Why not avoid all the redundancy by putting b1 in the next room aN+1?

If you were religious, you would know it's the Devil.

A sample portion of an infinite list as proposed by Cantor.


s1 000001...

s2 101010...

s3 001100...

s4 100000...

s5 100001...

s6 100111...

s7 101100...

s8 000110...

s9 101010...


Diagonal sequence formed from list starting with s1, then s2, then s3, etc.


d1 110110...

d2 011101...

d3 111011...

...

If d1 is not in the list, neither is d2, and all that follow.
For every s there is a corresponding d missing.
Therefore half of the list is missing!

We are discussing a point as being dimensionless. The stake, blob, pixel, is used to give the point some visibility. The point has to be associated with a physical object in order to be useful.

This seems similar to the Schrodinger's cat example.
The uncertainty lies in the radioactive sample, not the cat.
The uncertainty lies in the dynamics of the toss, not the die.

Metaphysician Undercover;

You should be aware that the mind is image oriented. It creates, analyzes, and stores images. Vision is the dominate sensory input from the external world. It's the nature of the mind to form the simplest images possible to represent things outside the mind.

The purpose of abstraction is to eliminate detail irrelevant for our purpose. A simple example, children use this form of abstract representation with their 'stick' figures for people. Thus we use ideal lines, circles, cubes, etc. to convey information to others. A social benefit is realized via storytelling.
This is most obvious in numbers used for counting, assessing the multiplicity of a collection of things. The numbers exclude all attributes of the things being counted.

As for the 'point', it too is an abstraction to serve as a location/coordinate. A surveyor places a stake as a marker/point for a property line. Being dimensionless, you can't see it, but an object is provided in the form of a marker, blob of medium on a surface, pixel on a screen. The point is somewhere within that marker/blob/pixel. This is not a problem since any calculations requiring the location will not vary to any significant degree. The same situation for the 'line' having no width. In graphics it's a continuous marker, in surveying it might be a laser. The line formed from points is a contradiction of terms since a point has no extent. How many zeros are added to a register to accumulate 1? We don't see trajectories or orbits either, but they still serve a purpose.

The continuum is another story.

An interesting quote by Poincare, The Measure of Time, 1898

"We helped ourselves with certain rules, which we usually use without giving us account over it [...] We choose these rules therefore, not because they are true, but because they are the most convenient,...
In other words, all these rules, all these definitions are only the fruit of an unconscious opportunism.“

They think aleph-naught is, or acts like, just another integer, — tim

Infinity is not an integer, or any type of number, and definitely not a quantifier.
It's a relation/condition of having no limit or boundary. Measurement requires boundaries. The application of 'rules of measurement for finite objects' cannot be applied to 'infinite objects. 'To measure an immeasurable object' is a contradiction of terms No one has yet explained how to measure a stick with one end (of infinite length). His extended alephs depend on his diagonal argument which is false, since he doesn't explain why the complementary sequence is already there before he constructs the one that isn't there! His work in other areas of math may be satisfactory, but there is a category for his 'supernatural revelation' of transfinite numbers, 'recreational mathematics'. He was a self appointed spokesman, and always anxious about his status in the mathematical community. It helps to research the person and their motivation before and during their theorizing. If you propose an idea from an authority figure, that would surely promote credibility. Then use symbols from the Hebrew alphabet! It sounds more like a public relations promotional strategy than a mathematical conception.
There is no issue of comprehension concerning 'infinity' as a useful abstraction, as in most of science, vs reality. The issue is knowing the difference.
I still haven't seen the average household with 2.3 children.

A quote by Cantor in his favor:
Ewald, W., From Kant to Hilbert, Oxford 1996.

"The old and oft-repeated proposition “Totum est majus sua parte” may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts “totum” and “pars”. Unfortunately, however, this “axiom” is used innumerably often without any basis and in neglect of the necessary distinction between “reality” and “quantity”, on the one hand, and “number” and “set”, on the other, precisely in the sense in which it is generally false. [An] example may help to explain. Let M be the totality (n) of all finite numbers n, and M¢ the totality (2n) of all even numbers 2n. Here it is undeniably correct that M is richer in its entity, than M¢; M contains not only the even numbers, of which M¢ consists, but also the odd numbers M¢¢ . On the other hand it is just as unconditionally correct that the same cardinal number belongs to both the sets M and M¢. Both of these are certain, and neither stands in the way of the other if one
heeds the distinction between reality and number."

The issue is the present day interpolation/misinterpretation 'there are as many even integers as integers'. The correspondence is a quantitative comparison of two sets. It is not a comparison of two classes of integers.
M could be the set of traditional married people.
W could be the set of married women.
P could be all male-female pairs formed from M
A 1-1 correspondence from P to W, does not mean 'there are as many women as people'.

If 'the cardinality of N and E are equal', means there are as many elements in
N as E, I can accept that. That however does not justify the statement 'there are as many even integers as integers. The first statement is about a measure of a set, the second is about the identity of the elements, which contradicts statistics. Math concepts should be consistent.

merely indicative that it is an illogical/impossible concept. — Devans99

That's the short version of my thoughts.

Indeed, but what he says is valid only as long as nothing travels faster than light. If we ever find something that travels faster than light, we could use it to measure the one-way speed of light, and then his postulate may not remain valid. — leo

The clock synch convention sends light signals in opposite directions to equally distant clocks. A typical outside observer describes the light transit times as a long out and short return forward and a short out and long return backward. This is 'skew' symmetry, which says, the forward path rotated 180 deg equals the backward path. Assuming in one direction (forward out) light speed <c with a delay dt, then the backward return will also have a delay dt. I.e.any variation is not detectable with this symmetry, regardless of propagation speed

The set of natural numbers N = {1, 2, 3, 4,...}

The set of even numbers E = {0, 2, 4, 6,...}
Putting all the fancy terms aside, by inspection, in some cases there are even integers in N corresponding to even integers in E. That is a one to one.correspondence only if you ignore the identity of the elements. Eg. there are as many pcs of trash in bag 1 as in bag 2. Then,removing E from both sides, D (odd) = {}..

If so, I would agree - infinity is impossible so God cannot be infinite. The bible says God is infinite - without any justification - and Aquinas ties himself in a logical knot trying to justify that claim. — Devans99

Eternal: without beginning and end.
That should qualify as ‘infinite’ or without limit.
Neither Aquinas nor any other human has any concepts to understand eternal or infinite.
Try explaining television to your dog, and you may get the idea.

No, it has to be one way or the other, there is a way that reality is even if we don’t see the whole of it, otherwise everything both happens and not happens at the same time, everything both exists and does not exist at the same time, and everything stops making sense. — leo

[You are conflating the existence of an object with the knowledge of that object. Since our primary sensory input is vision, our awareness requires light transit time. For 'local' events, there is no significant delay. For distant objects their existence becomes less certain, based on your knowledge. A natural disaster, terrorist attack, etc., could occur. A star 100 ly distant may not be there, after becoming a nova, 90 yr ago. People make an assumption based on the condition that nothing new happens. Someone you know passes away. You assumed he was alive, when he wasn't, until you got a call making you aware.]

[In the 1905 paper by A. Einstein, he uses a simple example of electromagnetism, requiring only relative motion of the magnet and coil to produce the effects. It's not rocket science, just fundamental physics.
1. The 2nd postulate states, 'the speed of light is constant and independent of its source'.
The 'independent' is the most significant property. It is equivalent to, 'events don't move', thus light is emitted as if from a fixed position in space, the same environment as the Lorentz ether, and allowing the 'fixed stars' or the cmb to serve as a ref. frame. There are no known experiments that can reveal any differences in SR or LET.
2. The 1st postulate states 'the laws of physics are the same for all inertial reference frames.'
When A and B are in relative motion, A will conclude The B-clock rate is slower than the A-clock, and B will conclude The A-clock rate is slower than the B-clock.
That is not a contradiction if you understand postulate 1. No one would complain if both clocks showed the same time, but it's not about whether the times differ, but whether each observes the same physics! In reading Einstein's work, he includes a disclaimer:]
"That light requires the same time to traverse the same path A to M as for the path B to M is in reality neither a supposition nor a hypothesis about the physical nature of light, but a stipulation which I can make of my own freewill in order to arrive at a definition of simultaneity."
Relativity The Special and the General Theory
Albert Einstein 1961 Crown Publishers Inc. pg 23
[If an observer in an inertial ref. frame can assume a pseudo rest frame, then he would expect light transit times to be equal out and back. The assessment of the distant clock requires a poll using light to get a clock reading. Both clocks run at a constant rate, but the transit times are not equal, so the assigned times vary. The observer is coincident with the emission and detection, thus knows the event times accurately. The more distant the reflection event, the more uncertain its time.]
--------------------------------
There is no absolute ref. frame. Light speed is finite, thus universal time has been replaced with subjective time. Position is relative therefore motion must be relative. Newton was wrong about two states, motion or rest. Rest is a special state of motion when two ref. frames have the same velocity. They can be moving relative to other ref. frames, while simultaneously being at rest relative to each other.
Motion modifies measurement and perception. The world wasn't ready for that, or to replace absolute values with relative ones, and all are still not convinced.

A measuring rod is divided into arbitrarily defined standard units.
Spatial measurements are made for an object by corresponding locations of the object to the rod.
A clock is a periodic standard event (tick) generator. It allows a measurement of the interval between events of interest in terms of ticks, in addition to an ordering of those events. Time is a human concept of convenience.

This is common to forums, which over time develop a collective attitude. It's a form of censorship which empowers people to remove whatever they don't understand,or don't want to accept.
In most cases, the questionable content dies a natural death, via lack of interest or disproof on the basis of established evidence. In a democracy, isn't everyone allowed a voice?

My notes from forum exchanges on Special Relativity, and it's role in 'time'.
What is time?

The operational definition of assigning a time to an event as mentioned by A. Einstein in his 1905 paper is essentially what it is, and how it's been done since humans appeared.
It is a correspondence convention, i.e., assigning events of interest to standard clock events, a measure and ordering of activity, with 'time' always increasing/accumulating.
It is an accounting scheme developed out of practical necessity, for human activities like agriculture, business, travel, science, etc. The unit of measure for time initially referred to relative positions of astronomical objects, stars, sun, and moon, which implies earth rotations and earth orbits. The year equates to the periodic motion of the earth relative to the sun, the month, the moon relative to the earth, and the day, the earth rotation relative to the stars. All units of time are by definition, involving spatial motion or distance. The clock further divides the day into smaller units of measure. The reference in the 1905 paper of the watch hand to a position on the watch face involves nothing more than counting hand cycles (hand motion of specific distances representing subdivisions of a day). Current scientific research requires clocks that generate smaller and more precise periods than those of the past. The second is defined as n wave lengths of a specific frequency of light. Note "n wave lengths" is a distance, but labeled as "time".

If we use a light based clock to time the speed of an object along a known distance x, what are we actually doing?
We are comparing the simultaneous motion of an object to the motion of light for a duration (number of ticks). The result is a ratio x/s = vt/ct = v/c or speed. It should be obvious that the ticks serve to correlate the positions of the object with the positions of the light signal, for simultaneous comparisons. If you use Minkowski space-time diagrams the vertical scale is not 'time', but ct, light path distance, i.e. they plot speed. This allows a simple comparison of equivalent entities, without consideration of the nature of those entities.
In summation: A clock provides a beat or rhythm via a periodic process, to coordinate and measure events.

quotes by the author of SR
From 'The Meaning of Relativity', Albert Einstein, 1956:
page 1.
"The experiences of an individual appear to us arranged in a series of events; in this series the single events which we remember appear to be ordered according to the criteria of "earlier" and "later", which cannot be analyzed further. There exists, therefore, for the individual, an I-time, or subjective time."
page 31.
"The non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space coordinates with the time coordinate."
page 32.
"Finally, with Minkowski, we introduce in place of the real time co-ordinate l=ct, the imaginary time co-ordinate..."

time and perception
Subjective time requires memory, which allows a comparison of a current state to a previous state for any changes, which lends itself to an interpretation of time flowing. Patients with brain damage to specific areas involved in maintaining a personal chronology, lose their ability to estimate elapsed time, short or long term. Consider the fact that people waking from a comatose state, have no memory of how much elapsed time, whether hrs, days, or even years.
Consider one of the greatest misnomers ever used, 'motion pictures' or 'movies', where a person observes a sequence of still photos and the mind melds them to produce moving objects where there is no motion. These cases show time as part of perception. Special Relativity then predicts alteration of measurement and perception via motion.

misc.
It was Minkowski who advocated the mathematical manipulation of the expression for the invariant interval from an equality to a generalized form of four variables, producing space-time. I refer to the Minkowski version of SR as a 'lines on paper' theory. Time is represented as a line, removing any attributes that would distinguish its identity from other variables, a line is a line.

Math equations that express a behavior as a function of time, are misleading when the time is interpreted as a causative factor. The time of an event must be assigned after the event occurs, i.e. after awareness! If a nova is observed in 2010, and is 100 ly distant, it didn't happen because it was 1910 on earth. It was the physical processes already in place that reacted to an unstable state. A person dies, not because it's his 'time', but because his biological system reaches a state that can't be maintained.

Which brings us to the real issue (for me) perpetuating the millenia of debating 'time'.
No one wants to be informed "atomic clock at NIST has a hole in it and time is running out". Time implies longevity. People gain some sense of security if they think there is an invisible entity behind the scenes arranging and scheduling more events.

Human experience does not contain anything without end/limit.
Finite provides boundaries that allow measurement.
Mathematics tries to manipulate the imaginary (mental) concepts using methods designed for the finite.
Am still waiting for anyone to measure a one ended stick.

Probability, in my opinion, has to be objective or real. By that I mean it is a property of nature just as mass or volume. So, when I say the probability of an atom of Plutonium to decay is 30% then this isn't because I lack information the acquisition of which will cause me to know exactly which atom will decay or not. Rather, radioactivity is objectively/really probabilistic. — the mad fool

[Probability is a human procedure based on statistics of past events, to predict future events. The mechanism (physical laws regulating behavior) of radioactive decay is not fully understood. We could speculate that space, full of radiation, is a factor. Probability compensates for lack of knowledge, by giving the most expected outcome.
In weather forecasting, there are so many variables, it isn't possible to know their current state, in such a dynamic system. This results in weather forecasts being very local and short term.]

If you agree with me so far let's go to my example: person A who doesn't have knowledge of the initial states of each dice throw and person B who has.

The fact of the matter is that, experimental probability? the outcomes of a throw of a dice, say done a 100 times, will be an almost perfect match with the calculated theoretical probability. For instance the probability of a dice throw with outcomes that are odd numbers is (3/6) or 50% and if you do throw the dice 100 times there will be 50 times the dice shows the numbers 1, 3, 5 (odd numbers).

This match between theoretical probability and experimental probability is "evidence" that the system (person A and the dice) is objectively/really probabilistic.

However, person B knows each initial state of the dice and can predict the exact outcome each time. — the mad fool

[No he can't. If B could predict the exact outcome, there would be no reason for probabilities, and there would be no 'game of chance'. To clarify the issue: in the process of throwing a die (singular), B does not KNOW the microscopic processes affecting the 'throw'. He hasn't refined his analysis to include factors he omits as insignificant, or there are factors he is not aware of (weather people only recently be came aware of ocean currents affecting weather patterns), or he can't monitor known factors fast enough to revise his initial prediction. Knowing the initial state does not determine an outcome with certainty. The outcomes of die tosses does not cluster around the 50% value, but has a range of +/- 49%. I see Leo has touched on this.
Let's focus on the fair coin toss, with H or T. If a coin is tossed 100 times, the outcome can be 100H or 100T, or any combination of the two totaling 100. Randomness requires that all the factors affecting the outcome are present to approx. the same degree, no bias, no dominate factor. Then the essential factor that must NOT be present, memory.
Each toss is independent of the others, and is independent of time. That means you can't predict when an H or a T will occur. This also allows for 100H in a row, with the popular response, 'but that can't happen if the odds are 50% for H'. The protester arguing 'it's such a rare event', he doesn't expect to see it. If it can't occur in his life time, or that of people before or after him, when can it happen? Time is not a factor, so a 'rare event' can happen anytime.]

The circle is present in many natural forms, since the beginning of humanity. The moon, an egg, a flower, a coin, pottery, fruit, tree rings, etc. It should be no surprise that the symbol 'o' should be used to represent a boundary (in its simplest form), as applied to some of the items listed. The idea of a boundary is easily applied to games, with all legal actions restricted to the interior. The idea is also applied in abstract logic (venn diagrams or sets). It's a simple step to containment, as in set theory. Quantity then is reduced to adding or removing elements from a container. Historical records show zero '0' used as a place holder by different cultures, at different times. A convention for accounting is difficult to trace since all cultures participated in trade, and all methods were not recorded. As in the modern world, the one who publishes first, gets the credit.
I agree with the set theory definition of {}, the empty set, but not operations applied to {}. The container is not equivalent to the elements it contains. A cup can contain water, wine, milk, coffee, etc., but the cup is none of these things. Additionally, the elements of a set are qualified by definition stating required attributes.

The weightage of some outcome is dependent upon the information you have about present conditions and the effects they leave in some future moment, and the amount of outcomes we are talking about - like rolling a six-sided dice vs a 20-sided dice. Because the 20-sided dice has more "possible outcomes" than the six-sided one, the probability of any particular side being on top decreases. This is all the result of our ignorance. If we weren't ignorant of the facts of the present conditions and the effects they lead to in the future, then there would be no such thing as possible outcomes. The one and only outcome would be known. — Harry Hindu

Experiments are typically done in an isolated environment to eliminate outside influences. Even with a mechanical tossing arm, at a microscopic level, it doesn't impart exactly the same impulse to a die each toss. Thus you don't have complete knowledge of the die state,unless you monitor the complete process, which itself introduces extraneous factors. You can only know the past!

The real numbers are at the center of both practice and intuition. While you have focused on Cantor, the nature of the real numbers as theoretical entities is just as strange and philosophically questionable. Why aren't you railing against the 'superstition' that 2 has a square root? Have you ever seen it? It's a theoretical construction — ee

I wouldn’t label it ‘superstition’, but an abstraction, like point, line, circle, the continuum, etc., all mental constructs for purposes of measurement. They are practical conveniences

I would just flip bits along the diagonal and have a sequence they didn't include on their list. — ee

[Direction is not a factor in forming a sequence.]

One can think of it as a game with symbols. — Eee — ee

[A well known mathematician, whose name escapes me, when asked to define mathematics replied "a manipulation of symbols". I was impressed by such a concise and accurate description.]

a quote by Cantor, Source:  Ewald, W., From Kant to Hilbert, Oxford 1996.
"[… Mathematics is in its development entirely free and is only bound in the self-evident respect that its concepts must both be consistent with each other, and also stand in exact relationships, ordered by definitions, to those concepts which have previously been introduced and are already at hand and established.  In particular, in the introduction of new numbers, it is only obligated to give definitions of them which will bestow such a determinacy and, in certain circumstances, such a relationship to the other numbers that they can in any given instance be precisely distinguished.  As soon as a number satisfies all these conditions, it can and must be regarded in mathematics as existent and real
"… the essence of mathematics lies entirely in its freedom".]"
We know what Cantor thought.
Cantor was most concerned with his standing in the mathematical community, beyond his bouts of depression. In publishing papers on the nature of infinity through his transfinite numbers, he claimed it was a supernatural revelation. Maybe expecting acceptance through authority, which does occur even in modern times. The well known mathematicians of that period didn't accept his ideas initially, so skepticism is not new.
He associated the truly infinite with GOD. He should have left it there.

An actuary can predict, from a group of 100 senior people, 10 will die within 15 yrs.
He just can't specify which individuals. Statistics is relative to group behavior.
Predicting die toss outcomes relies on the same type of statistics.

Tim;

'indoctrinate'
cause to believe something: to teach somebody a belief, doctrine, or ideology thoroughly and systematically, especially with the goal of discouraging independent thought or the acceptance of other opinions
Somehow this word seems appropriate, if applied to yourself.
First, there are no experts, just some people with more experience than others.

The example in question:
N={1 2 3 4 5 6 7 8 ...}
E={2 4 6 8 10 12 14 16...}
A one to one correspondence of E to N, with the conclusion 'there are as many even integers as there are integers'.
Why are the same integers on both sides of the correspondence?
1 is linked to 2, and 2 is linked to 4, i.e. all even integers have 2 links, all odd have 1 link, and each integer is unique.

And with that you might show us how you can 'approach infinity'.

Your post helped me to clarify my view.
If 'there are as many even integers as integers' is replaced with 'my new set has as many elements as E', then it makes sense.

I read Cantor's biography. It helps to understand his motivation to write.
I agree, ignorance is abundant.

"B knows the initial states" But he cannot know the future with certainty. Some factor that will intervene causing variation. The Neil Armstrong moon landing, a classic example of human intervention, when needed. There is an issue in quality control methods that repeated operations don't produce 'exactly' the same results every cycle. The variations are classified by a level of acceptance/significance.
This problem has its roots in an ideal world of absolute values. Current research denies this.
Re:quantum physics, 'It's not the physics that is strange, but the expectations of the researchers'.

Here is a paper that questions the 'diagonal argument'.
https://app.box.com/s/vdop6iqhi8azgoc2upd76ifu8zacq8e4

When JFK was assassinated, the general population could not accept that an ordinary individual could remove a popular public figure, so some thought it must be a conspiracy. I was never an advocate for that. Tragedies don't discriminate.
I said 'many', not 'all'. SR is a great theory, and has so much experimental support, why is it still questioned.

I don't presuppose a finite stick. They are the norm. Given the definition of infinite, without limit or boundary, and the fact that infinite is not a number or quantifier, a one ended stick is not measurable/quantifiable. You already described the futility of such an effort.
Cantor was an illusionist and fooled many people.

Can't elaborate on my response. There is no human experience with 'infinite' unbounbded/without limit entities. Cantor was an illusionist, who fooled many people. That's it.

The statement " there are as many even integers as integers",
is typically demonstrated using a 'one to one' correspondence as shown.
N: 1 2 3 4 5 6 ...
E: 2 4 6 8 10 12 ...
This is contradicted by:
1. Random sampling of integers results in an average of 50% even E, 50% odd D.
Statistics can be verified in the real world, and is useful in applications of probability.
2. In the above example, removing E from N leaves D, removing E from E leaves nothing, so where is the logic? An odd feature of this example is the appearance of the same integers in both sets.
The 'bijection' for example 1 defines y=2x, as a mapping from N to E. The results are not about the size of sets, but the definition used for mapping.

Representing the first 'one to one' correspondence above in a rectangular form, partitioned into subsets:
1 2 4 8...
3 6 12 24...
5 10 20 40...
...
The odd integers, all listed in column 1, are paired with the column of even integers to the right.
The remaining even integers in each column are paired with the column to the right.
The pairing is independent of direction.
The odd are paired 1 to 1 with an even.
The first subset of even (col 2) is paired with odd (col 1) and even (col 3).
The remaining even are paired with two columns.
Not all pairings are 1 to 1, and each integer appears only once.
A true example of a 1-1 correspondence, "there are as many even integers as odd integers "
D: 1 3 5 7 9 11 ...
E: 2 4 6 8 10 12 ...
A set without limit (infinite) is not measurable, since boundaries enable measurement.
I have a straight stick with one end. Can anyone tell me its length?

'S married to S' is not defined. It would allow same sex marriage.

"Why does a system whose outcomes we can actually predict behave as if we can't do that? That's what bothers me. Thanks."
In an ideal world, human knowledge would contain complete understanding of how the universe works.
Lacking that, we rely on models, representations, and approximations to the physical behavior of the world. The idea that knowing the current state of the universe, by knowing the positions and motions of every element, is an impossibility, since that awareness is always historical.
The rules (approximations of laws) for tossing dice are incomplete. Predictions are based on statistics, a history of past events.

It is a real physical object as it exists in the mind, a neural pattern, in the field of medicine. The problematic term is ‘infinite’. The mind has no experience with things without boundaries (the definition of that term). In the field of mathematics, ‘infinity’ is not a number or quantifier. It’s a relation for a set without a limit.
A more meaningful adjective for the set of integers is extendible. Using Peano type axioms of formation, we can always make a larger integer, but never a largest integer.
This suggests it is the process, not the object that is without limit. Since the symbol ‘1’ for the unit represents an immaterial abstraction, we can use as many as needed from an inexhaustible supply. The set of integers will manifest itself as a finite set for as long as the forming process continues. As you said, human thought is only familiar with things having boundaries. We can only measure things with boundaries.
Our world is based on abstractions of the mind, since we can’t comprehend the reality of it.

The reference set, eg. the set of integers, is a mental construct, used in the process of counting, a practical convenience. Counting is the most fundamental process of measurement, the answer to 'how many'. The nature/identity of the elements is a matter of definition, what attributes must the elements have to be a member of a set.

The symbol '4' represents the multiplicity/quantifier of a set of elements. The quantifier of a set, removes all attributes of the elements, color, gender, age, etc., i.e. any identity.
Thus '4' is a reference set to match one to one to an unknown set to determine its 'size' or quantity.

If position is relative to an object, then so is motion.

Newton stated an object at rest and an object in motion remain in that state unless acted on by a force.
Position of an object is relative to a reference object.
Motion is a change in position, thus relative to a ref. object.
Speed is the rate of change of motion, which has a range of 0 to light speed c.
We measure motion to determine rest, which is the absence of motion, just as dark is the absence of light, and dry is the absence of moisture. Thus there is no um for rest, and rest is also a relative state.
Newton defined two states of motion.
Let's redefine rest as the special case of two objects that have the same velocity.
Each object is at rest relative to the other, while simultaneously being in motion.

Neither Newton nor Lorentz suggested a unit of measure for 'rest'.
Measurement is the validation tool of science.
How do you measure 'rest'?