Am reading “the mathematical century” by unpronounceable.
Refers to 1900 to 2000.

Chapter 1 is “foundations of mathematics”

The evolution of what is considered the foundation of mathematics is interesting. In chronological sequence:

• Classical approach: numbers, algebra, geometry, analysis.
• Naive set theory of Frege (1879)
• Zermello-Fraenkel set theory (1921)
• ’Structure’ by Bouraki (1949)
• ’Category’ introduced in 1945 and found to underlie all of set theory in 1969.
• ’Lambda calculus’ introduced in 1933, formalised in 1969, and found to underlie all computing in the 1980s.

Lambda calculus is still considered a bit ‘out there’ as a foundation for all mathematics, but it does neatly avoid the paradox that killed naive set theory.

mollwollfumble said:

Am reading “the mathematical century” by unpronounceable.
Refers to 1900 to 2000.

Chapter 1 is “foundations of mathematics”

The evolution of what is considered the foundation of mathematics is interesting. In chronological sequence:

• Classical approach: numbers, algebra, geometry, analysis.
• Naive set theory of Frege (1879)
• Zermello-Fraenkel set theory (1921)
• ’Structure’ by Bouraki (1949)
• ’Category’ introduced in 1945 and found to underlie all of set theory in 1969.
• ’Lambda calculus’ introduced in 1933, formalised in 1969, and found to underlie all computing in the 1980s.

Lambda calculus is still considered a bit ‘out there’ as a foundation for all mathematics, but it does neatly avoid the paradox that killed naive set theory.

A problem with this book is that each part is impossible to understand unless you already know it. And if you already know it then you’re not learning anything new about it.

mollwollfumble said:

mollwollfumble said:

Am reading “the mathematical century” by unpronounceable.
Refers to 1900 to 2000.

Chapter 1 is “foundations of mathematics”

The evolution of what is considered the foundation of mathematics is interesting. In chronological sequence:

• Classical approach: numbers, algebra, geometry, analysis.
• Naive set theory of Frege (1879)
• Zermello-Fraenkel set theory (1921)
• ’Structure’ by Bouraki (1949)
• ’Category’ introduced in 1945 and found to underlie all of set theory in 1969.
• ’Lambda calculus’ introduced in 1933, formalised in 1969, and found to underlie all computing in the 1980s.

Lambda calculus is still considered a bit ‘out there’ as a foundation for all mathematics, but it does neatly avoid the paradox that killed naive set theory.

A problem with this book is that each part is impossible to understand unless you already know it. And if you already know it then you’re not learning anything new about it.

A common problem with maths books, in my experience.

What is the paradox that did away with naïve set theory (which I’ve not heard of before)?

I remember when they taught naive set theory at school and I thought has easy it would be and the teacher said that’s because of your lack of experience

The Rev Dodgson said:

mollwollfumble said:

mollwollfumble said:

Am reading “the mathematical century” by unpronounceable.
Refers to 1900 to 2000.

Chapter 1 is “foundations of mathematics”

The evolution of what is considered the foundation of mathematics is interesting. In chronological sequence:

• Classical approach: numbers, algebra, geometry, analysis.
• Naive set theory of Frege (1879)
• Zermello-Fraenkel set theory (1921)
• ’Structure’ by Bouraki (1949)
• ’Category’ introduced in 1945 and found to underlie all of set theory in 1969.
• ’Lambda calculus’ introduced in 1933, formalised in 1969, and found to underlie all computing in the 1980s.

Lambda calculus is still considered a bit ‘out there’ as a foundation for all mathematics, but it does neatly avoid the paradox that killed naive set theory.

A problem with this book is that each part is impossible to understand unless you already know it. And if you already know it then you’re not learning anything new about it.

A common problem with maths books, in my experience.

What is the paradox that did away with naïve set theory (which I’ve not heard of before)?

Naive set theory allows sets that are members of themselves and allows the set of all sets.

Bertrand Russell’s “the set of all sets that are members of themselves. Is that set a member of itself? If true then false and if false then true.”

https://en.m.wikipedia.org/wiki/Naive_set_theory

https://en.m.wikipedia.org/wiki/Begriffsschrift is where it was first introduced.

An introduction to Zakovian mathematics will interest you, turned the establishment on its head

Zarkov said:

An introduction to Zakovian mathematics will interest you, turned the establishment on its head

You’ll have to be more specific, or more vague, than that.

mollwollfumble said:

Zarkov said:

An introduction to Zakovian mathematics will interest you, turned the establishment on its head

You’ll have to be more specific, or more vague, than that.

It has concepts such as infinity plus 1 and almost infinite for very large numbers.

Zarkov said:

mollwollfumble said:

Zarkov said:

An introduction to Zakovian mathematics will interest you, turned the establishment on its head

You’ll have to be more specific, or more vague, than that.

It has concepts such as infinity plus 1 and almost infinite for very large numbers.

That’s already in my published monograph on infinite numbers. You’ll have to do better than that.

My latest thought on the matter relates to the difference between “potential” infinity that can only be approached but never reached and “actual” infinity that can be manipulated algebraically. Because actual infinity can be manipulated algebraically, infinity plus 1 does not have to equal infinity.

It was pointed out to me today that if actual infinity does not exist then real numbers do not exist, they would each just be the endpoints approached but never reached of a sequence of rational numbers.

Real numbers exist. Therefore actual infinity exists. Therefore it is perfectly valid to say that infinity plus 1 can differ from infinity.

As for “almost infinite”. That’s the radius of the universe. It has a less than 50% probability of being less than infinity.

Can Zarkov top that?

mollwollfumble said:

As for “almost infinite”. That’s the radius of the universe. It has a less than 50% probability of being less than infinity.

Please show working.

The Rev Dodgson said:

mollwollfumble said:

As for “almost infinite”. That’s the radius of the universe. It has a less than 50% probability of being less than infinity.

Please show working.

You asked that question before, i answered it before. You didn’t like the answer before. Are you sure you want a recap?

Starting point, the curvature of the universe is either positive, negative or zero. Occam’s razor suggests zero. Data suggests zero. Zero curvature means infinite. Negative curvature means infinite. Positive curvature means large finite.

mollwollfumble said:

The Rev Dodgson said:

mollwollfumble said:

As for “almost infinite”. That’s the radius of the universe. It has a less than 50% probability of being less than infinity.

Please show working.

You asked that question before, i answered it before. You didn’t like the answer before. Are you sure you want a recap?

Starting point, the curvature of the universe is either positive, negative or zero. Occam’s razor suggests zero. Data suggests zero. Zero curvature means infinite. Negative curvature means infinite. Positive curvature means large finite.

Ok, I thought we’d probably discussed it before, but I couldn’t remember the details.

Still don’t like it :)

The Rev Dodgson said:

mollwollfumble said:

The Rev Dodgson said:

Please show working.

You asked that question before, i answered it before. You didn’t like the answer before. Are you sure you want a recap?

Starting point, the curvature of the universe is either positive, negative or zero. Occam’s razor suggests zero. Data suggests zero. Zero curvature means infinite. Negative curvature means infinite. Positive curvature means large finite.

Ok, I thought we’d probably discussed it before, but I couldn’t remember the details.

Still don’t like it :)

On the asumption that curvature is constant, which it may not be. It may wander between positive and negative for instance.

If string theory is correct then nonzero curvature makes more sense. But if we take the demise of supersymmety as indicative of the demise of string theory then there’s no reason to suppose that the universe is anything other than flat.

mollwollfumble said:

The Rev Dodgson said:

mollwollfumble said:

A problem with this book is that each part is impossible to understand unless you already know it. And if you already know it then you’re not learning anything new about it.

A common problem with maths books, in my experience.

What is the paradox that did away with naïve set theory (which I’ve not heard of before)?

Naive set theory allows sets that are members of themselves and allows the set of all sets.

Bertrand Russell’s “the set of all sets that are members of themselves. Is that set a member of itself? If true then false and if false then true.”

https://en.m.wikipedia.org/wiki/Naive_set_theory

https://en.m.wikipedia.org/wiki/Begriffsschrift is where it was first introduced.

I have always been a bit puzzled as to why that sort of paradox is seen as a problem. A set of all sets that are not members of themself is clearly not a valid set. Why is that a problem?

mollwollfumble said:

The Rev Dodgson said:

mollwollfumble said:

You asked that question before, i answered it before. You didn’t like the answer before. Are you sure you want a recap?

Starting point, the curvature of the universe is either positive, negative or zero. Occam’s razor suggests zero. Data suggests zero. Zero curvature means infinite. Negative curvature means infinite. Positive curvature means large finite.

Ok, I thought we’d probably discussed it before, but I couldn’t remember the details.

Still don’t like it :)

On the asumption that curvature is constant, which it may not be. It may wander between positive and negative for instance.

If string theory is correct then nonzero curvature makes more sense. But if we take the demise of supersymmety as indicative of the demise of string theory then there’s no reason to suppose that the universe is anything other than flat.

Even if we temporarily accept that, a finite flat universe remains a possibility.

The Rev Dodgson said:

mollwollfumble said:

The Rev Dodgson said:

Ok, I thought we’d probably discussed it before, but I couldn’t remember the details.

Still don’t like it :)

On the asumption that curvature is constant, which it may not be. It may wander between positive and negative for instance.

If string theory is correct then nonzero curvature makes more sense. But if we take the demise of supersymmety as indicative of the demise of string theory then there’s no reason to suppose that the universe is anything other than flat.

Even if we temporarily accept that, a finite flat universe remains a possibility.

Not unless it’s a topological mutiverse, like a torus, where travelling in one direction eventually brings you back to the starting point. This seems pretty unlikely.

But you’re missing the point. The radius is defined as the inverse of the curvature. Flat is zero curvature. So even if the universe was finite and flat, it would still have infinite radius. OK, that sounds like a contradiction, so i’d better not pursue that thought any further.

The Rev Dodgson said:

mollwollfumble said:

The Rev Dodgson said:

A common problem with maths books, in my experience.

What is the paradox that did away with naïve set theory (which I’ve not heard of before)?

Naive set theory allows sets that are members of themselves and allows the set of all sets.

Bertrand Russell’s “the set of all sets that are members of themselves. Is that set a member of itself? If true then false and if false then true.”

https://en.m.wikipedia.org/wiki/Naive_set_theory

https://en.m.wikipedia.org/wiki/Begriffsschrift is where it was first introduced.

I have always been a bit puzzled as to why that sort of paradox is seen as a problem. A set of all sets that are not members of themself is clearly not a valid set. Why is that a problem?

Not sure i understand it myself. But let me try.

The concept of “set of all” is a necessary part of naive set theory. For example, R is the set of all real numbers. Each real number is also defined as a set.

Extending that idea, naive set theory allows “the set of all sets”.

Now “the set of all sets” contains itself. It has to or it wouldn’t be complete. But some sets do not contain themselves. Zero is defined as the null set, which contains nothing, so cannot contain itself.

So in naive set theory there must be a “set of all sets that do not contain themselves” that contains the null set but does not contain the “set of all sets”.

Hence the paradox.

It’s difficult to see a way out because eliminating “the set of all sets” without also eliminating “the set of all sets that a real numbers” is not trivial.

mollwollfumble said:

The Rev Dodgson said:

mollwollfumble said:

On the asumption that curvature is constant, which it may not be. It may wander between positive and negative for instance.

If string theory is correct then nonzero curvature makes more sense. But if we take the demise of supersymmety as indicative of the demise of string theory then there’s no reason to suppose that the universe is anything other than flat.

Even if we temporarily accept that, a finite flat universe remains a possibility.

Not unless it’s a topological mutiverse, like a torus, where travelling in one direction eventually brings you back to the starting point. This seems pretty unlikely.

But you’re missing the point. The radius is defined as the inverse of the curvature. Flat is zero curvature. So even if the universe was finite and flat, it would still have infinite radius. OK, that sounds like a contradiction, so i’d better not pursue that thought any further.

I don’t have any problem with a finite universe with infinite radius at all, at all.

mollwollfumble said:

So in naive set theory there must be a “set of all sets that do not contain themselves” that contains the null set but does not contain the “set of all sets”.

Hence the paradox.

It’s difficult to see a way out because eliminating “the set of all sets” without also eliminating “the set of all sets that a real numbers” is not trivial.

Why “must” there be such a set?

If a description of a set is self contradictory, then surely it isn’t a valid set, hence no paradox.

The Rev Dodgson said:

mollwollfumble said:

So in naive set theory there must be a “set of all sets that do not contain themselves” that contains the null set but does not contain the “set of all sets”.

Hence the paradox.

It’s difficult to see a way out because eliminating “the set of all sets” without also eliminating “the set of all sets that a real numbers” is not trivial.

Why “must” there be such a set?

If a description of a set is self contradictory, then surely it isn’t a valid set, hence no paradox.

“Must” because naive set theory says that we can partition “the set of all sets” into two disjoint halves, one that contains sets that are members of themselves, such as “the set of all sets” and one that contains everything else. Both halves are nonempty, and the one that isn’t “the set of all sets that are members of themselves” is the paradoxical one.

Perhaps i can find an analogy that will help. Take “the set of all sets that are real numbers” and partition it into two halves, one containing the rational numbers, and one containing everything else. Both halves are nonempty and infinite. The set containing everything that isn’t a rational number is the set of irrational numbers. That’s how an irrational number is defined in naive set theory.

If we eliminate the operation that produces “the set of all sets that are not members of themselves” from mathematics then we also eliminate “the set of irrational numbers” from mathematics, which is a very undesirable result.

mollwollfumble said:

The Rev Dodgson said:

mollwollfumble said:

So in naive set theory there must be a “set of all sets that do not contain themselves” that contains the null set but does not contain the “set of all sets”.

Hence the paradox.

It’s difficult to see a way out because eliminating “the set of all sets” without also eliminating “the set of all sets that a real numbers” is not trivial.

Why “must” there be such a set?

If a description of a set is self contradictory, then surely it isn’t a valid set, hence no paradox.

“Must” because naive set theory says that we can partition “the set of all sets” into two disjoint halves, one that contains sets that are members of themselves, such as “the set of all sets” and one that contains everything else. Both halves are nonempty, and the one that isn’t “the set of all sets that are members of themselves” is the paradoxical one.

Perhaps i can find an analogy that will help. Take “the set of all sets that are real numbers” and partition it into two halves, one containing the rational numbers, and one containing everything else. Both halves are nonempty and infinite. The set containing everything that isn’t a rational number is the set of irrational numbers. That’s how an irrational number is defined in naive set theory.

If we eliminate the operation that produces “the set of all sets that are not members of themselves” from mathematics then we also eliminate “the set of irrational numbers” from mathematics, which is a very undesirable result.

But that assumes that the set of all sets can be divided into “sets that are members of themselves” and “sets that are not members of themselves”.

That’s a strange assumption because we know that there is at least one set whose membership of itself cannot be defined, so when we remove the “set of all sets that are members of themselves” from the “set of all sets” we are left with two distinct sets, “the set of all sets that are not members of themselves” and “the set of all sets that have undefined membership of themselves”

If all numbers can be divided into rational or irrational, then the set of all numbers is not analogous to the set of all sets.

The Rev Dodgson said:

I have always been a bit puzzled as to why that sort of paradox is seen as a problem. A set of all sets that are not members of themself is clearly not a valid set. Why is that a problem?

A paradox indicates that the set of axioms is inconsistent.

The Rev Dodgson said:

But that assumes that the set of all sets can be divided into “sets that are members of themselves” and “sets that are not members of themselves”.

That’s actually the problem. The axioms of naive set theory admit the construction of such a division. It is worth noting that the set of sets that are members of themselves is just as invalid as the set of sets that are not members of themselves, even though it is not a paradox in the usual sense. The logical complement of a paradox is a circularity.

It should be noted that a paradox or circularity occurs because there are too many axioms or the set of axioms is too strong. Thus, the paradox or circularity can’t be removed by adding more axioms, but by removing axioms or reducing their strength.

KJW said:

The Rev Dodgson said:

I have always been a bit puzzled as to why that sort of paradox is seen as a problem. A set of all sets that are not members of themself is clearly not a valid set. Why is that a problem?

A paradox indicates that the set of axioms is inconsistent.

The Rev Dodgson said:

But that assumes that the set of all sets can be divided into “sets that are members of themselves” and “sets that are not members of themselves”.

That’s actually the problem. The axioms of naive set theory admit the construction of such a division. It is worth noting that the set of sets that are members of themselves is just as invalid as the set of sets that are not members of themselves, even though it is not a paradox in the usual sense. The logical complement of a paradox is a circularity.

But I don’t see why the apparent paradox is not removed simply by adding an axiom that the set of all sets that are not members of themselves is not a valid set.
I also don’t see why the set of all sets that are members of themselves is invalid.
And I’m not sure what “The logical complement of a paradox is a circularity” means.

KJW said:

It should be noted that a paradox or circularity occurs because there are too many axioms or the set of axioms is too strong. Thus, the paradox or circularity can’t be removed by adding more axioms, but by removing axioms or reducing their strength.

I don’t understand that either.
What is the proof that “the paradox or circularity can’t be removed by adding more axioms, but by removing axioms or reducing their strength”?
What is the problem with the proposed additional axiom given above?

The Rev Dodgson said:

But I don’t see why the apparent paradox is not removed simply by adding an axiom that the set of all sets that are not members of themselves is not a valid set.

By adding a new axiom, the original axioms remain inconsistent. Also, the new axiom will be inconsistent with the original axioms. Axioms don’t undo the effects of other axioms. Axioms are constraints. Inconsistencies can occur if there are too many constraints. For example, if fewer axioms define an object uniquely, then any additional axioms can either agree with the defined object, or they can disagree with the defined object. In the former case, the additional axioms are unnecessarily redundant, and in the latter case, they are inconsistent. One can’t remove an inconsistency by adding more constraints.

The Rev Dodgson said:

I also don’t see why the set of all sets that are members of themselves is invalid.
And I’m not sure what “The logical complement of a paradox is a circularity” means.

Consider the following paradox:

This statement is false.

The logical complement of this paradox is:

This statement is true.

The paradox is clearly inconsistent. But although the logical complement is not inconsistent, it is a circular statement that is meaningless. If one considers the set of all sets that are members of themselves, then one can’t determine if the set itself is a member.

KJW said:

The Rev Dodgson said:

But I don’t see why the apparent paradox is not removed simply by adding an axiom that the set of all sets that are not members of themselves is not a valid set.

By adding a new axiom, the original axioms remain inconsistent. Also, the new axiom will be inconsistent with the original axioms. Axioms don’t undo the effects of other axioms. Axioms are constraints. Inconsistencies can occur if there are too many constraints. For example, if fewer axioms define an object uniquely, then any additional axioms can either agree with the defined object, or they can disagree with the defined object. In the former case, the additional axioms are unnecessarily redundant, and in the latter case, they are inconsistent. One can’t remove an inconsistency by adding more constraints.

The Rev Dodgson said:

I also don’t see why the set of all sets that are members of themselves is invalid.
And I’m not sure what “The logical complement of a paradox is a circularity” means.

Consider the following paradox:

This statement is false.

The logical complement of this paradox is:

This statement is true.

The paradox is clearly inconsistent. But although the logical complement is not inconsistent, it is a circular statement that is meaningless. If one considers the set of all sets that are members of themselves, then one can’t determine if the set itself is a member.

OK then, rather than add an axiom, how about we remove the axiom: “All sets are either members of themself or not members of themself”.?

The Rev Dodgson said:

OK then, rather than add an axiom, how about we remove the axiom: “All sets are either members of themself or not members of themself”.?

“All sets are either members of themself or not members of themself” is not an axiom. Perhaps it would be instructive to look at the axioms of Zermello-Fraenkel set theory to see how they remove the paradox.

KJW said:

The Rev Dodgson said:

OK then, rather than add an axiom, how about we remove the axiom: “All sets are either members of themself or not members of themself”.?

“All sets are either members of themself or not members of themself” is not an axiom. Perhaps it would be instructive to look at the axioms of Zermello-Fraenkel set theory to see how they remove the paradox.

Agree. Challenging, though.

I can’t help wondering. We know from Godel that mathematics cannot be both consistent and complete. Zermello-Fraenkel set theory is known to be incomplete. Naive set theory is known to be inconsistent. Would it be so bad to use a system that is complete and inconsistent rather than one that is consistent and incomplete?

KJW said:

The Rev Dodgson said:

OK then, rather than add an axiom, how about we remove the axiom: “All sets are either members of themself or not members of themself”.?

“All sets are either members of themself or not members of themself” is not an axiom. Perhaps it would be instructive to look at the axioms of Zermello-Fraenkel set theory to see how they remove the paradox.

OK I’ll have a look (but didn’t you say we couldn’t get around this problem by adding axioms?).

If ““All sets are either members of themself or not members of themself” is not an axiom”, then why do we assume that all sets are either members of themself or not members of themselves?

mollwollfumble said:

KJW said:

The Rev Dodgson said:

OK then, rather than add an axiom, how about we remove the axiom: “All sets are either members of themself or not members of themself”.?

“All sets are either members of themself or not members of themself” is not an axiom. Perhaps it would be instructive to look at the axioms of Zermello-Fraenkel set theory to see how they remove the paradox.

Agree. Challenging, though.

I can’t help wondering. We know from Godel that mathematics cannot be both consistent and complete. Zermello-Fraenkel set theory is known to be incomplete. Naive set theory is known to be inconsistent. Would it be so bad to use a system that is complete and inconsistent rather than one that is consistent and incomplete?

That reminds me.

I think I have read that Gödel himself was not sure whether his work showed something really profound or really trivial. If that is the case, how come no-one else these days seems to share that doubt?