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The 1st rule of morals is to be symmetric. At the instance you have approved massacring, you have lost the right to object to being massacred
Topics
About:
truth
About:
fairness
The table of contents of this article
Starting Context
Target Context
-
The reader will understand that having been wronged never justifies wronging, rather at the instance you have approved wronging, you have lost the right to object to having been wronged or being wronged.
Orientation
There is an article on what being fair is.
There is an article on why bias is adamantly stuck to.
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
Stage Direction
Here is Special-Student-7 in a room in an old rather isolated house surrounded by some mountains in Japan.
1: Having Been Massacred Never Justifies Massacring, Rather Massacring Could Justify Having Been or Being Massacred
Special-Student-7-Hypothesizer
A massacring is going on on the Bias Planet.
Special-Student-7-Rebutter
Again ...
Special-Student-7-Hypothesizer
A tribe that were massacred in the past are massacring now.
Special-Student-7-Rebutter
... Have they not learned that massacring is bad through their having been massacred?
Special-Student-7-Hypothesizer
Rather they seem to be thinking that they are allowed to massacre because they were massacred.
Special-Student-7-Rebutter
... Is that supposed to be a revenge?
Special-Student-7-Hypothesizer
Whatever the culprits suppose, the act cannot be construed to be any revenge, because they are massacring a tribe who were not the past offenders.
Special-Student-7-Rebutter
Then, why are they allowed to massacre?
Special-Student-7-Hypothesizer
They have to survive.
Special-Student-7-Rebutter
Yes, I agree, and I also agree that also the tribe being massacred have to survive.
Special-Student-7-Hypothesizer
The pretext for their massacring is that they were attacked by some terrorists.
Special-Student-7-Rebutter
I agree that the terrorists must be brought to justice, but why can children or innocent persons be killed?
Special-Student-7-Hypothesizer
So-called "joint liability"?
Special-Student-7-Rebutter
Then, also the past massacring could be justified as "joint liability", could that not?
Special-Student-7-Hypothesizer
"joint liability" of what crime?
Special-Student-7-Rebutter
If there was even a single killer among the tribe, "joint liability" could be invoked.
Special-Student-7-Hypothesizer
It is quite unlikely that there was no killer among the tribe, as is so among each tribe. In fact, there is a grave fact that the tribe crucified a man on whose teachings one of the most major religion on the Bias Planet is based.
Special-Student-7-Rebutter
Then, we would have no reason to condemn the past massacring, if we accepted the principle of "joint liability".
Special-Student-7-Hypothesizer
We are claiming that the past massacring was absolutely unacceptable, but we are in danger of losing the reason to claim so.
Special-Student-7-Rebutter
What was the reason for the past massacring?
Special-Student-7-Hypothesizer
As an undertone, there was some enduring hatred against the tribe, who had a religion which assumed the tribe to be the chosen people, and also there were some circumstances like that there were many loan sharks among the tribe.
Special-Student-7-Rebutter
"chosen people"? What are "chosen people"?
Special-Student-7-Hypothesizer
I am not perfectly sure but they seem to be the special people chosen by God.
Special-Student-7-Rebutter
A quite barbarous idea, but we have to admit that many ignorant tribes imagined themselves to be "special". I guess that it used to be a rather prevalent phenomenon. Also many Japanese used to imagine that their land was the holy land and kamikaze (holy wind) was imagined to blow away unholy aliens.
Special-Student-7-Hypothesizer
Such idea comes from ignorance and certainly all were quite ignorant and all are still ignorant more or less.
Anyway, we condemn the past massacring as absolutely unacceptable, but that is not because the tribe is "special".
We are condemning any massacring or any killing because massacring or killing is bad, and if we approved (although we never do) your massacring or killing, we would have no reason to condemn your being massacred or killed.
If you bring up the absolutely-unacceptable idea of "joint liability", I point out that there is no tribe (especially yours) so impeccable to escape "joint liability".
We never say this, but a very expected voice is "The past massacring was the right thing to do after all, because they have proved themselves to be a massacring tribe.". I never say that, because I never want to become a beast approving massacring.
2: Stop Roughly Lumping Persons Together
Special-Student-7-Hypothesizer
In fact, "joint liability" is a root of much evil.
Special-Student-7-Rebutter
Do not roughly lump up persons together, we are saying ...
Special-Student-7-Hypothesizer
There are a variety of persons in any tribe or in any category, and we cannot say that the persons of a tribe or a category are such.
Special-Student-7-Rebutter
All the persons of a gender are certainly of the gender, but other than that, we cannot say that all the persons of a gender have a same taste or are stupid or are weak or are bad or are inferior or something, without any firm evidence.
Special-Student-7-Hypothesizer
Certainly, there are differences of averages, minimums, maximums between categories, but there can be the most superior person in a category of a low average, and it is absolutely unjust to treat the most superior person based on the average.
Special-Student-7-Rebutter
We are repeatedly saying that we have to evaluate any individual by looking at the individual, not by roughly lumping up some individuals in a category and assuming that all the individuals in the category are such.
Special-Student-7-Hypothesizer
That die-hard practice of judging by categories comes from the inveterate laziness of most people, who refuse to bother to look at individuals, because that is a bother.
In fact, laziness is the root of all the evil.
Whatever we say, the lazy people will refuse to bother.
Special-Student-7-Rebutter
But respecting the dignity of a human being is to look at the individual, without treating the human being as just a member of a category.
Special-Student-7-Hypothesizer
For the massacring case, do not regard an individual to deserve to be killed just because the individual is lumped up in a tribe.
3: But Persons Who Have Not Raised Objections Against Terrors Are Not Innocent
Special-Student-7-Hypothesizer
But I do not agree to the claim that the every person who did not directly commit the terror is innocent.
If you disagree with the terror, you have to raise objections, otherwise, you are an accomplice.
If you see someone bullied and are silent, you are an accomplice of the bullying.
Special-Student-7-Rebutter
That is a very important point: some bullies inevitably appear statistically speaking, and the issue is whether the others stop the bullies.
Special-Student-7-Hypothesizer
If the massacring tribe claim that being massacred are not innocent because they were silent watching terrors, I in fact agree about the persons who were really silent, although I do not agree that killing is the right punishment, and I point out that there may be some persons who raised objections, who are indeed innocent, and killing such innocent persons is an absolute atrocity, and the children cannot be blamed.
Special-Student-7-Rebutter
Again, do not roughly lump up persons together, we are saying. If there are some (even if many) silent accomplices, punish only them, not the innocent who raised objections.
Special-Student-7-Hypothesizer
I observe that many persons are thinking that being silent is a safe thing to do, but in fact, being silent watching any injustice is being an accomplice of the injustice, and you have lost the right to claim being innocent, which is the most dangerous thing to do in my opinion.
Special-Student-7-Rebutter
You should think of the fact that if you are silent watching a terror, you could be killed as an accomplice of the terror, is that safe?
Special-Student-7-Hypothesizer
In fact, we are publishing this article in order to refuse being an accomplice of the massacring. If you do not do likewise, you are an accomplice of the massacring.
4: Talking Government-Wise
Special-Student-7-Hypothesizer
Talking government-wise, the country who has produced a terrorist against another country has to be the first to condemn the terrorist and try the hardest to bring the culprit to justice.
If the government fails to do that, the government cannot object to be regarded as the boss of the terrorist.
If the people see the government failing, it is the people's duty to condemn the government, and if the people fail to condemn the government, the people cannot object to be regarded as the supporters of the terrorist.
Special-Student-7-Rebutter
Obviously, the victim country would resort to retaliation according to the logic.
Special-Student-7-Hypothesizer
We repeat that massacring is not the right punishment, but we would like to remind you that being a silent bystander could bring you the consequences.
5: The 1st Rule of Morals Is to Be Symmetric
Special-Student-7-Hypothesizer
I have to observe that many people on the Bias Planet do not follow the 1st rule of morals: be symmetric.
Special-Student-7-Rebutter
I wonder whether they even know the rule.
Special-Student-7-Hypothesizer
If you need to object to being killed, you need to object to killing, which is symmetry, which they do not seem to understand.
Special-Student-7-Rebutter
I wonder whether that is a matter of lack of imagination of imagining in others' shoes or just sheer selfishness.
Special-Student-7-Hypothesizer
Are they selfish because they lack imagination or do they lack imagination because they are selfish? ... Well, probably, they just lack imagination and are selfish.
Special-Student-7-Rebutter
Or their logic seems involving their being "special".
Special-Student-7-Hypothesizer
If they are "special", also any other tribe is "special", which is symmetry.
Special-Student-7-Rebutter
They are observing "God"'s commandments, so, they are "special".
Special-Student-7-Hypothesizer
That "God" they call is just their "God"; another tribe are observing the tribe's "God", so, the another tribe are "special", which is symmetry.
Special-Student-7-Rebutter
Someone may say that we are just promoting our own agenda.
Special-Student-7-Hypothesizer
Yes, we are, and so what? Certainly, our agenda is being truthful and fair, and we despise the people who are untruthful or unfair, but we admit their right to promote their agenda and to despise us, which is symmetry.
Look, there is nothing wrong in each one's promoting one's own agenda as far as one does not begin to harm others' basic human rights.
Special-Student-7-Rebutter
One can legitimately despise anyone, because that is about one's own mind, but one is not allowed to harm someone's basic human rights (especially kill) someone just because one despise someone.
6: It Is Indeed Our Business
Special-Student-7-Hypothesizer
There may be someone who say "It is none of your business.", but if we were silent watching a massacre, we would be an accomplice of the massacre, which is very much our business.
Special-Student-7-Rebutter
There are many people who deem that being silent is safe, but that is a blatant short sight: they are risking becoming accomplices of wrongdoings.
Special-Student-7-Hypothesizer
We definitely do not want to be any accomplice of any massacring, and so, we are forced to speak up.
Certainly, we also condemn the terrors that are claimed to be the justification of the massacring.
Special-Student-7-Rebutter
I have heard a claim that condemning the massacring is justifying the terrors, but that is absolutely a perverted argument.
Special-Student-7-Hypothesizer
We are saying that any basic human rights violation is not allowed, whether that is a terror or a massacring.
References
<The previous article in this series | The table of contents of this series |
<The previous article in this series | The table of contents of this series |
It is high time to exorcise the ghost of Protagoras.
Topics
About:
truth
About:
physics
The table of contents of this article
Starting Context
Target Context
-
The reader will be released from the epistemological nonsense: humans determine the reality.
Orientation
There is an article on that what we are supposed to do is to maintain the unboundedly consistent hypotheses system.
There is an article on that to establish the unbounded consistency is the only way to near truths.
There is an article on the distinctions between the reality, observations, and interpretations of observations, and where relativity, ambiguity, or unpredictability could exist.
There is an article on the real trick of Zeno's paradox and how to resist any trick
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
Stage Direction
Here is Special-Student-7 in a room in an old rather isolated house surrounded by some mountains in Japan.
1: The Ghost of Protagoras
Special-Student-7-Hypothesizer
There was a man named Protagoras.
Special-Student-7-Rebutter
"was"? So, he is not now.
Special-Student-7-Hypothesizer
He has been dead for about 2500 years.
But his ghost seems to be still haunting the planet.
Special-Student-7-Rebutter
We should not be any persons who believe the existence of any ghost.
Special-Student-7-Hypothesizer
Of course, I am not talking about any literal ghost, but a mentality promoted by him.
Special-Student-7-Rebutter
What did he promote?
Special-Student-7-Hypothesizer
He is known to have said "Man is the measure of all things.".
Special-Student-7-Rebutter
Well, aside from that it should have been "Man, woman, Klingon, and any other sentient being are the measures of all things.", it does not seem any problem.
Special-Student-7-Hypothesizer
I deem not it any problem, but he is said to have claimed that there is no truth except subjective judgments.
Special-Student-7-Rebutter
We should know that that claim is not any deduction from "Man is the measure of all things.".
Special-Student-7-Hypothesizer
Of course. As the reality is not automatically measured, a sentient being has to measure the reality in order for a measurement to happen, but the reality does not need to be measured in order to be as it is.
The reality is nothing but the truth, which does not require being measured by any "Man".
Special-Student-7-Rebutter
Has Big Bang begun to exist only after "Man" began to measure it? Does the inside of a black hole not exist because "Man" cannot measure it? Will the universe vanish just because humans perished, because the universe will not be measured any more?
Special-Student-7-Hypothesizer
Oddly enough, the ghost is die-hard on the Bias Planet, ruling many Biasians, even many intellectuals like physicists, even now.
2: Certainly, any Measurement Appears Only When Someone Does a Measuring
Special-Student-7-Hypothesizer
A rock exists underground somewhere.
Special-Student-7-Rebutter
Well ...
Special-Student-7-Hypothesizer
It is ridiculous to believe that the rock begins to exist only at the instant a "Man" finds the rock.
Special-Student-7-Rebutter
Well, we need to be clear about what "the rock" means.
Special-Student-7-Hypothesizer
What do you mean?
Special-Student-7-Rebutter
Is a sand grain attached to "the rock" a part of "the rock"?
Special-Student-7-Hypothesizer
I think not.
Special-Student-7-Rebutter
But there are some sand grains that are adhering fast to "the rock" to be took off "the rock".
Special-Student-7-Hypothesizer
Well, it is just a matter of definition.
Special-Student-7-Rebutter
In fact, it will be quite subtle to determine which grains belong to "the rock", because it is a matter of how strongly we try to take off the grain, while if we try strongly enough, any grain will come off.
Special-Student-7-Hypothesizer
Anyway, it is just a matter of definition.
Special-Student-7-Rebutter
I am saying that "the rock" appears only after a "Man" makes the definition.
Special-Student-7-Hypothesizer
That is only about what the "Man" calls what; the reality underground is not changed any bit if the "Man" changes what part he call "the rock".
Special-Student-7-Rebutter
I know.
Any measurement of the diameter of "the rock" (after "the rock" has been defined) appears only after a "Man" chooses a unit.
Special-Student-7-Hypothesizer
Of course: "31.2 cm" appears only after a "Man" measures "the rock", but the diameter is not changed any bit just because the "Man" has chosen 'inch' instead.
Special-Student-7-Rebutter
But strictly speaking, the atoms in "the rock" are incessantly wobbling, and what is the diameter indeed?
Special-Student-7-Hypothesizer
And of course the boundary of each atom is not clear: the electrons are moving around.
But that is only about what a "Man" call what; the reality underground is not changed any bit just because the "Man" has changed the definition of the diameter.
Special-Student-7-Rebutter
I know.
3: It Is a Matter of Interpretation, Which of Course Appears Only When a "Man" Does the Interpretation
Special-Student-7-Hypothesizer
In fact, "31.2 cm" is an interpretation of the reality and does not appear until a "Man" does the interpretation.
It is like a book translated into a language from an original book in another language. The translated book has existed only after someone did the translation.
And importantly, the translated book cannot be completely accurate.
In fact, any coordinate appears only after a "Man" spreads a net as a coordinates system.
Obviously, any interpretation exists only because a "Man" did the interpretation.
Some discourses like ""the rock" begins to exist only after a "Man" finds it and defines what it is." and ""the diameter" begins to exist only after a "Man" measures it by determining the unit and the boundary." is about interpretations, not about the reality.
4: Our Interpretations Are Trapped in Our Intuition
Special-Student-7-Hypothesizer
When we make an interpretation, the interpretation is (at least mostly) trapped in our intuition.
When we think of the spacetime, we interpret it based on the pathetic human intuition built on pathetic human daily experiences.
The coordinates of an electron is an interpretation of "the location" of the electron, where "the location" is a part of the reality, which does not require being interpreted by any "Man".
"The coordinates appear only after a "Man" measured them." is indeed correct, because the interpretation appears only after the "Man" made the interpretation, but "the location" exists independent from any human interpretation.
Special-Student-7-Rebutter
The issue is that "the location" in the reality tends to be confused with the location by an interpretation.
Special-Student-7-Hypothesizer
Probably, "the location" in the reality is something that is quite hard (or impossible) to be imagined accurately by pathetic human intuition.
Something called "spin" by physicists is a somewhat inaccurate interpretation of something in the reality.
5: It Is the Interpretation of the Status of the Reality Determined By Humans, Not the Status of the Reality
Special-Student-7-Hypothesizer
While I often hear the opinion that the status of the reality is determined by humans by some physicists, I find it quite absurd.
Special-Student-7-Rebutter
Probably, the wordings are bad.
Special-Student-7-Hypothesizer
Maybe. As the reality exists without requiring any "Man", the way the reality is in without any intervention by any "Man" is the status of the reality.
Special-Student-7-Rebutter
As the reality as it is cannot be grasped by "Man", "Man" tries to build a somewhat inaccurate model of the reality, and the status of the model seems what they are calling "the status of the reality".
Special-Student-7-Hypothesizer
In fact, there is no such thing as "spin with respect to a direction" in the reality, but they are saying that the status of the reality is determined by humans because "spins with respect to a direction" are determined by human measurements. Certainly, "spin with respect to a direction" is not determined until a "Man" measures it, because it is an interpretation, which does not exist until someone makes the interpretation.
Special-Student-7-Rebutter
As the interpretation is somewhat inaccurate, although the interpretation is somewhat determined by the original, it is somewhat determined by the interpreter, which seems to be what is happening in Quantum Mechanics.
6: It Would Be Meaningless to Do Physics if Humans Were Determining the Reality
Special-Student-7-Rebutter
If humans were determining the reality, why would we not just arbitrarily determine the reality?
Special-Student-7-Hypothesizer
As we suppose that there is the reality that objectively exists, we humbly try to grasp the reality, do we not? Otherwise, we could determine the reality however we liked.
For example, why would we not make a measuring apparatus that read any "spin" 42, and establish the law, "Every spin is 42."?
Special-Student-7-Rebutter
If humans were determining the reality, there does not seem any reason why we should not do so.
Special-Student-7-Hypothesizer
Or we would determine any length to be 42, because why not? as we determine the reality, why should we not determine so?
Special-Student-7-Rebutter
Physics would be meaningless.
7: The Status of the Reality Is the Reality As It Is
Special-Student-7-Hypothesizer
The status of the reality is the reality as it is.
Special-Student-7-Rebutter
Humans measure "the spin" of a particle and some people seem to be calling "the spin" "the status of the reality".
Special-Student-7-Hypothesizer
The reality does not need to be measured by any human, but humans are trying to push the reality into a human model, and some people are calling the distorted reality, which is not really the reality, "the reality".
That distorted pseudo-reality is certainly determined by humans.
References
<The previous article in this series | The table of contents of this series |
<The previous article in this series | The table of contents of this series | The next article in this series>
What do you really mean by "chicken" in "Which came first, the egg or the chicken?"? The exact "chicken" or the species that bore the first egg?
Topics
About:
truth
The table of contents of this article
Starting Context
Target Context
-
The reader will know that the egg came first for chickens, a creature should have come first for the entire life forms, and a creature should have come first for the species that first began to bear eggs.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
Stage Direction
Here is Special-Student-7 in a room in an old rather isolated house surrounded by some mountains in Japan.
1: The Egg Came First, if "Which Came First, the Egg or the Chicken?" Means Really What It Is Saying
Special-Student-7-Hypothesizer
There is a question, "Which came first, the egg or the chicken?".
Special-Student-7-Rebutter
The answer seems obvious, if the question means really what it is saying.
Special-Student-7-Hypothesizer
The question has been regarded to be a very difficult question on the Bias Planet.
Special-Student-7-Rebutter
How so?
Special-Student-7-Hypothesizer
"Any egg should have been born from a chicken, which should have been hatched from an egg, which . . .. Oh, my! Which came first?"
Special-Student-7-Rebutter
Well, is the question really talking about "chicken" as the specific species?
Special-Student-7-Hypothesizer
What if it is?
Special-Student-7-Rebutter
"Any (chicken) egg should have been born from a chicken" is simply wrong, because the first chicken egg can have been (and most probably was) a cross of 2 different species.
Special-Student-7-Hypothesizer
Was the first egg not a mutation?
Special-Student-7-Rebutter
'chicken' was not created in 1 leap by 1 crossing or 1 mutation, but in multiple leaps by multiple crossings and possibly some mutations, while the demarcation of 'chicken' is a matter of definition.
Special-Student-7-Hypothesizer
Anyway, the egg should have come first?
Special-Student-7-Rebutter
It should be so, because the genes are determined in the egg, and the creature has no option but to become as the genes dictate.
Special-Student-7-Hypothesizer
Did not a mutation happen to a creature?
Special-Student-7-Rebutter
A mutation may have happened to an elbow cell of a creature, but such a mutation should not have been propagated to descendants.
Special-Student-7-Hypothesizer
A mutation may have happened to the oviduct.
Special-Student-7-Rebutter
The mutation did not happen to the oviduct, but to a cell of the oviduct, and while it is doubtful that such a mutation to a single cell bore the first chicken egg, even if it did, I will not call the mother the first chicken, because the mother was not particularly a chicken except the single cell.
Special-Student-7-Hypothesizer
So, the egg was the first of chickens, anyway.
2: It Should Have Been a Rather Common Knowledge Since a Long Time Ago
Special-Student-7-Rebutter
I am rather amazed by the fact that the question has been deemed to be difficult.
Special-Student-7-Hypothesizer
I understand that some Biasians have been doing crossings since a long time ago, and probably, 'chicken' was created intentionally by some breeders (of many generations), who should have known that the egg comes first.
The situation is the same for plants: we get a seed as the first of a new species by pollinating a pistil of a species by some pollens of anther species; no new species grows from a seed of an existing species.
Special-Student-7-Rebutter
I cannot help but assume that the knowledge was rather common among some people, even before the knowledge of genetics.
3: What if "Chicken" Really Means 'Creature'?
Special-Student-7-Hypothesizer
"chicken" in the question does not really mean "chicken" but 'creature', I guess.
Special-Student-7-Rebutter
If so, as far as we are based on the evolution, the question should be really "Was the first life form an egg or a creature?".
Special-Student-7-Hypothesizer
Also the answer to that question seems obvious, considering the fact that all the primitive species do not propagate by eggs. Obviously, 'egg' was a later invention as a way of propagation.
Special-Student-7-Rebutter
As far as we are based on the evolution theory, it seems an agreed hypothesis that the first life form was a single cell life form, which we do not call egg, and the life form propagated by just dividing.
Special-Student-7-Hypothesizer
So, the creature was the first.
4: What if "Chicken" Really Means 'the First Species That Bore the First Egg'?
Special-Student-7-Hypothesizer
'chicken' may mean 'the first species that bore the first egg'.
Special-Student-7-Rebutter
Then, it was not really chicken, but a kind of fish or an even more primitive species.
Special-Student-7-Hypothesizer
Whatever the species was, there was the first mother that bore the first egg.
Special-Student-7-Rebutter
The issue is which of the mother and the egg should be called to be the beginning of the species.
Special-Student-7-Hypothesizer
While it is a matter of definition, ultimately speaking, but it will be far more reasonable that the mother is called the beginning.
Special-Student-7-Rebutter
Why so?
Special-Student-7-Hypothesizer
Because the mother should have had some genes that made the species propagate by eggs, because why else did the mother bear the egg?: as we discussed before, a mutation to a cell of the mother is unlikely to have made the mother bear an egg, and even if it did, the mechanism had to be encoded into the egg as genes, which is quite unlikely unless the mother already had the genes.
The natural hypothesis is that the genes were generated in the mother when it was born, as an accident of the dividing operation, and so, the mother bore the egg which included the genes..
Special-Student-7-Rebutter
The reason why the hypothesis that the genes appeared first in the egg is unlikely is that then, the mother just accidentally bore the egg and the egg just accidentally contained the new genes, which is a very unlikely accumulation of double accidents.
On the other hand, the hypothesis that the mother got the genes at born was a single accident.
Special-Student-7-Hypothesizer
So, we are almost sure that the creature was first.
5: The Morals
Special-Student-7-Hypothesizer
The reason why we have talked about the question is rather the morals which we can draw from it than the answer itself.
The first moral is that we have to be clear about any question: it is very futile to talk about a question that is not clear what it means.
Special-Student-7-Rebutter
Especially, every concept in the question has to be clear; what does "chicken" mean?, for example.
Special-Student-7-Hypothesizer
A very typical, but very prevalent type of futile arguments is that the 2 parties are talking about 2 different things, typically using the same term.
An obvious major cause is that most terms are defined very loosely on the Bias Planet.
Another obvious major cause is that most Earthians use (already loose) terms very loosely, while they even boast using terms intentionally out of scope, a method called simile.
Special-Student-7-Rebutter
The 2nd cause seems to be the root cause: they are fine with loose definitions because they use terms loosely anyway.
Special-Student-7-Hypothesizer
Biasians' justification seems to be that "People will understand things without the things' being so explicitly stated!", but that does not seem to be the case, as far as we judge from many cross-purpose arguments among Biasians we witness.
Special-Student-7-Rebutter
I guess that there are some cases in which "People will understand", but most Earthians do not seem to be able to adopt to the other cases.
In fact, I understand that it is natural: languages were created for daily usages which were (at least) mostly cases in which "People will understand", not for precise arguments, and being economical was a major priority: they did not want to have multiple terms if 1 term could be understood in multiple meanings depending on the contexts.
The problem is that that strategy does not work well when things get intricate
Special-Student-7-Hypothesizer
The second moral is that we have to view things in a longer and wider perspective.
Special-Student-7-Rebutter
The assumption "Any chicken egg should have been born from a chicken." may seem to be true in a view in which chickens daily bear chicken eggs and chickens are daily hatched from chicken eggs, but the view is by a very narrow perspective from which the humans history or the life forms evolution is excluded.
Special-Student-7-Hypothesizer
The problem of "alternative truths" is that such "truths" are views in some narrow perspectives and opposing parties are just insisting their own views in their own narrow perspectives.
Special-Student-7-Rebutter
The real truth is only one, certainly is difficult for us to grasp it in its entirety, but what we should do is to endeavor to broaden our perspectives in order to grasp the truth in a more entire form, which is what we are advocating in some of our articles like this and this.
References
<The previous article in this series | The table of contents of this series | The next article in this series>
<The previous article in this series | The table of contents of this series | The next article in this series>
Russell's "paradox" seems just that the naive comprehension axiom was too naive in the 1st place. Why cannot we start with the natural definition of set?
Topics
About:
truth
About:
mathematics
The table of contents of this article
Starting Context
Target Context
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The reader will have a material for thinking about definition of set.
Orientation
There is an article on that what we are supposed to do is to maintain the unboundedly consistent hypotheses system.
There is an article on that to establish the unbounded consistency is the only way to near truths.
There is an article on the distinctions between the reality, observations, and interpretations of observations, and where relativity, ambiguity, or unpredictability could exist.
There is an article on the real trick of Zeno's paradox and how to resist any trick
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
Stage Direction
Here is Special-Student-7 in a room in an old rather isolated house surrounded by some mountains in Japan.
0: A Note
Special-Student-7-Hypothesizer
We are talking about set theories in this article.
Note that we are students of set theories and are not saying that we know everything of set theories; it is very possible that we are mistaken on some or many points.
But a major merit of a student's speaking up what he or she thinks is that professors will know how their discourses can be misunderstood and what more elaborations their discourses can use in order to prevent such misunderstandings. Another merit is that other students can clarify their concerns if they have the same or similar concerns with ours or can solidify their correct understandings if they know we are wrong, by indicating how we are wrong.
As Sherlock Holmes said "I am afraid, my dear Watson, that most of your conclusions were erroneous. When I said that you stimulated me I meant, to be frank, that in noting your fallacies I was occasionally guided towards the truth." in 'The Hound of the Baskervilles', honest erroneous reasoning can be very more useful than just parroting what are taught by authorities.
1: What Is Russell's "Paradox"?
Special-Student-7-Hypothesizer
There is something called Russell's paradox.
Special-Student-7-Rebutter
Good for you.
Special-Student-7-Hypothesizer
Not particularly good for me.
Let us define the set, \(R\), as the collection of all the sets each of which is not any member of itself.
Is \(R\) a member of \(R\) or not?
Special-Student-7-Rebutter
I understand that it is a trick.
Special-Student-7-Hypothesizer
Let you be tricked for a while.
Special-Student-7-Rebutter
Well, if we suppose that \(R\) is a member of \(R\), \(R\) is not any member of \(R\) by the definition of the "set"; if we suppose that \(R\) is not any member of \(R\), \(R\) is a member of \(R\) by the definition of the "set"; oh, my!. . . . Was I tricked well?
Special-Student-7-Hypothesizer
. . . Not well: do not put 'set' in double quotes please, as though you know that it is a fraud.
Special-Student-7-Rebutter
"paradox"? It seems just that not every definition is well-defined: it seems just that such a "set" does not exist.
If that is a paradox, I can think of many other paradoxes easily: for example, "A man has only 2 legs and has only 3 legs.".
Special-Student-7-Hypothesizer
If the man has only 2 legs, the man cannot have 3 legs; if the man has 3 legs, the man cannot have only 2 legs; oh, a paradox! a mystery!
Special-Student-7-Rebutter
Such a man just does not exist.
As we have reminded ourselves in an article concerning Zeno's paradox, implicitly assuming the existence of something that does not exist is a typical type of frauds.
2: It Seems Just That the Naive Comprehension Axiom Was Too Naive
Special-Student-7-Rebutter
Actually, I understand that there was a historical significance in the "paradox": the so-called "naive comprehension axiom" was believed to be valid and the attempts on mathematics based on the axiom were in jeopardy.
Special-Student-7-Hypothesizer
I have heard that there was a movement that was attempting to rebuild the whole mathematics on a set theory.
Special-Student-7-Rebutter
And the set theory at that time was based on the naive comprehension axiom. I understand that the "paradox" was significant.
Special-Student-7-Hypothesizer
For your information, 'naive comprehension axiom' is "Any precisely specified property of member can be used to define a set."; for example, "member is not any member of itself" is accepted to be a precisely specified property.
But in hind sight, what I feel is "Was the naive comprehension axiom not too naive, in the 1st place?".
Special-Student-7-Rebutter
A proposed axiom can be found out to include contradictions, and that is what happened to the naive comprehension axiom.
Special-Student-7-Hypothesizer
Russell's "paradox" seems a matter of "As the naive comprehension axiom includes contradictions, we need to dispose the axiom." rather than any paradox.
Special-Student-7-Rebutter
In fact, there is no contradiction in the universe, and if something appears to be a contradiction, that is just because there is a surreptitious wrong premise you are supposing. When that surreptitious wrong premise is difficult to be identified, the appears-to-be contradiction is called paradox, but in the Russell's "paradox" case, the wrong premise is too obvious for the "paradox" to be called paradox.
3: Why Cannot We Start with the Natural Definition of Set?
Special-Student-7-Hypothesizer
In rather lower education, 'set' is typically taught to be simply any collection for which the membership is unambiguously determined, which means that whatever object in the universe is determined whether it is in the collection or not.
Special-Student-7-Rebutter
It seems a very natural definition.
Special-Student-7-Hypothesizer
By that natural definition, Russell's "set" is just not any set, because the membership is ambiguous.
In fact, this is what anyone who is based on the natural definition will feel: Huh? "paradox"? Isn't it just that the "set" is not any set based on the definition?
Note that Russell's "paradox" is a refutation of the naive comprehension axiom, not of the natural definition of set.
Special-Student-7-Rebutter
All the text books I have read (I do not say "all the text books in the world") do not explain why they deem the natural definition to be inadequate.
I mean, if the natural definition is inadequate, it is fine, but the reason should be disclosed, and we point out that the reason is not being shared, as far as I know.
Special-Student-7-Hypothesizer
In fact, what mainstream set theories are calling "set" is not 'set' by the natural definition, and we need to distinguish the 2 different concepts. So, we will call 'set' in the natural definition 'n-set' hereafter.
Special-Student-7-Rebutter
Is 'n-set' different from 'collection'?
Special-Student-7-Hypothesizer
Well, 'n-set' is in fact the same with 'collection' in our notion of 'collection'; in fact, "collection for which the membership is unambiguously determined" is a redundant expression for us, because the membership of any collection is unambiguously determined, for us.
But I am not sure what 'collection' means for the other persons, because I have never seen 'collection' defined in any mathematical textbook.
Special-Student-7-Rebutter
"collection" seems to be being used to-get-out-of-the-pinch when they cannot use "set".
Special-Student-7-Hypothesizer
Anyway, 'n-set' and 'collection' are the same for us, and we will use one or the other in the same meaning.
4: 'Class' Is Not 'N-Set', Because ...
Special-Student-7-Hypothesizer
Note that 'n-set' is different from so-called 'class'.
Special-Student-7-Rebutter
'class' seems to be used in some different meanings in some different set theories.
Special-Student-7-Hypothesizer
Certainly: 'class' is sometimes informally used in the ZFC theory as a way of making sloppy expressions, which can be really legitimately expressed without 'class' but with 'formula'; 'class' is formerly an essential entity in the NBG theory or in the MK theory, but 'class' in the NBG theory and 'class' in the MK theory seem different.
'class' in the ZFC theory seems something to be just dismissed, because it is needless and just misleading.
We will here mean only 'class' in the NBG theory or in the MK theory, but each such 'class' is different from 'n-set'.
That is because 'class' allows only sets as members and furthermore, is required to have a formula, which is the major point that annoys me about the mainstream set theories.
'class' seems to be avoiding Russell's "set" and similars by the restrictions of allowing only sets as members and having a formula, and so, any class seems to be an n-set, but an n-set may not be a class, because the n-set may contain a non-set as a member or the n-set may not be able to be expressed with a formula.
Special-Student-7-Rebutter
Let us clarify the relation between 'class' and 'formula'.
Special-Student-7-Hypothesizer
If any n-set with only set members can be expressed with a formula, the n-set is a class, which is because of the axiom scheme of class comprehension or the class existence theorem.
And if any n-set is a class, the class can be expressed with a formula, which is because the class can be represented by a variable, \(C\), and the class can be expressed as \(\{p \vert p \in C\}\).
Special-Student-7-Rebutter
Well, is that not a vicious cycle?
Special-Student-7-Hypothesizer
I do not think so: the point is that the n-set is presupposed to be a class and there is no reason why the class should not be represented by the variable.
Of course, we cannot do like that when the n-set is not known to be a class, because the n-set is not guaranteed to be able to be represented by a variable (any variable can represent only a class). So, of course, a n-set may not have any formula.
Special-Student-7-Rebutter
So, we can say that any class is nothing but any n-set with only set members that has a formula.
5: Not Every Sub-Collection of a Set Is Guaranteed to Be a Set
Special-Student-7-Hypothesizer
What is annoying about at least most textbooks on or based on set theories (we have not read all the textbooks, but all the textbooks we have read are so more or less) is that they begin to say like "As this is a sub-collection of a set, this is a set by the axiom scheme of subset.". ... Huh? a formula is required, right?
Special-Student-7-Rebutter
The authors may be supposing that the formulas are obvious.
Special-Student-7-Hypothesizer
But there are many cases for which the formulas are not even remotely obvious, at least for us.
Special-Student-7-Rebutter
Probably we are too dull for the authors.
Special-Student-7-Hypothesizer
I see .... Well, aside from the obviousness of the cases in the textbooks, why are we not allowed to call an unambiguously determined sub-collection of a set without any formula 'set'?
Special-Student-7-Rebutter
Is there such a sub-collection?
Special-Student-7-Hypothesizer
Let us think of an example. Let us think of a diameter of Sun as \([-r, r]\) and the cylinder, \(S^1 \times [-r, r]\), where \(S^1\) is the 1-sphere (the unit circle). Let us think of the sequence of the open cylinders, \(S^1 \times (-r, 0), S^1 \times (-r + r / 2^1, 0 + r / 2^1), S^1 \times (-r + r / 2^1 + r / 2^2, 0 + r / 2^1 + r / 2^2), ..., S^1 \times (-r + r / 2^1 + ... + r / 2^j, 0 + r / 2^1 + ... + r / 2^j), ...\). Let us take the number of the Helium atoms contained inside each cylinder at a specified instant. The collection of the numbers should form a sub-collection of the natural numbers set. The point is that the sub-collection is unambiguously determined. Then, please present the formula for the sub-collection.
Special-Student-7-Rebutter
Obviously, I am too dull.
Special-Student-7-Hypothesizer
Note that although the NBG or MK theory says that every subclass of any set is a set, a sub-collection of a set may not be a set there, because the sub-collection may not be a subclass: if a formula is not presented for the sub-collection, the sub-collection may not be a class, so, not a subclass.
There may be someone who thinks that the power set axiom is guaranteeing the existences of all the sub-collections, but that is not so: the power set is the set of all the subsets, not of all the sub-collections, while there may be a sub-collection that is not any subset. All the members of the power set are certainly sets, but that is just because all the non-set sub-collections are excluded from the power set.
6: Let Us Dismiss the Epistemological Nonsense
Special-Student-7-Rebutter
There are some people who claim that such a sub-collection does not exist because humans cannot measure the numbers of the Helium atoms.
Special-Student-7-Hypothesizer
That is a most disgustingly selfish claim! Does something vanish from the reality just because humans cannot measure it?
Does an electron not exist just because humans cannot measure it (because, for example, the electron is in a black hole)?
How about a cockroach on another planet that cannot be seen, because the planet is so far from the Bias Planet such that the cockroach will have dissolved to dusts before you reach the planet even in the light speed? Then, does the cockroach not exist, just because humans cannot see it?
While according to General Relativity, the vast area of the spacetime manifold is unreachable by humans (because of the maximum speed), does the area not exist by virtue of humans' inability to reach it?
Special-Student-7-Rebutter
Oddly, there are many Earthians who find no problem in insisting that whatever they cannot know do not exist.
Special-Student-7-Hypothesizer
Well, they are the same with a blind man who is insisting that there is no light because he cannot see any light.
Special-Student-7-Rebutter
It is of course ridiculous for the cockroach, but someone may say that any mathematical object is different from a physical object in that any physical object exists without any interaction with humans but any mathematical object is essentially a human construction, so, any mathematical object does not exist without being constructed by a human.
Special-Student-7-Hypothesizer
That is an odd view, because for example, not all the subsets of the natural numbers set have been constructed by humans, so, do only the subsets that have been constructed by humans so far exist? What does the power set of the natural numbers set contain? Does the power set wobble as someone has constructed a subset? Does a subset constructed by a hermit count? What if the only paper sheet that recorded a subset is burned?
Special-Student-7-Rebutter
In fact, even constructivists will not say that any mathematical object exists only if the object has been constructed: they will say that any mathematical object exists only if the object CAN be constructed. If they claimed the former, the power set would indeed wobble.
Special-Student-7-Hypothesizer
Anyway, the collection of the numbers of the Helium atoms has been constructed by us: the members of the collection are unambiguously determined and the collection exists as a physical fact.
Special-Student-7-Rebutter
They may be some people who claim that humans should humbly refrain from talking about what they cannot know.
Special-Student-7-Hypothesizer
But we know the existence of the sub-collection, even if we cannot know the members of the sub-collection, so, we should admit the existence of the sub-collection. For example, if we see a box, we know that the set of the balls in the box exists even if we cannot see inside the box (because the box cannot be opened somehow).
7: Is 'Set' a Small Enough Collection?
Special-Student-7-Hypothesizer
They seem to be saying like "'set' is a small enough collection.".
Special-Student-7-Rebutter
No, they are not saying so unless in mistakes; they are saying "'set' is a small enough class.", which is a totally different statement.
Special-Student-7-Hypothesizer
Ah .... "'set' is a small enough collection." is not correct, right?
Special-Student-7-Rebutter
That should not be correct, because a sub-collection of a set may not be a set without having any formula.
Special-Student-7-Hypothesizer
But "'set' is a small enough class." is correct?
Special-Student-7-Rebutter
That seems so, because that means "Every subclass of any set is a set.", which is correct, because being a subclass means having a formula.
Special-Student-7-Hypothesizer
So, there may be a non-set that is smaller than a set?
Special-Student-7-Rebutter
The mainstream set theories should not be denying that there may be a non-set collection that is "smaller" than a set.
Special-Student-7-Hypothesizer
We must note that just being small does not guarantee being a set.
8: A Collection Is Not Generally Defined by Properties of Members
Special-Student-7-Hypothesizer
Russell's "paradox" aside, the fundamental characteristic of mainstream set theories seems to be that they are relying on properties of members in order to define any set.
Special-Student-7-Rebutter
That seems the basic principle.
Special-Student-7-Hypothesizer
But a collection is not generally defined by properties of members.
When a collection is a bag and the members are the objects thrown into the bag by me, I do not necessarily throw objects into the bag because the objects have some properties like being red or something, but throw objects into the bag just because I happened to have laid my hands on them.
What distinguishes the members of the collection from the non-members is only that the members happened to have been touched by my hands when I blindly groped for objects.
Ultimately, the members are members just because they were chosen by the collection.
Special-Student-7-Rebutter
'were chosen by the collection' is certainly not anything inherent in the members, but someone may say that it is a property of the members anyway.
Special-Student-7-Hypothesizer
I am saying that such a wording is detrimental.
For example, a nutcase somewhere may have contrived a collection that contains me as a member, but I do not admit being chosen by the nutcase as my property. It is essentially foreign to me that the nutcase has contrived the collection.
Special-Student-7-Rebutter
Someone will say that you can just distinguish 'inherent property' and 'external property'.
Special-Student-7-Hypothesizer
I am saying that "external property" is fundamentally a oxymoron and should be dismissed.
It is important to clarify what property belongs to what entity, and a wording like "external property" happens because the property is attributed to a wrong entity.
Membership of collection is really a property of the collection, but as the membership is erroneously attributed to the members, the bad wording as "external property" happens.
Special-Student-7-Rebutter
It seems a typical mentality of Earthians to regard belonging to a collection, for example a club, as their property, or even their identity.
Special-Student-7-Hypothesizer
There is a confusion there: if someone regards belonging to a club as his or her property, that is because he or she has changed internally because of belonging to the club, not because just he or she belongs to the club.
Belonging to the club usually causes at least some changes in his of her conciousness, and such changes are his or her properties, not belonging to the club itself is. For example, he or she is proud (or ashamed) of belonging to the club or has some experience gained by belonging to the club, which is his or her property.
On the other hand, when a ball is thrown into a bag, the ball does not change internally.
9: After All, Mainstream Set Theories Are Theories Only on a Limited Kind of Collections, but Where Is a Theory on General Collections?
Special-Student-7-Rebutter
But there are some collections that are really defined by (inherent) properties of the members.
Special-Student-7-Hypothesizer
Of course there are, and mainstream set theories are theories only on that kind of collections, or really only on a more limited kind of collection, because not every (inherent) property but only any property expressed with a formula is accepted.
Special-Student-7-Rebutter
Probably, we were just wrong expecting otherwise.
Special-Student-7-Hypothesizer
It seems so: we were expecting that a set theory was a theory on general collections, but that was not the case.
Of course, it is fine that such a theory exists, especially for studying numbers, but I am wondering where is a theory on general collections.
Special-Student-7-Rebutter
Do we need a theory on general collections?
Special-Student-7-Hypothesizer
Of course, because, for example, there is no reason why the spacetime is based on a set as a limited kind of collection (why does the collection underlying the spacetime have to condescend to humans to have a formula?).
Special-Student-7-Rebutter
That seems the point that our dissatisfaction for mainstream set theories hinges on.
Special-Student-7-Hypothesizer
We are expecting mathematics as a tool for describing the reality. As mathematics limits itself in some limited concepts (by the reason that such limitations are convenient for humans), we are fettered in describing the reality. Especially, as 'manifold' is defined based on 'set', we cannot really model the spacetime as a manifold. Or more plainly, we cannot even talk about a set of 2 electrons, because mainstream set theories require members to be sets, but an electron is not particularly any set. Of course, we cannot talk about the set of the Helium atom numbers.
Special-Student-7-Rebutter
Probably, the problem is that 'manifold' is defined based on 'set' instead of on 'collection'.
Special-Student-7-Hypothesizer
More generally, too much of mathematics seems to be based on 'set', which seems to be the remnant of the attempts to build the entire mathematics on a set theory.
For example, why does a general function have to be from a set into a set instead of from a collection into a collection?
The base of the entire mathematics has to be a theory on general collections, which mainstream set theories gave up to be, so, mainstream set theories are fine by themselves if they are just some little theories in a corner of mathematics, but too much of mathematics seems to be relying on mainstream set theories.
Special-Student-7-Rebutter
Probably, that is because most mathematicians are concerned with only pure mathematics: pure mathematics are simplified with only sets.
10: What Is Set in the ZFC Set Theory?
Special-Student-7-Hypothesizer
What is the definition of 'set' in the ZFC set theory?
In fact, I understand that the theory does not define 'set'; the body of axioms in the theory does not define 'set', but just cites the single set, the empty set, and says that IF some things are ALREADY sets, all the things constructed from the sets in certain ways are sets.
In that discourse, the things constructed according to the axioms from the empty set are certainly sets, but that does not mean that there is no other set.
So, the theory is citing what are safely sets, but is not saying what exactly are sets.
Special-Student-7-Rebutter
In fact, the axioms also say that some things cannot be sets: for example, the collection of all the sets cannot be any set, because if it was a set, the axiom scheme of subset would say that the Russell's "set" would be a set, a contradiction.
Special-Student-7-Hypothesizer
So, the theory is saying that some things are definitely sets and some other things are definitely non-sets, but the other things are not declared to be sets or non-sets.
Special-Student-7-Rebutter
A point to be clarified is that whether every element must be a set.
Special-Student-7-Hypothesizer
Ah, the mainstream schools are saying that every element has to be a set, but that is not because it must be inevitably so, but just because that simplification is convenient for mathematicians. In fact, some schools legitimately allow so-called urelements, which are not sets.
The mainstream schools say "Mathematics doesn't need to talk about a set of 2 people.", but applications of mathematics need to talk about sets with non-set-members.
Special-Student-7-Rebutter
The mainstream schools seem to be saying that "If you want to talk about a set of 2 people, you can just relate the set of 2 people to a proper set of 2 set-elements.", but . . .
Special-Student-7-Hypothesizer
Being told to "relate", the problem is that as the set of 2 people does not exist at all in their theories, we cannot even talk about the set of 2 people. . . . How can we "relate" what we cannot talk about?
I mean, it is OK that they study mainly sets of sets, but at least sets of non-set-elements have to exist in their theories, if only in order for us to be able to "relate" them to sets of sets.
Special-Student-7-Rebutter
Well, as the term, "set", is used for multiple concepts, our talk becomes confusing; we should distinguish different concepts with different terms.
Special-Student-7-Hypothesizer
We have already defined 'n-set', which is nothing but 'collection'.
We will call any n-set that satisfies the axioms without the restriction of having to have only set-elements, 'a-set' (artificial set).
We will call any a-set that has only set-elements, 'r-set' (restricted set).
We will call any r-set that can be constructed from the empty set, 'c-set' (constructible set).
Special-Student-7-Rebutter
OK.
Special-Student-7-Hypothesizer
Any 'set' in the ZFC theory (at least in the most mainstream school of it) should be an r-set.
Special-Student-7-Rebutter
That should mean that the theorems in the theory are guaranteed to hold only for r-sets.
Special-Student-7-Hypothesizer
They should be so; probably many theorems will hold also for a-sets, and less theorems will hold also for n-sets, but we cannot use the theorems for a-sets or n-sets unless we rigorously check that the theorems hold also for a-sets or n-sets.
Special-Student-7-Rebutter
We have to be aware that the fact that there is a bijection from an n-set onto an r-set does not guarantee that a theorem holds for the n-set.
Special-Student-7-Hypothesizer
Obviously, a theorem that uses the supposition that any element of a "set" is a "set" does not necessarily hold for an n-set, even if the n-set is bijective to an r-set. For example, a theorem that any element of any "set" is a "set" does not hold for an n-set, even if the n-set is bijective to an r-set.
11: The Burali-Forti "Paradox"
Special-Student-7-Hypothesizer
The Burali-Forti "paradox" is that the collection of all the ordinal numbers is not any set.
Special-Student-7-Rebutter
. . . Why is that a paradox?
Special-Student-7-Hypothesizer
To state in our terminology, it is that the collection of all the ordinal numbers is not any r-set, which is not any paradox for us.
The "paradox" was thought to be a paradox, because by the naive comprehension axiom, the collection should be a set, but is not any set.
Special-Student-7-Rebutter
Why is the collection of all the ordinal numbers not any r-set?
Special-Student-7-Hypothesizer
As it is a conclusion reached via a rather long chain of deductions from the axioms, it is not so easily explained, but to state the gist, it has been proven that any transitive r-set that is well-ordered by membership is an ordinal number; it has been proven that the collection of all the ordinal numbers is transitive and is well-ordered by membership; so, if the collection was an r-set, it would be an ordinal number; but it has been proved that any ordinal number cannot be an element of itself; so, a contradiction, so, the collection cannot be any r-set.
Special-Student-7-Rebutter
Well, for us, it is just a matter of that the collection is an n-set, but not any r-set, not particularly any paradox.
12: Being 1st-Order Logic Is Not the Purpose, Right?
Special-Student-7-Hypothesizer
An answer I heard to the question, "Why can not every sub-collection of any set be admitted to be a set without being required to have a formula?", is that it is inevitable because the mainstream set theories are 1st-order logic.
Special-Student-7-Rebutter
Then, why do they not give up 1st-order logic, while 1st-order logic seems insufficient?
Special-Student-7-Hypothesizer
Probably, they do not admit 1st-order logic's being insufficient.
Special-Student-7-Rebutter
That should mean that they deem 'Not every sub-collection of a set can be admitted to be a set.' no problem.
Special-Student-7-Hypothesizer
That Helium atom numbers collection cannot be dealt with, but why is that no problem?
Special-Student-7-Rebutter
Probably, because most mathematicians are concerned with only pure mathematics, not with applications for other disciplines, especially physics.
Special-Student-7-Hypothesizer
In consideration of applications, it should be a problem, because the collection of the spacetime points does not have any formula, probably.
Special-Student-7-Rebutter
And the collection of some 2 persons cannot be dealt with.
Special-Student-7-Hypothesizer
My impression is that they stick to 1st-order logic just because it is convenient for humans, but is science not an endeavor to approach truths however inconvenient for humans?
Not admitting a collection to be a set seems akin to not admitting an irrational number to be a number, just because it is inconvenient for humans.
Special-Student-7-Rebutter
As it is not particularly bad to study rational numbers, it is not particularly bad to study "sets" (r-sets), but 'manifold', etc. should be defined based on collection, not on "set", which may be what we should say.
13: To Repeat, Prevalent Discourses Are Insufficient
Special-Student-7-Hypothesizer
As we said in the 1st section, we guess that we are mistaken on some or many points.
But we are very sure that prevalent discourses are insufficient enough to naturally cause such misunderstandings.
So, we are saying that more appropriate explanations are due.
References
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