so far. It is much easier
in binary because only 8 rows are needed. If this was in decimal
then 1000 rows would have been required. Using binary makes the
pattern easier to see. Rather than writing fractional decimal
values, you can imagine writing these in fractional binary instead.
In decimal you can imagine first listing all numbers with only one digit
after the decimal point, then two digits, then three digits, and so on
indefinitely. Every single sequence is then eventually listed. There are
some subtleties to consider, but needless to say, I have written this all out
fully in its own paper, which is of course only a click away.
So the reals are in fact enumerable. But does that actually mean that they
have the same cardinality as the naturals? NO! Theorem 14.3 (Book of
Proof) is wrong on this point too. Why can’t infinite lists have different
sizes (cardinalities)? The argument seems to be that infinities are either
enumerable (countably infinite) or uncountable (having a greater size than
an enumerable set). Essentially this is prejudging the outcome of the
continuum hypothesis.
And the continuum hypothesis also neglects
the idea of strict subsets of the naturals having smaller cardinality than
the naturals (something which we have adequately demonstrated earlier
in this paper).
Enumerating the Continuum, (2019), Green. L.O. (v1.20, 2019)
Book of Proof, page 289. Are there infinite sets with cardinalities between 0 and ||?
EXPOSED! The bogus Mathematics of Infinity, (2019), Green. L.O. (v2.10, 2021)
Misuse of Bijection
Leslie Green CEng MIEE 8 of 11 v1.00: May 2022
Geometric Bijections
If, like me, you have found some of the previous bijection examples to be
epic fails in logic, these next ones may feel laughable in their stupidity.
This first one is again (adapted) from the Book of Proof, Figure 14.1, page
In the original, the red line and the intervals d and e are not shown. The
original idea was to prove that the cardinality of the unit interval is equal
to the cardinality of the (positive) continuum, a standard result! Again this
is remarkably careless thinking.
It is very reasonable to ask how many little fixed sub-intervals of length d
there are in the open interval from 0 to 1. It is also very reasonable to
consider d shrinking in size to some infinitesimal value. It is entirely
unreasonable to consider either d or e as mathematical points.
In geometry the fundamental definition of a line is something with one
dimension, length. A point has no dimension, by definition. It does not
Book of Proof, page 273, Example 14.3: Show that |(0,)| = |(0,1)|
even have an infinitesimal dimension. Infinitely many points of no-
dimension cannot fill the open interval (0, 1).
By neglecting the dimension of these sub-intervals, it has been possible to
construct a (spurious) bijection between point-like sub-intervals in the
interval (0, 1) and weird extensible non-points in the interval (0, ). The
only way you can equate these two things is to continuously vary the size
of the (unmentioned) intervals! This ‘proof’ is so bad that it was worthy of
making a problem for school kids.
The ‘proof’ also prompted a complete
paper on the subject,
again only a click away.
Another egregious example of the bijection between points ‘proving’ that
different infinities have the same size (cardinality) is found by constructing
a radial line through a pair of concentric circles. Clearly there is a 1:1
correspondence between points on the pair of circles, so the cardinalities
are equal (in some alternate universe where the different arc-segment
lengths are neglected). This fallacy is also discussed in the paper just cited.
I hope I have convincingly made the point that enumerations and
(supposed) bijections are not any sort of way of comparing the
cardinalities of infinite sets. In fact all of this overcomplicated nonsense
hides the simplicity of how to count the elements in infinite sets.
We can now proceed to compare the cardinality of infinite sets properly.
The Fundamental Basis of the Real Number Line (the Continuum), (2022), Green. L.O.
(v1.00, 2022)
Misuse of Bijection
Leslie Green CEng MIEE 9 of 11 v1.00: May 2022
an infinite number
We need to make one tiny change by defining (please pronounce this as
Fi) as a number, much larger than any finite number you care to pick, this
finite number not itself being a power or function of .
If you like, is infinite but in a definite rational way.
|| = |{ 1, 2, 3, 4, 5, 6, 7, 8, 9, … , }| =
Be careful here: 0 ( and )
All of Cantor’s notations for infinities, cardinal numbers, countable
infinities, enumerable sets, and so forth are rejected, never to be
mentioned again. does not obey the rules of transfinite cardinals as it is
to be considered as a natural number. The set of even numbers, E, is
trivially counted as follows:
|E| = |{ n : n 0 (mod 2) }| = /2
We no longer have to consider density as a measure of the size of infinite
sets as we can evaluate the results directly.
|A| = |{ n : (a 0, b ) a < b, n a (mod b) }| = /b
You could trivially explain this to school kids (albeit without the set-builder
notation), and they would be very happy that ‘infinity’ behaved quite
When I first devised as a means of solving a problem involving infinite
it was not clear that could be considered as a natural
number. Initially it was a more ‘aloof’ symbol which followed its own set
A New Understanding of Numbers, (2021), Green. L.O. (v1.01, 2021)
Re-Learning to Count to Infinity, (2015), Green. L.O. (v2.00, 2021)
of rules, the -rules. Over the intervening years the -rules have had to
change only slightly as difficulties arose, and were suitably dealt with.
Fundamentally, the -rules are based on the idea that is a very big
(‘infinitely’ big) number. As an example, ² + = ². This is not
considered as an approximation since the relative error is smaller than any
value you wish to consider as significant. Some may prefer to write this as:
² + ²
using the asymptotic equality sign.
When you subtract equal multiples of equal powers of , you can expose
smaller powers of that were hidden by the first order simplification
shown above. These technicalities are fully written up elsewhere,
need not concern us here.
What we are interested in here is the quantity of real numbers in the open
interval (0, 1), and in the open interval (0, ). The first thing to note is that
(0, ) is strictly not an interval on the real number line!
(0, ) is not a sensible or valid interval. It is essentially meaningless,
mathematically, although it is evidently ‘accepted’ usage. However, we
can re-frame the interval into valid mathematical language simply by
writing (0, ). We can then write the following powerful and informative
|(0, 1)| = |(0, )|
The TRUE Mathematics of Infinity, (2019), Green. L.O. (v1.20, 2021)
Misuse of Bijection
Leslie Green CEng MIEE 10 of 11 v1.00: May 2022
|(0, 1)| gives the count of half-open intervals which constitute the open
interval shown. (They are not ‘points’, as mentioned earlier, since
points have no measure, and therefore cannot fill an interval of
measure 1.)
That equation was not a choice, it was a consequence of three earlier
(1) The density of reals is constant in each unit interval of the
(2) The cardinality of the set of natural numbers is .
(3) The natural number line is of the same length as the real
number line. (In effect, is simultaneously the maximum
natural number, the maximum rational number, and the
maximum real number.)
By way of contrast, there is no fixed or pre-ordained value for |(0, 1)|,
even though our reference text claims
that || = |P()|
This same text is remarkably reticent at stating the value of the cardinality
of the power set of an infinite set (the set of all subsets of an infinite set).
For a finite set, A, we have |P(A)| = 2|A| given as Fact 1.4 (Book of Proof,
page 15).
So we come back to the open interval (0, 1) and consider what quantity of
real numbers there should be in that interval. Certainly the amount is
some infinite value, but hopefully you have realised by now that this is
only a vague and qualitative description.
Book of Proof, Theorem 14.11, page 288.
Which of , /10, /2, , , ², ³ is the most reasonable infinite
value to pick? Any positive power of is infinite. Even log() is infinite.
|(0, 1)| is a choice, a definition.
Back in 2015 I chose |(0, 1)| = , for simplicity, and for symmetry with the
count of natural numbers. It means that there are ‘infinitely many’ real
numbers in the interval between 0 and 1. Why would you need any more?
It is also agnostic as to base. Having sub-intervals does not mean that
the reals have decimal digits, or binary digits. Those are different
levels of infinity.
We therefore have |+| = ² (and not 2 or 10).
With half-open intervals forming the real number line from 0 to 1 it is
clear that we ‘only’ have log10() decimal digits. On the other hand,
log10() is as ‘infinite’ as you care to make it. There is no possible case
where you are lacking in digits.
Just to wrap things up we could mention that it is reasonable for a natural
number to be represented by a mathematical point on the number line.
However, since the continuum is made up of infinitesimal sub-intervals,
you might like to think of a real number as being some slightly more
spread-out value:
The Hierarchy of Infinities, (2019), Green. L.O. (v2.10, 2019)
Misuse of Bijection
Leslie Green CEng MIEE 11 of 11 v1.00: May 2022
There is no case where a bijection between infinite sets can be relied upon
to compare their cardinalities. However, infinite sets can be compared
directly in terms of their element-count using the infinite number, . The
result is then directly comparable with the result obtained using the
asymptotic density method.
Whilst it is hard to intellectually understand the difference between
cardinalities of say log(), , , ², ³, 2, 10 and so forth, it is
easy to say that: log() <  < < ² < ³ < 2 < 10, a steadily
increasing hierarchy of infinities.
The count of real numbers in the interval (0, 1) is a choice, not an absolute
The Continuum Hypothesis is an outdated idea since the Chart of
plots a large array of both smaller and larger infinities than the
cardinality of the natural numbers, as well as the infinite values of the
Euler Zeta function.
The Hierarchy of Infinities, (2019), Green. L.O. (v2.10, 2019)
Version History
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