Japanese schools talk about "shugo"s, but are they sets or collections?
Topics
About: elementary school mathematics
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: What Does "shugo" (Used in Japanese Schools) Mean?
- 2: Caution: There Are Some Multiple Set Theories
- 3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?
- 4: In Fact, a Collection in General Is Not Defined by Any Property of Member
- 5: "shugo" Means 'Collection'
- 6: The Concept of 'Collection' Lives on
- 7: Mathematics Is Not Exactly Built on the Set Theory
Starting Context
- The reader knows the background of this site.
Target Context
- The reader will know the distinction between sets and collections.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
There is an article on definition of set and Russel's "paradox".
Main Body
1: What Does "shugo" (Used in Japanese Schools) Mean?
Special-Student-7-Hypothesizer
Japanese schools talk about "shugo" s.
For example, they think of the "shugo" of some 3 apples, the "shugo" of the students of a class, etc..
Special-Student-7-Rebutter
What is "shugo" exactly?
Special-Student-7-Hypothesizer
Any "shugo" is any collection of some objects such that the membership is unambiguous.
Special-Student-7-Rebutter
Is there ever a collection such that the membership is ambiguous?
Special-Student-7-Hypothesizer
I do not think there is: the concept of 'collection' itself includes the requirement that the membership is unambiguous; the qualification is just for emphasis.
Special-Student-7-Rebutter
So, is "shugo" nothing but 'collection' in English?
Special-Student-7-Hypothesizer
I can just guess so, because Japanese textbooks do not say what "shugo" is called in English, but dictionaries usually say that "shugo" in mathematics is "set" in English.
Special-Student-7-Rebutter
But 'set' and 'collection' are 2 different things in mathematics.
2: Caution: There Are Some Multiple Set Theories
Special-Student-7-Hypothesizer
As a caution, there are some multiple set theories, and what 'set' is in each theory depends on the theory.
Special-Student-7-Rebutter
Very confusing; cannot the theories use different names?
Special-Student-7-Hypothesizer
It is very confusing, but as each theory exists because it regards the other theories unsatisfactory, each theory says "'set' we call is the real 'set'!", and each theory uses "set".
Special-Student-7-Rebutter
I kind of understand, but ...
Special-Student-7-Hypothesizer
Anyway, arguably the most popular one is the ZFC set theory, and we will mean the ZFC set theory by "the set theory" and mean 'set' in the ZFC set theory by "set" hereafter.
Note that what we are going to say here are not particularly canonical; as we have not seen any convincing argument, we are presenting a hypothesis that is convincing for us. Nobody should accept something just because someone (whoever he or she is) says so or just because many people say so; check yourself whether the hypothesis is unboundedly consistent, which is the only way to approach truths.
3: How Have 'Set' and 'Collection' Ended Up Meaning Different Things?
Special-Student-7-Hypothesizer
Most textbooks cite Russel's "paradox" as the reason why the set theory needs to be as it is, but the arguments are not convincing at all, at least for us.
Special-Student-7-Rebutter
What are the arguments like?
Special-Student-7-Hypothesizer
Historically, there was the so-called "naive set theory", which is based on the so-called "naive comprehension axiom", which is "Any precisely specified property of member can be used to define a set.".
Then, Russel's "paradox" came along and refuted the "naive comprehension axiom", and so refuted the "naive set theory".
So, now, the ZFC set theory admits only the empty set and the things constructed from the empty set as "set".
So, the collection of some 2 electrons are not admitted to be "set", because the collection is not constructed from the empty set.
Special-Student-7-Rebutter
There seems a wide gap in the reasoning: why would the collection of the 2 electrons not be "set" just because the "naive comprehension axiom" has been refuted?
Special-Student-7-Hypothesizer
I have not seen any convincing explanation.
In fact, why will we not have the modified axiom that "Any precisely specified property of member that (the property) determines membership unambiguously can be used to define a set."?
Special-Student-7-Rebutter
What Russel's "paradox" says is that just "precisely specified property of member" does not guarantee the unambiguous-ness of membership, so, why will we not add the unambiguous-ness as an additional requirement?
Special-Student-7-Hypothesizer
I have not seen any convincing argument why; it seems just that as checking it is not easy in general, they want a way to be able to define sets without bothering to do checking.
Special-Student-7-Rebutter
Well, is that an attitude to humbly approach truths? That seems an attitude to ignore whatever are inconvenient for humans.
Special-Student-7-Hypothesizer
Whether the unambiguous-ness can be checked easily is just a matter of human convenience, and the essence of the ZFC set theory is that it gave up covering the concept of "collection" and has decided to degenerate to deal with only convenient-for-humans kind of collections, which are now called "sets", which is my understanding.
In fact, I do not say at all that having a theory about such limited kind of collections is bad, but we need to be aware what the theory is really doing.
Special-Student-7-Rebutter
We need to avoid some confusions: "naive set theory" means the theory with the "naive comprehension axiom" not the concept of 'collection'; while the ZFC set theory excludes most collections like a collection of some 2 electrons, that is not because the collection of 2 electrons are inappropriate but because the set theory has degenerated.
Special-Student-7-Hypothesizer
The main cause of confusions is that the ZFC set theory keeps using the term, "set", while the concept of 'set' has degenerated from equaling 'collection' to 'convenient-for-humans collection'.
4: In Fact, a Collection in General Is Not Defined by Any Property of Member
Special-Student-7-Hypothesizer
In fact, I think that the "naive comprehension axiom" is more fundamentally problematic than Russel's "paradox".
The problem is that it tries to define any collection by a property of member.
For example, let us think of the collection of some balls I put into a bag.
I put a ball into the bag just because my groping hand happened to touch the ball, not because the ball is red or blue or something.
Special-Student-7-Rebutter
You mean that the collection is not defined by any property of ball, like being red or something?
Special-Student-7-Hypothesizer
Yes.
Special-Student-7-Rebutter
Your collection is usually defined by enumerating the balls, as the collection is finite: you cannot put infinitely many balls into the bag by groping.
Special-Student-7-Hypothesizer
Yes, when the collection is finite, there is the escape.
But what if the collection is infinite?
Special-Student-7-Rebutter
How will you define an infinite collection not by property of member?
Special-Student-7-Hypothesizer
If we ignore Relativity, let us take a \(\mathbb{R}^3\) Cartesian coordinates system for the Universe, and take the 1-radius open ball around each rational-coordinates point. Then, the numbers of the electrons in each open ball at a time, which is a subcollection of the natural numbers set, is my infinite collection.
Special-Student-7-Rebutter
You mean that the collection is not defined by any property of natural number, like being even or something?
Special-Student-7-Hypothesizer
Yes.
Special-Student-7-Rebutter
It does not seem to be really infinite, supposing that there are only some finite number of electrons in the Universe.
Special-Student-7-Hypothesizer
Then, let us take the pair of the center-coordinates and the number of the electrons, like ((1.2, 3.4, 5.6), 7).
Special-Student-7-Rebutter
Then, certainly, the collection is infinite.
What if we are not allowed to ignore Relativity?
Special-Student-7-Hypothesizer
Let us take a chart for the spacetime manifold, whose (the chart's) domain is homeomorphic to \(\mathbb{R}^4\), and let us do the same thing for that \(\mathbb{R}^4\).
Special-Student-7-Rebutter
So, you are saying that such collections exist in the reality but the "naive comprehension axiom" cannot grasp such collections.
Special-Student-7-Hypothesizer
In fact, that is also the problem of the "restricted comprehension axiom", which is in fact the same with the "naive comprehension axiom" except that the "restricted comprehension axiom" thinks of only the elements of an already-known-to-be-set collection.
Special-Student-7-Rebutter
Is "the number of the electrons in the open ball" not 'property of member'?
Special-Student-7-Hypothesizer
At least, the ZFC set theory does not admit such a property: the "restricted comprehension axiom" requires that the property is expressed as a formula that allows only the specified operators like \(\in\) and some already-known-to-be-set collections.
Special-Student-7-Rebutter
Did the "naive comprehension axiom" admit such a property?
Special-Student-7-Hypothesizer
Maybe, but such a usage of the term, "property of member", would be quite harmful (a too-far-fetched interpretation, I would say), in my opinion. Being "the number of the electrons in the open ball" is not any property of the natural number but is a property of the Universe: I mean, being "the number of the electrons in the open ball" is not about the natures of the natural number (like being even, prime, or something) but about the natures of the Universe.
Special-Student-7-Rebutter
Anyway, the "restricted comprehension axiom" does not allow your collection.
Special-Student-7-Hypothesizer
So, there is the collection in the reality, but the ZFC set theory refuses to cope with the collection.
5: "shugo" Means 'Collection'
Special-Student-7-Hypothesizer
So, "shugo" in Japanese textbooks means 'collection' not "set".
And when a Japanese student later hears of Russel's "paradox" and hears '"set" of some 2 electrons' refuted, the "shugo" of the 2 electrons is not refuted at all, because the collection of the 2 electrons is not refuted at all.
And when he or she later hears that "naive set theory" is inappropriate, the concept of "shugo" (the concept of 'collection') is not inappropriate at all, because the concept of 'collection' is not based on the naive comprehension axiom.
6: The Concept of 'Collection' Lives on
Special-Student-7-Hypothesizer
The concept of 'collection' is still valid and is indeed used in mathematics.
As a typical example, in the category theory, a category in general is not any set but a collection. For example, the category of all the sets, \(Set\), is a collection but not any set.
Special-Student-7-Rebutter
\(Set\) is prevalently called "class".
Special-Student-7-Hypothesizer
Yes, 'class' is a concept wider than 'set' and narrower than 'collection', but anyway, a class is not any set in general.
Special-Student-7-Rebutter
How is 'class' narrower than 'collection'?
Special-Student-7-Hypothesizer
Roughly speaking, any class is still constructed from the empty set and is still defined by property of member with a formula.
Anyway, if something was inappropriate just because it was not "set" in the ZFC set theory, the whole category theory would be inappropriate.
And "collection" is still frequently used in many mathematical textbooks: while a reader may wonder why "collection" is being used instead of "set", that is because the mentioned object is not or is not necessarily any "set" according to the ZFC set theory, and if 'collection' was inappropriate, such textbooks would be inappropriate, which is not the case.
7: Mathematics Is Not Exactly Built on the Set Theory
Special-Student-7-Hypothesizer
A rather prevalent misconception is that "whole mathematics is built on the set theory.".
Special-Student-7-Rebutter
That is what the naive set theory aspired to do.
Special-Student-7-Hypothesizer
But the attempt failed (by mainly Russel's "paradox") and because the set theory gave up dealing with general collections and has degenerated to be a theory on a very limited kind of collections, the aspiration is given up.
Certainly, the set theory is still a very important part of mathematics, but it is not exactly the basis of whole mathematics.
Special-Student-7-Rebutter
A proof is that "collection" is still used in many mathematical textbooks.
