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
- Orientation
- Main Body
- 0: A Note
- 1: What Is Russell's "Paradox"?
- 2: It Seems Just That the Naive Comprehension Axiom Was Too Naive
- 3: Why Cannot We Start with the Natural Definition of Set?
- 4: 'Class' Is Not 'N-Set', Because ...
- 5: Not Every Sub-Collection of a Set Is Guaranteed to Be a Set
- 6: Let Us Dismiss the Epistemological Nonsense
- 7: Is 'Set' a Small Enough Collection?
- 8: A Collection Is Not Generally Defined by Properties of Members
- 9: After All, Mainstream Set Theories Are Theories Only on a Limited Kind of Collections, but Where Is a Theory on General Collections?
- 10: What Is Set in the ZFC Set Theory?
- 11: The Burali-Forti "Paradox"
- 12: Being 1st-Order Logic Is Not the Purpose, Right?
- 13: To Repeat, Prevalent Discourses Are Insufficient
Starting Context
- The reader knows the background of this site.
Target Context
- 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 DirectionHere 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.
