Russell's paradox can be quite confusing to start with. It is based around an abstract question, based on an already abstract idea of a set which is far removed from the way most people think about and reason about things.
Yet, it's actually far more than just an issue for mathematicians. If you try to develop any foundations for clear logical reasoning in a way that lets you talk about a collection or set of several other things, then this paradox is just a few logical steps away.
I think myself that it's rather remarkable that nobody ever thought about it until the C20. Somehow you'd expect the Ancient Greeks or the Chinese or Indian mathematicians and logicians and philosophers to come up with this one. But for some reason, they never did as far as we know. But it is easy of course to say that with hindsight :).
Anyway I'm not going to say anything new here, just try to explain the idea itself.
So, first of all, it arises as a result of trying to clarify the idea of a set or collection. The idea that you can have clubs and have members of them for instance. Or you can have a bucket with things in it. But more generally also when you start reasoning e.g. about mountains - then you don't have to think this way - but it is quite natural to extrapolate to a more abstract level and think about all mountains. Either all actual mountains on the Earth - or more generally even about all possible mountains.
So the idea is - can we make this idea of a set or collection of things logically precise? Can we make it so precise that you always know exactly what someone means when they talk about a collection of things?
Well if we do make the idea totally precise like that, precise enough even for logicians, we have to say how we understand any questions anyone may ask about these collections.
Mathematicians and logicians usually call them sets, so let's use the word "set" here for our new logically precise word for a collection with members.
I think a good example of this concept in ordinary use is a football team. Or any other team or club.
Nobody would confuse a football team with a footballer, so we need the idea of a collection of footballers which in some way is different from a individual footballer.
And then you will also want to talk about sets of sets. E.g. premier league football teams. Is the premier league a football team? Is it a footballer? Surely not. So it's something new, and it seems most natural to understand it as a set of sets of footballers.
So, it's natural to have hierarchies of sets like that as well in ordinary language. Of course in maths it gets much more elaborate but we have the beginnings of it already in ordinary language and some things would be almost impossible to say if we couldn't talk about sets of sets.
So then we are into the territory where you can ask questions about these sets.
So for instance, for the set of oranges, you would never ask this in ordinary conversation I know, but is the set of all oranges itself an orange?
Orange Blossom and oranges, photo by Ellen Levy Finch
The members of the set of all oranges are of course oranges - but what about the set of all of them?
If the set of all oranges was an orange - you'd be able to eat it (and what happens to your logic after someone eats the set of all oranges :) ).
If the set of all mountains was itself a mountain, you'd be able to answer questions such as "where is it located, and how long does it take to climb it and what is the easiest route to its summit?".
You can ask those questions about individual mountains but not about the set of all mountains, so it doesn't seem possible to build a logic where the set of all oranges is an orange or the set of all mountains is a mountain.
And actually, you have to build rather contrived sets to get one that's a member of itself according to reasonably convincing logical interpretations of the naive logic of sets and collections.
But what happens if you go all the way and consider the set of all abstract concepts themselves? What if we gather together all these ideas of abstraction that anyone has come up with or could ever come up with?
Is that itself an abstract concept? If it is, then it's a member of itself, in naive set theory.
For the set of all abstract concepts to fail to be a member of itself, you'd have to say that the set of all abstract concepts is not an abstract concept which sounds rather paradoxical.
So you can find sets that are members of themselves in naive set theory, but they are not so common and it is quite hard to come up with good examples - that is one of the best.
Russell's paradox applies no matter whether you think hardly any sets are members of themselves, or even, that none are, or that almost all sets are.
It makes no difference what you think there.
Just as we did with the set of all abstract concepts - now that we have this idea of a set or collection, it makes sense to talk about the "set of all sets".
This set would include all football teams. But also the premier league. Also all ice hockey teams. All Olympics teams. All clubs, all nations.
Even temporary sets, for instance the collection of all people walking along a particular street at a particular moment. All possible ways of picking out atoms from our universe. On and on, all possible sets, physical or abstract.
Or as a mathematician, you could decide to restrict it to just purely mathematical notions. E.g. sets of numbers, so it includes all finite sets like
{1, 2, 3} or {5, 717} or whatever.
Also, say, the set of all lines, faces and vertices of a cube, octahedron, dodecahedron etc.
As well as all those finite sets, to make a set theory of all of maths you need also all infinite sets, e.g. the set of all even numbers, all odd numbers all primes etc - endlessly many different ways of picking out an infinite set of numbers.
And on and on, many other mathematical sets.
At first all seems fine, you talk about the set of all sets, and it is natural to say, that just like abstract concepts, that the set of all sets is itself a set.
And Frege of course famously spent years of is life building up a logical approach to set theory on that assumption. His aim was to make a theory where everything in mathematics can be described using sets.
But then Russell comes along and asks Frege to look at the "set of all sets that are not a member of themselves".
So, from our reasoning about a set of oranges not being an orange - most sets are not members of themselves, so this qualifies them as members of your new set.
So, naively (once you've thought it through for a while) the natural assumption would be that this new set consists of nearly all sets - perhaps even all sets except the set of all sets itself. No problem there so far.
But then - you ask his question "Is this set a member of itself"?
If Russell had asked if his new set was a member of the set of all sets, it would be easy - just answer "yes", that would be the naive answer.
But instead, he asked, the more subtle question, is it a member of itself?
If it is a member of itself - well by definition it consists of all sets that are not a member of themselves. So if it is a member of itself, it has to be one of the sets that aren't a member of themselves, so then that means it isn't a member of itself.
And if it isn't a member of itself, then this property of not being a member of itself immediately qualifies it for membership.
So you keep going round and round in circles like that. If it is a member of itself, it isn't. And if it isn't a member of itself, then it is.
So you are left with a paradox. This is a paradox of logic that you get if you try to write out the assumptions of naive set theory in clear logical statements.
The thing is that we start off with these fuzzy concepts where we identify things as of particular types and so on. E.g. we think we know what a banana or orange is. And you can't do mathematics rigorously at that level, you run into all sorts of problems.
This idea that you need rigor and logical precision in maths started with Euclid's axioms of geometry - and gradually developed through many other axioms and reasoning systems. It came to its head at the end of the nineteenth century - actually in study of physical problems such as heat flow in metals - where mathematicians found that the concepts they were using so far lead to many paradoxes. Things that seemed to make no sense and eventually, direct contradictions. So they needed some way to make their ideas more precise.
So you have to define your concepts with logical and mathematical precision.
And one way to do that is to use the idea of a set and membership which is what mathematicians usually find most helpful. So if you do that you have to define clearly what is meant by a set and what is meant by membership.
There might be other ways of clarifying concepts. Don't have to take set and membership as fundamental.
You can think in terms of predicates for instance - an orange is something that is round, grows on a particular type of tree, tastes sweet, has a thick rind etc etc - but one way or another you still find you have marked out a particular collection of objects as instances of your concept and others as not.
And then an orange is something that satisfies all those predicates. But sweetness is not an orange, it's a quality of oranges. (And as philosophers one can argue endlessly about what this means exactly)
So once you get that far, you end up with something that looks like sets and membership one way or another. And then once you do that, then Russell's paradox comes up.
To avoid it you'd need some language that doesn't split things into collections like that, which is rather hard to imagine how it would work.
You could try developing a logical language where the set of all oranges is an orange, and the set of all bananas is a banana, and a set consisting of an orange and a banana is itself in some way - an orange with a banana (here it gets a bit tricky).
And a football team is a footballer and the premier league is a football team and also a footballer.
(Except, maybe for the empty set which has no members),
It might be possible to develop such a theory. I don't know, you could try at least.
And then ask then if the set of all sets is a set - and your answer will be yes.
But that doesn't change anything, you can still ask: "what about the set of all sets that are not members of themselves?"
How can this set be a member of itself, when the defining characteristic needed for membership is that it isn't a member of itself?
Maybe you try to make this an exceptional case in your theory - this is the only set that isn't a member of itself
But if it isn't a member of itself, then that immediately qualifies it to be a member of itself, so then it has to be a member of itself.
So you are still faced with Russell's paradox.
It seems clear that there is no getting around it in a naive theory that has a notion of set and membership except to restrict your reasoning in some way.
One way around it, quite natural, is to say that you simply can't ask if the set of all oranges is an orange. That's quite reasonable sounding. Maybe it is a nonsensical question, and that's why we got into trouble?
That was Russell's solution in his type theory. This works but is rather cumbersome. Because as you follow through the implications you find you have to have an infinite number of layers of types, to avoid paradox.
Or you could look for a way to develop a logic that doesn't have any notion of membership or set built into it - and find some other primitive concept in its place.
But if so what? There are many areas of discourse where we can get by without much use of sets and collections. But sometimes we do need them.
How are you going to talk about football teams and footballers and the premier league without using the concept of membership of a group?
And once you have got as far as defining membership - even if it is not a primitive concept of your theory - then if you just "follow your nose" with naive ideas - you are not far from defining the set of all sets and reasoning about it. Before long you end up with Russell's paradox.
This arises even more so in mathematics where similar ideas are needed all the time.
The standard way to get around it is to restrict your formal logical reasoning in some way so that you can't discuss this question in your theory.
The most common approach is to say that you can't talk about "sets" in this perfectly general way. Instead you can only talk about particular types of set, and when talk about the "set of all sets" you have to introduce new terms - call it something else, e.g. a class, or a set of type 2 or whatever.
Russell's type theory was a start there with his infinitely many types.
Nowadays we have a simpler approach where you just two types of collection - sets, and then classes. When you talk about classes they can't be members of anything and questions of membership are meaningless. So then the paradox doesn't arise for the classes.
Another, less common approach is to permit the set of all sets to be a member of itself and restrict your logical reasoning in other ways, restrict the language you use to construct sets so that statement "set of all sets that are not a member of themselves" can't be said in your language - that's Quine's solution in his "new foundations".
In Quine's logic, then being "a member of itself" is not a meaningful concept in his logic. He has the requirement that all properties like that have to be "stratified" - it's a complicated requirement - but basically it rules out properties that relate a set directly to itself by membership.
So, you just can't discuss sets and whether they are members of themselves or not in his theory at all. Any question like that gets "lost in translation" when you try to ask it. So then Russell's question can't be asked so the paradox doesn't arise.
In some ways Quine's solution is closer to intuitive set theory, because it permits you to have a set of all sets which is a set. Let's you talk about the "set of all sets" with the same language as more ordinary sets.
But one way or another you have to restrict what you are permitted to talk about in your logical language you use to reason about sets, or restrict interpretation of the word "set", or you run into Russell's paradox.
Up to Frege, nobody developed a theory to use as a foundation for logic that avoided Russell's paradox.
Now we can avoid it with hindsight. But it's not easy. Why did nobody come up with any of these ideas at all until the early twentieth century?
It usually seems at least a bit artificial - like they have obviously restricted the language in order to avoid the paradox.
Why is that? This is not an issue just for maths but also for all attempts to formalize intuitive logic whenever we use it to reason about collections of things.
Sometimes proponents of various ways around it will argue that their solution is a natural one to use.
For instance, proponents of Quine's approach can say that philosophically it is more natural. But then if that's what you think, you are still faced with the question - if it is so natural - why did we need Russell to point out his paradox first before anyone came up with a way of thinking about sets that avoids it?
It is based on simple ideas. If it is so natural, why did no philosopher in the entire history of human civilizations before Russell's paradox come up with Quine's approach or anything similar?
I think this is also a curiosity.
This paradox is so simple to state and uses a basic concept that everyone is familiar with - the idea of membership of a group or collection. Even back to the ancient Greeks and Chinese - many of Euclid's theorems are far more complex than this. Or later on the Indian and the Arabic mathematicians and philosophers. They developed many intricate chains of logical reasoning - but not this one.
And you don't even need the concept of zero, or negative numbers, or ratios or any of the new concepts of modern mathematics to understand it.
It may well be the most simple genuinely totally new chain of pure logical reasoning to be developed in modern times, One that got missed by everyone right up to 1901 when Russell came up with the paradox.
So why didn't some of the ancient mathematicians or philosophers think of this. Even the nineteenth century mathematicians, came up with many elaborate ideas about sets, and various very complicated to state paradoxes, and worked on these ideas continuously for years in the case of Frege and Cantor - but for some reason never came up with this one.
We know that it is possible for humans to miss simple concepts. For thousands of years, entire civilizations rose and fell without developing the modern idea of a negative number for instance. Right up to the middle ages, mathematicians would treat what we now consider to be a single equation (e.g. the quadratic equation) as several different cases, because they didn't have the concept of negative numbers in the modern sense.
And same for the notion of zero for positional notation. Or ratios with arbitrary numbers at the top and bottom (for instance the Babylonians had "unit fractions" with the requirement that the top number in the fraction had to be 1, and did clumsy arithmetic where everything had to be expressed in unit fractions).
So - perhaps it's an example of this? And if so, I can't help but wonder, are there any other remaining really simple concepts that we haven't yet thought of?
I wonder, as there are surely many ideas that we haven't yet thought of - how simple is the simplest concept that we haven't yet thought of?
No way to answer that I suppose.
Yet, it's actually far more than just an issue for mathematicians. If you try to develop any foundations for clear logical reasoning in a way that lets you talk about a collection or set of several other things, then this paradox is just a few logical steps away.
I think myself that it's rather remarkable that nobody ever thought about it until the C20. Somehow you'd expect the Ancient Greeks or the Chinese or Indian mathematicians and logicians and philosophers to come up with this one. But for some reason, they never did as far as we know. But it is easy of course to say that with hindsight :).
Anyway I'm not going to say anything new here, just try to explain the idea itself.
WHAT IS A SET OR COLLECTION?
So, first of all, it arises as a result of trying to clarify the idea of a set or collection. The idea that you can have clubs and have members of them for instance. Or you can have a bucket with things in it. But more generally also when you start reasoning e.g. about mountains - then you don't have to think this way - but it is quite natural to extrapolate to a more abstract level and think about all mountains. Either all actual mountains on the Earth - or more generally even about all possible mountains.
So the idea is - can we make this idea of a set or collection of things logically precise? Can we make it so precise that you always know exactly what someone means when they talk about a collection of things?
Well if we do make the idea totally precise like that, precise enough even for logicians, we have to say how we understand any questions anyone may ask about these collections.
Mathematicians and logicians usually call them sets, so let's use the word "set" here for our new logically precise word for a collection with members.
WHY WE NEED SETS - OR SOMETHING LIKE THEM
I think a good example of this concept in ordinary use is a football team. Or any other team or club.
Nobody would confuse a football team with a footballer, so we need the idea of a collection of footballers which in some way is different from a individual footballer.
And then you will also want to talk about sets of sets. E.g. premier league football teams. Is the premier league a football team? Is it a footballer? Surely not. So it's something new, and it seems most natural to understand it as a set of sets of footballers.
So, it's natural to have hierarchies of sets like that as well in ordinary language. Of course in maths it gets much more elaborate but we have the beginnings of it already in ordinary language and some things would be almost impossible to say if we couldn't talk about sets of sets.
So then we are into the territory where you can ask questions about these sets.
IS THE SET OF ORANGES AN ORANGE?
So for instance, for the set of oranges, you would never ask this in ordinary conversation I know, but is the set of all oranges itself an orange?
The members of the set of all oranges are of course oranges - but what about the set of all of them?
If the set of all oranges was an orange - you'd be able to eat it (and what happens to your logic after someone eats the set of all oranges :) ).
If the set of all mountains was itself a mountain, you'd be able to answer questions such as "where is it located, and how long does it take to climb it and what is the easiest route to its summit?".
You can ask those questions about individual mountains but not about the set of all mountains, so it doesn't seem possible to build a logic where the set of all oranges is an orange or the set of all mountains is a mountain.
And actually, you have to build rather contrived sets to get one that's a member of itself according to reasonably convincing logical interpretations of the naive logic of sets and collections.
IS THE SET OF ABSTRACT CONCEPTS AN ABSTRACT CONCEPT?
Dissolving of nature into abstract concepts in Mondrian's art:In art, music, writing, then we look at various ways of abstracting from things. Lots of particular abstractions.
- The Flowering Apple Tree
- Pier and Ocean
- Tableau No. IV
See Piet Mondrian
But what happens if you go all the way and consider the set of all abstract concepts themselves? What if we gather together all these ideas of abstraction that anyone has come up with or could ever come up with?
Is that itself an abstract concept? If it is, then it's a member of itself, in naive set theory.
For the set of all abstract concepts to fail to be a member of itself, you'd have to say that the set of all abstract concepts is not an abstract concept which sounds rather paradoxical.
So you can find sets that are members of themselves in naive set theory, but they are not so common and it is quite hard to come up with good examples - that is one of the best.
RUSSELL'S PARADOX
Russell's paradox applies no matter whether you think hardly any sets are members of themselves, or even, that none are, or that almost all sets are.
It makes no difference what you think there.
Just as we did with the set of all abstract concepts - now that we have this idea of a set or collection, it makes sense to talk about the "set of all sets".
This set would include all football teams. But also the premier league. Also all ice hockey teams. All Olympics teams. All clubs, all nations.
Even temporary sets, for instance the collection of all people walking along a particular street at a particular moment. All possible ways of picking out atoms from our universe. On and on, all possible sets, physical or abstract.
Or as a mathematician, you could decide to restrict it to just purely mathematical notions. E.g. sets of numbers, so it includes all finite sets like
{1, 2, 3} or {5, 717} or whatever.
Also, say, the set of all lines, faces and vertices of a cube, octahedron, dodecahedron etc.
Comparison of truncated icosahedron and soccer ball
Amongst all the many sets of mathematics, you can describe a truncated icosahedron as a set of edges, faces and vertices
As well as all those finite sets, to make a set theory of all of maths you need also all infinite sets, e.g. the set of all even numbers, all odd numbers all primes etc - endlessly many different ways of picking out an infinite set of numbers.
And on and on, many other mathematical sets.
At first all seems fine, you talk about the set of all sets, and it is natural to say, that just like abstract concepts, that the set of all sets is itself a set.
And Frege of course famously spent years of is life building up a logical approach to set theory on that assumption. His aim was to make a theory where everything in mathematics can be described using sets.
RUSSELL'S SET OF ALL SETS THAT AREN'T A MEMBER OF THEMSELVES
But then Russell comes along and asks Frege to look at the "set of all sets that are not a member of themselves".
So, from our reasoning about a set of oranges not being an orange - most sets are not members of themselves, so this qualifies them as members of your new set.
So, naively (once you've thought it through for a while) the natural assumption would be that this new set consists of nearly all sets - perhaps even all sets except the set of all sets itself. No problem there so far.
But then - you ask his question "Is this set a member of itself"?
If Russell had asked if his new set was a member of the set of all sets, it would be easy - just answer "yes", that would be the naive answer.
But instead, he asked, the more subtle question, is it a member of itself?
If it is a member of itself - well by definition it consists of all sets that are not a member of themselves. So if it is a member of itself, it has to be one of the sets that aren't a member of themselves, so then that means it isn't a member of itself.
And if it isn't a member of itself, then this property of not being a member of itself immediately qualifies it for membership.
So you keep going round and round in circles like that. If it is a member of itself, it isn't. And if it isn't a member of itself, then it is.
So you are left with a paradox. This is a paradox of logic that you get if you try to write out the assumptions of naive set theory in clear logical statements.
OUR FUZZY CONCEPTS
The thing is that we start off with these fuzzy concepts where we identify things as of particular types and so on. E.g. we think we know what a banana or orange is. And you can't do mathematics rigorously at that level, you run into all sorts of problems.
This idea that you need rigor and logical precision in maths started with Euclid's axioms of geometry - and gradually developed through many other axioms and reasoning systems. It came to its head at the end of the nineteenth century - actually in study of physical problems such as heat flow in metals - where mathematicians found that the concepts they were using so far lead to many paradoxes. Things that seemed to make no sense and eventually, direct contradictions. So they needed some way to make their ideas more precise.
So you have to define your concepts with logical and mathematical precision.
And one way to do that is to use the idea of a set and membership which is what mathematicians usually find most helpful. So if you do that you have to define clearly what is meant by a set and what is meant by membership.
There might be other ways of clarifying concepts. Don't have to take set and membership as fundamental.
You can think in terms of predicates for instance - an orange is something that is round, grows on a particular type of tree, tastes sweet, has a thick rind etc etc - but one way or another you still find you have marked out a particular collection of objects as instances of your concept and others as not.
And then an orange is something that satisfies all those predicates. But sweetness is not an orange, it's a quality of oranges. (And as philosophers one can argue endlessly about what this means exactly)
So once you get that far, you end up with something that looks like sets and membership one way or another. And then once you do that, then Russell's paradox comes up.
To avoid it you'd need some language that doesn't split things into collections like that, which is rather hard to imagine how it would work.
WHAT IF YOU GO THE OTHER WAY AND TRY TO MAKE ALL SETS MEMBERS OF THEMSELVES
You could try developing a logical language where the set of all oranges is an orange, and the set of all bananas is a banana, and a set consisting of an orange and a banana is itself in some way - an orange with a banana (here it gets a bit tricky).
And a football team is a footballer and the premier league is a football team and also a footballer.
(Except, maybe for the empty set which has no members),
It might be possible to develop such a theory. I don't know, you could try at least.
And then ask then if the set of all sets is a set - and your answer will be yes.
But that doesn't change anything, you can still ask: "what about the set of all sets that are not members of themselves?"
How can this set be a member of itself, when the defining characteristic needed for membership is that it isn't a member of itself?
Maybe you try to make this an exceptional case in your theory - this is the only set that isn't a member of itself
But if it isn't a member of itself, then that immediately qualifies it to be a member of itself, so then it has to be a member of itself.
So you are still faced with Russell's paradox.
It seems clear that there is no getting around it in a naive theory that has a notion of set and membership except to restrict your reasoning in some way.
UNASKING THE QUESTION
One way around it, quite natural, is to say that you simply can't ask if the set of all oranges is an orange. That's quite reasonable sounding. Maybe it is a nonsensical question, and that's why we got into trouble?
That was Russell's solution in his type theory. This works but is rather cumbersome. Because as you follow through the implications you find you have to have an infinite number of layers of types, to avoid paradox.
Or you could look for a way to develop a logic that doesn't have any notion of membership or set built into it - and find some other primitive concept in its place.
But if so what? There are many areas of discourse where we can get by without much use of sets and collections. But sometimes we do need them.
How are you going to talk about football teams and footballers and the premier league without using the concept of membership of a group?
And once you have got as far as defining membership - even if it is not a primitive concept of your theory - then if you just "follow your nose" with naive ideas - you are not far from defining the set of all sets and reasoning about it. Before long you end up with Russell's paradox.
This arises even more so in mathematics where similar ideas are needed all the time.
STANDARD WAYS AROUND IT
The standard way to get around it is to restrict your formal logical reasoning in some way so that you can't discuss this question in your theory.
The most common approach is to say that you can't talk about "sets" in this perfectly general way. Instead you can only talk about particular types of set, and when talk about the "set of all sets" you have to introduce new terms - call it something else, e.g. a class, or a set of type 2 or whatever.
Russell's type theory was a start there with his infinitely many types.
Nowadays we have a simpler approach where you just two types of collection - sets, and then classes. When you talk about classes they can't be members of anything and questions of membership are meaningless. So then the paradox doesn't arise for the classes.
- There is no such thing in your language as a class of all classes.
- The sets can be members of sets or classes - but you have a carefully designed axiom system that makes it impossible to construct a set of all sets.
QUINE'S LOGIC - RUSSELL'S QUESTION GETS "LOST IN TRANSLATION"
Another, less common approach is to permit the set of all sets to be a member of itself and restrict your logical reasoning in other ways, restrict the language you use to construct sets so that statement "set of all sets that are not a member of themselves" can't be said in your language - that's Quine's solution in his "new foundations".
In Quine's logic, then being "a member of itself" is not a meaningful concept in his logic. He has the requirement that all properties like that have to be "stratified" - it's a complicated requirement - but basically it rules out properties that relate a set directly to itself by membership.
So, you just can't discuss sets and whether they are members of themselves or not in his theory at all. Any question like that gets "lost in translation" when you try to ask it. So then Russell's question can't be asked so the paradox doesn't arise.
In some ways Quine's solution is closer to intuitive set theory, because it permits you to have a set of all sets which is a set. Let's you talk about the "set of all sets" with the same language as more ordinary sets.
But one way or another you have to restrict what you are permitted to talk about in your logical language you use to reason about sets, or restrict interpretation of the word "set", or you run into Russell's paradox.
THOUGH WE CAN SOLVE IT, OUR SOLUTIONS ARE RATHER CONTRIVED
Up to Frege, nobody developed a theory to use as a foundation for logic that avoided Russell's paradox.
Now we can avoid it with hindsight. But it's not easy. Why did nobody come up with any of these ideas at all until the early twentieth century?
It usually seems at least a bit artificial - like they have obviously restricted the language in order to avoid the paradox.
Why is that? This is not an issue just for maths but also for all attempts to formalize intuitive logic whenever we use it to reason about collections of things.
Sometimes proponents of various ways around it will argue that their solution is a natural one to use.
For instance, proponents of Quine's approach can say that philosophically it is more natural. But then if that's what you think, you are still faced with the question - if it is so natural - why did we need Russell to point out his paradox first before anyone came up with a way of thinking about sets that avoids it?
It is based on simple ideas. If it is so natural, why did no philosopher in the entire history of human civilizations before Russell's paradox come up with Quine's approach or anything similar?
WHY DID NOBODY THINK OF RUSSELL'S PARADOX BEFORE?
I think this is also a curiosity.
This paradox is so simple to state and uses a basic concept that everyone is familiar with - the idea of membership of a group or collection. Even back to the ancient Greeks and Chinese - many of Euclid's theorems are far more complex than this. Or later on the Indian and the Arabic mathematicians and philosophers. They developed many intricate chains of logical reasoning - but not this one.
And you don't even need the concept of zero, or negative numbers, or ratios or any of the new concepts of modern mathematics to understand it.
It may well be the most simple genuinely totally new chain of pure logical reasoning to be developed in modern times, One that got missed by everyone right up to 1901 when Russell came up with the paradox.
So why didn't some of the ancient mathematicians or philosophers think of this. Even the nineteenth century mathematicians, came up with many elaborate ideas about sets, and various very complicated to state paradoxes, and worked on these ideas continuously for years in the case of Frege and Cantor - but for some reason never came up with this one.
We know that it is possible for humans to miss simple concepts. For thousands of years, entire civilizations rose and fell without developing the modern idea of a negative number for instance. Right up to the middle ages, mathematicians would treat what we now consider to be a single equation (e.g. the quadratic equation) as several different cases, because they didn't have the concept of negative numbers in the modern sense.
And same for the notion of zero for positional notation. Or ratios with arbitrary numbers at the top and bottom (for instance the Babylonians had "unit fractions" with the requirement that the top number in the fraction had to be 1, and did clumsy arithmetic where everything had to be expressed in unit fractions).
So - perhaps it's an example of this? And if so, I can't help but wonder, are there any other remaining really simple concepts that we haven't yet thought of?
I wonder, as there are surely many ideas that we haven't yet thought of - how simple is the simplest concept that we haven't yet thought of?
No way to answer that I suppose.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
Lives in Isle of Mull
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