What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Is there more than one mathematical universe?
Joel David Hamkins Professor of Logic Sir Peter Strawson Fellow
University of Oxford University College
Wijsgerig Festival DRIFT Amsterdam, 11 May 2019
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What does it mean to make existence assertions in mathematics?
Is there an ideal mathematical reality that mathematical assertions are about?
Is there more than one such universe? None?
What are mathematical objects? What are numbers?
Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is there an ideal mathematical reality that mathematical assertions are about?
Is there more than one such universe? None?
What are mathematical objects? What are numbers?
Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics.
What does it mean to make existence assertions in mathematics?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is there more than one such universe? None?
What are mathematical objects? What are numbers?
Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics.
What does it mean to make existence assertions in mathematics?
Is there an ideal mathematical reality that mathematical assertions are about?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What are mathematical objects? What are numbers?
Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics.
What does it mean to make existence assertions in mathematics?
Is there an ideal mathematical reality that mathematical assertions are about?
Is there more than one such universe? None?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics.
What does it mean to make existence assertions in mathematics?
Is there an ideal mathematical reality that mathematical assertions are about?
Is there more than one such universe? None?
What are mathematical objects? What are numbers?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Ontology in mathematics The theme of this conference is ontology. I shall discuss ontology in mathematics.
What does it mean to make existence assertions in mathematics?
Is there an ideal mathematical reality that mathematical assertions are about?
Is there more than one such universe? None?
What are mathematical objects? What are numbers?
Does every mathematical problem have a definite answer? Does every mathematical assertion have a definite truth value?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve.
Roman numerals LVII: take fifty, add five, then two more.
We should distinguish between the number and the numeral, between the mathematical object and the description of that object.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a number?
Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Roman numerals LVII: take fifty, add five, then two more.
We should distinguish between the number and the numeral, between the mathematical object and the description of that object.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a number?
Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven.
Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford We should distinguish between the number and the numeral, between the mathematical object and the description of that object.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a number?
Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven.
Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve.
Roman numerals LVII: take fifty, add five, then two more.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a number?
Consider the number 57. What is it? The notation 57 is a recipe: take five tens and add seven.
Binary 111001 is another: start with thirty-two, fold in sixteen and eight; garnish with one on top. Chill and serve.
Roman numerals LVII: take fifty, add five, then two more.
We should distinguish between the number and the numeral, between the mathematical object and the description of that object.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford In the mines, they found the largest number!
A gigantic number 3. Over four meters tall; made of stone.
Broken numbers from the mine were used for fractions.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The largest number
In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford A gigantic number 3. Over four meters tall; made of stone.
Broken numbers from the mine were used for fractions.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The largest number
In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis.
In the mines, they found the largest number!
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Broken numbers from the mine were used for fractions.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The largest number
In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis.
In the mines, they found the largest number!
A gigantic number 3. Over four meters tall; made of stone.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The largest number
In the children’s novel, The phantom tollbooth, numbers come from the number mine in Digitopolis.
In the mines, they found the largest number!
A gigantic number 3. Over four meters tall; made of stone.
Broken numbers from the mine were used for fractions.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Triangular numbers can be arranged in a triangle.
1 3 6 10 15
One proceeds to the hexagonal numbers and so forth.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Kinds of numbers Square numbers can be arranged in a square.
1 4 9 16 25
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Kinds of numbers Square numbers can be arranged in a square.
1 4 9 16 25
Triangular numbers can be arranged in a triangle.
1 3 6 10 15
One proceeds to the hexagonal numbers and so forth.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford But, this depends on the base we use to represent the number.
Thus, palindromicity is not a property of the number, but of the numeral. Every number is a palindrome in any sufficiently large base. It becomes a single digit.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Palindromic numbers
Palindromic numbers have digits reading the same forwards and backwards.
121 523323325
Like the phrase, “I prefer pi.”
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Every number is a palindrome in any sufficiently large base. It becomes a single digit.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Palindromic numbers
Palindromic numbers have digits reading the same forwards and backwards.
121 523323325
Like the phrase, “I prefer pi.”
But, this depends on the base we use to represent the number.
Thus, palindromicity is not a property of the number, but of the numeral.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Palindromic numbers
Palindromic numbers have digits reading the same forwards and backwards.
121 523323325
Like the phrase, “I prefer pi.”
But, this depends on the base we use to represent the number.
Thus, palindromicity is not a property of the number, but of the numeral. Every number is a palindrome in any sufficiently large base. It becomes a single digit.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Yet, we say that they are equal
1 3 = . 2 6
But how can two things be identical, if one has a property the other does not? Being in lowest terms is not a property of numbers, but of representations of numbers, of numerals.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A confounding case Consider the fractions 1 3 2 6 The first is in lowest terms; the second is not.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford But how can two things be identical, if one has a property the other does not? Being in lowest terms is not a property of numbers, but of representations of numbers, of numerals.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A confounding case Consider the fractions 1 3 2 6 The first is in lowest terms; the second is not. Yet, we say that they are equal
1 3 = . 2 6
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Being in lowest terms is not a property of numbers, but of representations of numbers, of numerals.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A confounding case Consider the fractions 1 3 2 6 The first is in lowest terms; the second is not. Yet, we say that they are equal
1 3 = . 2 6
But how can two things be identical, if one has a property the other does not?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A confounding case Consider the fractions 1 3 2 6 The first is in lowest terms; the second is not. Yet, we say that they are equal
1 3 = . 2 6
But how can two things be identical, if one has a property the other does not? Being in lowest terms is not a property of numbers, but of representations of numbers, of numerals.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a number?
Philosophers of mathematics describe a variety of philosophical attitudes to take towards the nature of mathematical existence. platonism formalism intuitionism constructivism logicism structuralism nominalism fictionalism
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford A particular line or circle, drawn on paper, is imperfect.
In the platonic realm, there are ideal forms, perfect lines and circles.
And numbers, functions and sets.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Platonism
According to platonism, numbers and other mathematical objects exist as abstract objects in a realm of ideal forms.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford In the platonic realm, there are ideal forms, perfect lines and circles.
And numbers, functions and sets.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Platonism
According to platonism, numbers and other mathematical objects exist as abstract objects in a realm of ideal forms.
A particular line or circle, drawn on paper, is imperfect.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford And numbers, functions and sets.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Platonism
According to platonism, numbers and other mathematical objects exist as abstract objects in a realm of ideal forms.
A particular line or circle, drawn on paper, is imperfect.
In the platonic realm, there are ideal forms, perfect lines and circles.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Platonism
According to platonism, numbers and other mathematical objects exist as abstract objects in a realm of ideal forms.
A particular line or circle, drawn on paper, is imperfect.
In the platonic realm, there are ideal forms, perfect lines and circles.
And numbers, functions and sets.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What matters is the structural roles played by numbers in a system exhibiting a certain overall structure.
The integer number zero, for example, is the additive identity in the ring of integers.
It is also the only additive integer idempotent, the only number z for which z + z = z.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Structuralism
According to structuralism, it doesn’t matter what numbers are, as individuals.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford The integer number zero, for example, is the additive identity in the ring of integers.
It is also the only additive integer idempotent, the only number z for which z + z = z.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Structuralism
According to structuralism, it doesn’t matter what numbers are, as individuals.
What matters is the structural roles played by numbers in a system exhibiting a certain overall structure.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Structuralism
According to structuralism, it doesn’t matter what numbers are, as individuals.
What matters is the structural roles played by numbers in a system exhibiting a certain overall structure.
The integer number zero, for example, is the additive identity in the ring of integers.
It is also the only additive integer idempotent, the only number z for which z + z = z.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford zero is not a successor 0 6= S(x). The successor operation is one-to-one
S(x) = S(y) ←→ x = y.
Every number is generated from 0 by successor. That is, if A is a set of numbers, with 0 ∈ A and x ∈ A =⇒ S(x) ∈ A, then A has all numbers.
All the familiar arithmetic truths follow as consequences.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Dedekind arithmetic For a structuralist account of the natural numbers
N = { 0, 1, 2,... } Dedekind identified the following axioms:
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford The successor operation is one-to-one
S(x) = S(y) ←→ x = y.
Every number is generated from 0 by successor. That is, if A is a set of numbers, with 0 ∈ A and x ∈ A =⇒ S(x) ∈ A, then A has all numbers.
All the familiar arithmetic truths follow as consequences.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Dedekind arithmetic For a structuralist account of the natural numbers
N = { 0, 1, 2,... } Dedekind identified the following axioms: zero is not a successor 0 6= S(x).
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Every number is generated from 0 by successor. That is, if A is a set of numbers, with 0 ∈ A and x ∈ A =⇒ S(x) ∈ A, then A has all numbers.
All the familiar arithmetic truths follow as consequences.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Dedekind arithmetic For a structuralist account of the natural numbers
N = { 0, 1, 2,... } Dedekind identified the following axioms: zero is not a successor 0 6= S(x). The successor operation is one-to-one
S(x) = S(y) ←→ x = y.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Dedekind arithmetic For a structuralist account of the natural numbers
N = { 0, 1, 2,... } Dedekind identified the following axioms: zero is not a successor 0 6= S(x). The successor operation is one-to-one
S(x) = S(y) ←→ x = y.
Every number is generated from 0 by successor. That is, if A is a set of numbers, with 0 ∈ A and x ∈ A =⇒ S(x) ∈ A, then A has all numbers.
All the familiar arithmetic truths follow as consequences.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford This means that the axioms determine a unique structure.
Any two models of Dedekind arithmetic are isomorphic copies of one another.
Given two Dedekind systems of arithmetic, one matches up their 0s, and then S(0) and S(S(0)) so on, proving ultimately that this provides an isomorphism.
According to structuralism, we needn’t say anything more about what natural numbers are, except that they fulfill the Dedekind axioms.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity
Dedekind established that his axioms were categorical.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Given two Dedekind systems of arithmetic, one matches up their 0s, and then S(0) and S(S(0)) so on, proving ultimately that this provides an isomorphism.
According to structuralism, we needn’t say anything more about what natural numbers are, except that they fulfill the Dedekind axioms.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity
Dedekind established that his axioms were categorical.
This means that the axioms determine a unique structure.
Any two models of Dedekind arithmetic are isomorphic copies of one another.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford According to structuralism, we needn’t say anything more about what natural numbers are, except that they fulfill the Dedekind axioms.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity
Dedekind established that his axioms were categorical.
This means that the axioms determine a unique structure.
Any two models of Dedekind arithmetic are isomorphic copies of one another.
Given two Dedekind systems of arithmetic, one matches up their 0s, and then S(0) and S(S(0)) so on, proving ultimately that this provides an isomorphism.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity
Dedekind established that his axioms were categorical.
This means that the axioms determine a unique structure.
Any two models of Dedekind arithmetic are isomorphic copies of one another.
Given two Dedekind systems of arithmetic, one matches up their 0s, and then S(0) and S(S(0)) so on, proving ultimately that this provides an isomorphism.
According to structuralism, we needn’t say anything more about what natural numbers are, except that they fulfill the Dedekind axioms.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford The real numbers are Dedekind complete: every cut is filled.
And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it contin- uous; [Ded63, §3, Continuity and Irrational numbers, 1872]
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
Dedekind observed how every real number is determined by the cut it makes in the rational numbers.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
Dedekind observed how every real number is determined by the cut it makes in the rational numbers.
The real numbers are Dedekind complete: every cut is filled.
And if we knew for certain that space was discontinuous there would be nothing to prevent us, in case we so desired, from filling up its gaps, in thought, and thus making it contin- uous; [Ded63, §3, Continuity and Irrational numbers, 1872]
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford The method of “postulating” what we want has many advan- tages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil. [Rus19, p. 71]
He undertakes the ‘honest toil’ of placing an ordered field structure on the set of Dedekind cuts, and proving that they thereby become Dedekind complete. On this account, a real number is a Dedekind cut.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
R is the Dedekind completion of Q
Russell explained how to construct the real numbers using Dedekind cuts.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford He undertakes the ‘honest toil’ of placing an ordered field structure on the set of Dedekind cuts, and proving that they thereby become Dedekind complete. On this account, a real number is a Dedekind cut.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
R is the Dedekind completion of Q
Russell explained how to construct the real numbers using Dedekind cuts.
The method of “postulating” what we want has many advan- tages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil. [Rus19, p. 71]
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
R is the Dedekind completion of Q
Russell explained how to construct the real numbers using Dedekind cuts.
The method of “postulating” what we want has many advan- tages; they are the same as the advantages of theft over honest toil. Let us leave them to others and proceed with our honest toil. [Rus19, p. 71]
He undertakes the ‘honest toil’ of placing an ordered field structure on the set of Dedekind cuts, and proving that they thereby become Dedekind complete. On this account, a real number is a Dedekind cut.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford On this account, a real number is an equivalence class of Cauchy sequences.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Cauchy completion
An alternative continuity concept is provided by Cauchy, inspired by the idea that every real number is the limit of the various rational sequences.
Cauchy forms the Cauchy-completion of the rational line using equivalence classes of Cauchy sequences.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Cauchy completion
An alternative continuity concept is provided by Cauchy, inspired by the idea that every real number is the limit of the various rational sequences.
Cauchy forms the Cauchy-completion of the rational line using equivalence classes of Cauchy sequences.
On this account, a real number is an equivalence class of Cauchy sequences.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford In contemporary language, he specified that the reals should be: a complete ordered field.
It turns out that all complete ordered fields are isomorphic.
This is a categorical characterization of the real numbers.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity for the real numbers
Hilbert identified the natural properties we want in the real numbers.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford It turns out that all complete ordered fields are isomorphic.
This is a categorical characterization of the real numbers.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity for the real numbers
Hilbert identified the natural properties we want in the real numbers.
In contemporary language, he specified that the reals should be: a complete ordered field.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford This is a categorical characterization of the real numbers.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity for the real numbers
Hilbert identified the natural properties we want in the real numbers.
In contemporary language, he specified that the reals should be: a complete ordered field.
It turns out that all complete ordered fields are isomorphic.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity for the real numbers
Hilbert identified the natural properties we want in the real numbers.
In contemporary language, he specified that the reals should be: a complete ordered field.
It turns out that all complete ordered fields are isomorphic.
This is a categorical characterization of the real numbers.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is it a certain Dedekind cut? Is it an equivalence class of Cauchy sequences? A geometric length? Something else?
For the structuralist, it doesn’t matter. The reals are any complete ordered field, and it doesn’t matter which one, since they are all isomorphic.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
What is the number π, as a mathematical object?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is it an equivalence class of Cauchy sequences? A geometric length? Something else?
For the structuralist, it doesn’t matter. The reals are any complete ordered field, and it doesn’t matter which one, since they are all isomorphic.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
What is the number π, as a mathematical object?
Is it a certain Dedekind cut?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford A geometric length? Something else?
For the structuralist, it doesn’t matter. The reals are any complete ordered field, and it doesn’t matter which one, since they are all isomorphic.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
What is the number π, as a mathematical object?
Is it a certain Dedekind cut? Is it an equivalence class of Cauchy sequences?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford For the structuralist, it doesn’t matter. The reals are any complete ordered field, and it doesn’t matter which one, since they are all isomorphic.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
What is the number π, as a mathematical object?
Is it a certain Dedekind cut? Is it an equivalence class of Cauchy sequences? A geometric length? Something else?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
What is a real number?
What is the number π, as a mathematical object?
Is it a certain Dedekind cut? Is it an equivalence class of Cauchy sequences? A geometric length? Something else?
For the structuralist, it doesn’t matter. The reals are any complete ordered field, and it doesn’t matter which one, since they are all isomorphic.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford √ The structural role played by 2, for example, in any complete ordered field, is that it is the unique positive number that squares to 2, where 2 is 1 + 1, where 1 is the unique multiplicative identity.
In any complete ordered field, every rational number is definable, and every real number is characterized by how it cuts the rational numbers.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Structuralism in the reals
According to structuralism, real numbers are comprehended by their roles within a larger structure.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford In any complete ordered field, every rational number is definable, and every real number is characterized by how it cuts the rational numbers.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Structuralism in the reals
According to structuralism, real numbers are comprehended by their roles within a larger structure. √ The structural role played by 2, for example, in any complete ordered field, is that it is the unique positive number that squares to 2, where 2 is 1 + 1, where 1 is the unique multiplicative identity.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Structuralism in the reals
According to structuralism, real numbers are comprehended by their roles within a larger structure. √ The structural role played by 2, for example, in any complete ordered field, is that it is the unique positive number that squares to 2, where 2 is 1 + 1, where 1 is the unique multiplicative identity.
In any complete ordered field, every rational number is definable, and every real number is characterized by how it cuts the rational numbers.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Complex numbers
Given the real numbers, one proceeds to the complex numbers C, motivated by the enticing, yet perhaps terrifying, possibility of the imaginary unit √ i = −1.
We want to consider complex numbers of the form a + bi, where a and b are real.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford But which one? After all, −i also is a square root of −1.
(−i)2 = (−1)2i2 = i2 = −1.
Indeed, complex conjugation is an automorphism of C.
a + bi 7→ a − bi.
The numbers i and −i have exactly same structural role in C. Is this a problem for structuralism? Perhaps my i is your −i. How would we know?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A small problem of reference We said that the imaginary unit i is the square root of −1.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford After all, −i also is a square root of −1.
(−i)2 = (−1)2i2 = i2 = −1.
Indeed, complex conjugation is an automorphism of C.
a + bi 7→ a − bi.
The numbers i and −i have exactly same structural role in C. Is this a problem for structuralism? Perhaps my i is your −i. How would we know?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A small problem of reference We said that the imaginary unit i is the square root of −1. But which one?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Perhaps my i is your −i. How would we know?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A small problem of reference We said that the imaginary unit i is the square root of −1. But which one? After all, −i also is a square root of −1.
(−i)2 = (−1)2i2 = i2 = −1.
Indeed, complex conjugation is an automorphism of C.
a + bi 7→ a − bi.
The numbers i and −i have exactly same structural role in C. Is this a problem for structuralism?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A small problem of reference We said that the imaginary unit i is the square root of −1. But which one? After all, −i also is a square root of −1.
(−i)2 = (−1)2i2 = i2 = −1.
Indeed, complex conjugation is an automorphism of C.
a + bi 7→ a − bi.
The numbers i and −i have exactly same structural role in C. Is this a problem for structuralism? Perhaps my i is your −i. How would we know?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford We may define a suitable addition and multiplication on such pairs. (a, b) · (c, d) = (ac − bd, ad + bc)
Thus, we build a copy of the complex numbers C using pairs of real numbers. There is nothing so mysterious about the complex numbers after all.
Furthermore, C is interpretable in R, not just as a field, but as a field with a coordinatization.
This extra structure distinguishes i from −i.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Constructing the complex numbers
Associate complex number a + bi with point (a, b) in the plane.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Furthermore, C is interpretable in R, not just as a field, but as a field with a coordinatization.
This extra structure distinguishes i from −i.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Constructing the complex numbers
Associate complex number a + bi with point (a, b) in the plane.
We may define a suitable addition and multiplication on such pairs. (a, b) · (c, d) = (ac − bd, ad + bc)
Thus, we build a copy of the complex numbers C using pairs of real numbers. There is nothing so mysterious about the complex numbers after all.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford This extra structure distinguishes i from −i.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Constructing the complex numbers
Associate complex number a + bi with point (a, b) in the plane.
We may define a suitable addition and multiplication on such pairs. (a, b) · (c, d) = (ac − bd, ad + bc)
Thus, we build a copy of the complex numbers C using pairs of real numbers. There is nothing so mysterious about the complex numbers after all.
Furthermore, C is interpretable in R, not just as a field, but as a field with a coordinatization.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Constructing the complex numbers
Associate complex number a + bi with point (a, b) in the plane.
We may define a suitable addition and multiplication on such pairs. (a, b) · (c, d) = (ac − bd, ad + bc)
Thus, we build a copy of the complex numbers C using pairs of real numbers. There is nothing so mysterious about the complex numbers after all.
Furthermore, C is interpretable in R, not just as a field, but as a field with a coordinatization.
This extra structure distinguishes i from −i.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Complex numbers in Heaven
Imagine that at your death you are astonished to meet God in Heaven, who informs you, “Yes, you were completely right about platonism for the real numbers—there they are!”
He points across the way, and indeed there you see them, the real numbers, each of them√ a perfect platonic ideal of its kind. You find the numbers π, e, 2, each where you expect them.
“But,” God continues, “you were wrong about platonism for the complex numbers; you have to construct them from the reals as pairs (a, b), with the parentheses and comma and everything.”
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford In type theory, one makes these distinctions.
Type-changing translations: (57)Z,Q translates integer 57 to rational number; (57)Z,R translates to real number.
Are our arithmetic calculations ridden-through with invisible type-adjusting morphisms?
When programming, one changes from type int to float, to enable certain operations.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Type-theoretic account of numbers Every natural number is commonly also an integer, a real, etc.
N ⊆ Z ⊆ Q ⊆ R ⊆ C
Is the integer 57 the same number as the real number 57?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Type-changing translations: (57)Z,Q translates integer 57 to rational number; (57)Z,R translates to real number.
Are our arithmetic calculations ridden-through with invisible type-adjusting morphisms?
When programming, one changes from type int to float, to enable certain operations.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Type-theoretic account of numbers Every natural number is commonly also an integer, a real, etc.
N ⊆ Z ⊆ Q ⊆ R ⊆ C
Is the integer 57 the same number as the real number 57?
In type theory, one makes these distinctions.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Are our arithmetic calculations ridden-through with invisible type-adjusting morphisms?
When programming, one changes from type int to float, to enable certain operations.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Type-theoretic account of numbers Every natural number is commonly also an integer, a real, etc.
N ⊆ Z ⊆ Q ⊆ R ⊆ C
Is the integer 57 the same number as the real number 57?
In type theory, one makes these distinctions.
Type-changing translations: (57)Z,Q translates integer 57 to rational number; (57)Z,R translates to real number.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford When programming, one changes from type int to float, to enable certain operations.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Type-theoretic account of numbers Every natural number is commonly also an integer, a real, etc.
N ⊆ Z ⊆ Q ⊆ R ⊆ C
Is the integer 57 the same number as the real number 57?
In type theory, one makes these distinctions.
Type-changing translations: (57)Z,Q translates integer 57 to rational number; (57)Z,R translates to real number.
Are our arithmetic calculations ridden-through with invisible type-adjusting morphisms?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Type-theoretic account of numbers Every natural number is commonly also an integer, a real, etc.
N ⊆ Z ⊆ Q ⊆ R ⊆ C
Is the integer 57 the same number as the real number 57?
In type theory, one makes these distinctions.
Type-changing translations: (57)Z,Q translates integer 57 to rational number; (57)Z,R translates to real number.
Are our arithmetic calculations ridden-through with invisible type-adjusting morphisms?
When programming, one changes from type int to float, to enable certain operations.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Is this a problem for structuralism? How can we secure a definite number concept upon the comparatively murky concept of ‘arbitrary’ set?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity: based upon set theory
I should like to emphasize that the categoricity arguments for N and R, important for structuralism, take place in set theory. Natural numbers N characterized by induction principle, concerned with arbitrary A ⊆ N. Real numbers R characterized as complete ordered field, but completeness involves arbitrary A ⊆ R. Thus, the uniqueness of our number concepts is riding upon our set-theoretic conceptions.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Categoricity: based upon set theory
I should like to emphasize that the categoricity arguments for N and R, important for structuralism, take place in set theory. Natural numbers N characterized by induction principle, concerned with arbitrary A ⊆ N. Real numbers R characterized as complete ordered field, but completeness involves arbitrary A ⊆ R. Thus, the uniqueness of our number concepts is riding upon our set-theoretic conceptions.
Is this a problem for structuralism? How can we secure a definite number concept upon the comparatively murky concept of ‘arbitrary’ set?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Gradually, the capacity of set theory to express essentially any abstract mathematical structure emerged.
Set theory began to serve as an ontological foundation for mathematics.
To be precise in twentieth-century mathematics often came to mean to specify one’s mathematical structure set-theoretically.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Set theory as a foundation
Set theory was initially a tool brought into other mathematical domains. Dedekind considered sets of numbers; Cantor had sets of reals.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford Set theory began to serve as an ontological foundation for mathematics.
To be precise in twentieth-century mathematics often came to mean to specify one’s mathematical structure set-theoretically.
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Set theory as a foundation
Set theory was initially a tool brought into other mathematical domains. Dedekind considered sets of numbers; Cantor had sets of reals.
Gradually, the capacity of set theory to express essentially any abstract mathematical structure emerged.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Set theory as a foundation
Set theory was initially a tool brought into other mathematical domains. Dedekind considered sets of numbers; Cantor had sets of reals.
Gradually, the capacity of set theory to express essentially any abstract mathematical structure emerged.
Set theory began to serve as an ontological foundation for mathematics.
To be precise in twentieth-century mathematics often came to mean to specify one’s mathematical structure set-theoretically.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Paradise
In light of the power and utility of set theory, Hilbert had said, Aus dem Paradies, das Cantor uns geschaffen, soll uns niemand vertreiben konnen.¨ No-one shall cast us from the paradise that Cantor has created for us.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Faithful representation
Moschovakis summarizes the attitude: ...we will discover within the universe of sets faithful rep- resentations of all the mathematical objects we need, and we will study set theory on the bases of the lean axiomatic system of Zermelo as if all mathematical objects were sets. The delicate problem in specific cases is to formulate precisely the correct definition of a “faithful representation” and to prove that one such exists. [Mos06, p. 34, emphasis original]
Thus, set theory had become a grand unified theory of mathematics.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Set theory as uniform framework The rise of set theory as a unifying mathematical foundation was an important historical development.
While mathematics diversified with ever-greater specialized complexity, mathematicians would sometimes seek to apply results from one area within another. For example, one might use notions and theorems from topology and analysis in order to prove results in algebra or conversely.
This practice would be logically incoherent unless the subjects were part of one logical framework. By providing a unifying context, therefore, a single theory in which one can view all mathematical arguments as taking place, set theory thereby logically facilitated this transfer.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Balkanization
Imagine in contrast a balkanized mathematics, divided into realms. Might not geometry, algebra, and analysis have been totally separate efforts?
Mathematics does exhibit some balkanization today: the axiom of choice, for example, is routinely assumed in many subjects, but resisted in others.
One can easily imagine much worse conflicts.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The Set-Theoretical Universe
Sets accumulate transfinitely to form the universe of all sets. The orthodox view among set theorists thereby exhibits a two-fold realist or platonist nature: First, mathematical objects (can) exist as sets, and Second, these sets enjoy a real mathematical existence, accumulating to form the universe of all sets.
A principal task of set theory, on this view, is to discover the fundamental truths of this cumulative set-theoretical universe. These truths will include all mathematical truths.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Uniqueness of the universe
On this traditional platonist view, the set-theoretic universe is unique: it is the universe of all sets.
In particular, on this view every set-theoretic question, such as the Continuum Hypothesis and others, has a definitive final answer in this universe.
With the ontological problem thus settled on this view, what remains is the epistemological problem: how shall we discover these final set-theoretic truths?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The Universe View
Let me therefore describe as the universe view, the position that:
There is a unique absolute background concept of set, instantiated in the cumulative universe of all sets, in which set-theoretic assertions have a definite truth value.
Thus, the universe view is one of determinism for set-theoretic truth, and hence also for mathematical truth.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Daniel Isaacson: The reality of mathematics
In support of this view, Isaacson sharply distinguishes between particular vs. general mathematical structure. Two fundamentally different uses of axioms. Axioms express our knowledge about a particular structure, such as the natural numbers N, real numbers R. Axioms define a general class of structures, such as class of groups, fields, orders. Axioms for particular structures often have character of self-evident truths. Typically characterize the structure up to isomorphism. Categoricity. hN, Si satisfies Peano’s axioms; R is a complete ordered field. Axioms for general structures are more like definitions.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Isaacson: mathematical experience
Particular structures are found by mathematical experience, and then characterized as unique.
We come to know particular mathematical structures—the natural numbers N, the reals R, and so on—by a process of informal rigour, establishing their coherence, often accompanied by a second-order categoricity result.
For Isaacson, the point then is that the cumulative universe of set theory is a particular mathematical structure, characterized in second-order logic.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Martin’s categoricity argument
Donald Martin argues that there is at most one structure meeting the concept of set.
Assuming what he calls the ‘Uniqueness Postulate’, asserting that every set is determined uniquely by its members, any two structures meeting the “weak” concept of set must agree. They will have the same ordinal stages of construction and will construct the same sets at each stage.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Mathematical support for the universe view
The universe view is often combined with consequentialism as a criterion for truth.
For example, set theorists point to the increasingly stable body of regularity features flowing from the large cardinal hierarchy, such as determinacy consequences and uniformization results in the projective hierarchy for sets of reals.
Because these regularity features are mathematically desirable and highly explanatory, the large cardinal perspective seems to provide a coherent unifying theory.
This is taken as evidence for the truth of those axioms.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Main challenge for the universe view
A difficulty for the Universe view. The central discovery in set theory over the past half-century is the enormous range of set-theoretic possibility. The most powerful set-theoretical tools are most naturally understood as methods of constructing alternative set-theoretical universes, universes that seem fundamentally set-theoretic.
forcing, ultrapowers, canonical inner models, etc.
Much of set-theory research has been about constructing as many different models of set theory as possible. These models are often made to exhibit precise, exacting features or to exhibit specific relationships with other models.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The Continuum Hypothesis
The Continuum Hypothesis (CH) is the assertion that every set of real numbers is either countable or equinumerous with R. This was a major open question from the time of Cantor, and appeared at the top of Hilbert’s famous list of open problems at the dawn of the 20th century.
(1938) Godel¨ proved that CH holds in the constructible universe L. (1962) Cohen proved that L has a forcing extension L[G] with ¬CH.
Thus, the Continuum Hypothesis is now known to be formally independent of the axioms of set theory. It is neither provable nor refutable in ZFC.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
An imaginary alternative history Imagine that set theory had followed a different history: Imagine that as set theory developed, theorems were increasingly settled in the base theory. ...that the independence phenomenon was limited to paradoxical-seeming meta-logic statements. ...that the few true independence results occurring were settled by missing natural self-evident set principles. ...that the basic structure of the set-theoretic universe became increasingly stable and agreed-upon. Such developments could have constituted evidence for the Universe view.
But the actual history is not like this...
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Actual history: an abundance of universes
Over the past half-century, set theorists have discovered a vast diversity of models of set theory, a chaotic jumble of set-theoretic possibilities.
Whole parts of set theory exhaustively explore the combinations of statements realized in models of set theory, and study the methods supporting this exploration.
Would you like CH or ¬CH? How about CH + ¬♦? Do you want ℵn 2 = ℵn+2 for all n? Suslin trees? Kurepa trees? Martin’s Axiom?
Set theorists build models to order.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Category-theoretic nature As a result, the fundamental object of study in set theory has become: the model of set theory.
We have L, L[0]], L[µ], L[E~ ]; we have models V with large cardinals, forcing extensions V [G], ultrapowers M, cut-off universes Lδ, Vα, Hκ, universes L(R), HOD, generic ultrapowers, boolean ultrapowers, etc. Forcing especially has led to a staggering variety of models.
Set theory has discovered an entire cosmos of set-theoretical universes, connected by forcing or large cardinal embeddings, like lines in a constellation filling a dark night sky.
Set theory now exhibits a category-theoretic nature.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A challenge to the universe view
The challenge is for the universe view to explain this central phenomenon, the phenomenon of the enormous diversity of set-theoretic possibilities.
The universe view seems to be fundamentally at odds with the existence of alternative set theoretic universes. Will they be explained away as imaginary?
In particular, it does not seem sufficient, when arguing for the universe view, to identify a particularly robust or clarifying theory, if the competing alternatives still appear acceptably set-theoretic. It seems that one must still explicitly explain (or explain away) the pluralistic illusion.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The Multiverse View
A competing position accepts the alternative set concepts as fully real.
The Multiverse view. The philosophical position holding that there are many set-theoretic universes.
The view is that there are numerous distinct concepts of set, not just one absolute concept of set, and each corresponds to the universe of sets to which it gives rise.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Diverse set concepts
The various concepts of set are simply those giving rise to the universes we have been constructing. A key observation From any given concept of set, we are able to generate many new concepts of set, relative to it.
From a set concept giving rise to a universe W , we describe W other universes LW , HOD , L(R)W , K W , forcing extensions W [G], W [H], ultrapowers, and so on. Each such universe amounts to a new concept of set described in relation to the concept giving rise to W .
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
A philosophical enterprise becomes mathematical
Many of these concepts of set are closely enough related to be analyzed together from the perspective of a single set concept.
So what might have been a purely philosophical enterprise—comparing different concepts of set—becomes in part a mathematical one.
And the subject known as the philosophy of set theory thus requires a pleasing mix of (sometimes quite advanced) mathematical ideas with philosophical matters.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Multiverse view is realism
The multiverse view is a brand of realism. The alternative set-theoretical universes arise from different concepts of set, each giving rise to a universe of sets fully as real as the Universe of sets on the Universe view. The view in part is that our mathematical tools—forcing, etc.—have offered us glimpses into these other mathematical worlds, providing evidence that they exist. A platonist may object at first, but actually, this IS a kind of platonism, namely, platonism about universes, second-order realism. Set theory is mature enough to adopt and analyze this view mathematically.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Plenitudinous platonism
The multiverse view has strong affinities with Mark Balaguer’s view:
“The version of platonism that I am going to develop in this book—I will call it plenitudinous platonism, or alternatively, full-blooded platonism (FBP for short)— differs from traditional versions of platonism in several ways, but all of the dif- ferences arise out of one bottom-level difference concerning the question of how many mathematical objects there are. FBP can be expressed very intuitively, but also rather sloppily, as the view that all possible mathematical objects exist.” (p. 5)
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The analogy with Geometry Geometry studied concepts—points, lines, planes—with a seemingly clear, absolute meaning; but those fundamental concepts shattered via non-Euclidean geometry into distinct geometrical concepts, realized in distinct geometrical universes. The first consistency arguments for non-Euclidean geometry presented them as simulations within Euclidean geometry (e.g. ‘line’ = great circle on sphere). In time, geometers accepted the alternative geometries more fully, with their own independent existence, and developed intuitions about what it is like to live inside them. Today, geometers have a deep understanding of these alternative geometries, and no-one now regards the alternative geometries as illusory.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Set theory – geometry Geometers reason about the various geometries: externally, as embedded spaces. internally, by using newly formed intuitions. abstractly, using isometry groups.
Extremely similar modes of reasoning arise with forcing: We understand the forcing extension from the perspective of the ground model, via names and the forcing relation. We understand the forcing extension by jumping inside: “Argue in V [G]” We understand the forcing extension by analyzing automorphisms of the Boolean algebra.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Isaacson on analogy with geometry
“The independence of the fifth postulate reflects the fact. . . that there are different geometries, in one of which the fifth postulate holds (is true), in others of which it is false. It makes no sense to ask whether the fifth pos- tulate is really true or not. Whether it holds or not is a matter of which geometry we are in. The truth or falsity of the fifth postulate is not an open question, and is not something that can be overcome by finding a new ax- iom to settle it. By contrast, the independence of the continuum hypothesis does not establish the existence of a multiplicity of set theories. In a sense made precise and established by the use of second-order logic, there is only one set theory of the continuum. It remains an open question whether in that set theory [ CH holds or not].” (p. 38)
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The dream solution template for CH
Set theorists yearn for a definitive solution to CH, what I call the dream solution:
Step 1. Produce a set-theoretic assertion Φ expressing a naturally manifest set-theoretic principle.
Step 2. Prove that Φ determines CH. That is, prove that Φ =⇒ CH, or prove that Φ =⇒ ¬CH.
And so, CH would be settled, since everyone would accept Φ and its consequences.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Dream solution will never be realized
I have argued that the dream solution is unworkable.
I argue that our rich experience in worlds having CH and others having ¬CH, worlds that seem fully set-theoretic to us, will prevent us from ever accepting a principle Φ as manifestly true, if it decides CH.
In other words, success in the second step exactly undermines the first step.
My prediction is that any specific dream solution proposal will be rejected from a position of deep mathematical experience with the contrary.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
CH in the Multiverse
CH is independent, yes, but more than this...
Every set-theoretic universe has a forcing extension in which CH holds, and another in which it fails.
We can turn CH on and off like a lightswitch.
We have a rich experience in the resulting models, which seem fully set-theoretic to us.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Turning the tables on Isaacson So it is not merely that CH is formally independent and we have no additional knowledge.
Rather, we have an informed, deep understanding of how CH could hold or fail, of how to build such worlds from one another.
If a proposed axiom Φ settles CH, then we will not look upon it as intrinsic to the concept of set, since we already know how it can fail.
It would be like someone having an axiom implying that only Brooklyn existed, while we already know about Manhattan and the other boroughs of New York.
Thus, the dream solution will not succeed.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
The CH is settled
On the multiverse perspective, the CH question is settled.
It is incorrect to describe it as an open question.
The answer consists of our detailed understanding of how the CH both holds and fails throughout the multiverse, of how these models are connected and how one may reach them from each other while preserving or omitting certain features.
Fascinating open questions about CH remain, of course, but the most important essential facts are known.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford But still, mathematicians want to know: is CH true?
The question may seem innocent, but interpreting it sensibly leads to the extremely difficult philosophical issues I have been discussing in this talk.
Ultimately, the question becomes: do we have just one mathematical world or many?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Truth
Fine, we can neither prove nor refute the continuum hypothesis; it is independent of the ZFC axioms.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford The question may seem innocent, but interpreting it sensibly leads to the extremely difficult philosophical issues I have been discussing in this talk.
Ultimately, the question becomes: do we have just one mathematical world or many?
What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Truth
Fine, we can neither prove nor refute the continuum hypothesis; it is independent of the ZFC axioms.
But still, mathematicians want to know: is CH true?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford What is a number? Set-theoretic foundations The multiverse view Dream Solution of CH is unattainable
Truth
Fine, we can neither prove nor refute the continuum hypothesis; it is independent of the ZFC axioms.
But still, mathematicians want to know: is CH true?
The question may seem innocent, but interpreting it sensibly leads to the extremely difficult philosophical issues I have been discussing in this talk.
Ultimately, the question becomes: do we have just one mathematical world or many?
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford References
References
Richard Dedekind. Essays on the theory of numbers. I:Continuity and irrational numbers. II:The nature and meaning of numbers. Authorized translation by Wooster Woodruff Beman. Dover Publications, Inc., New York, 1963, pp. iii+115.
Victoria Gitman and Joel David Hamkins. “A natural model of the multiverse axioms”. Notre Dame J. Formal Logic 51.4 (2010), pp. 475–484. ISSN: 0029-4527. DOI: 10.1215/00294527-2010-030. arXiv:1104.4450[math.LO]. http://wp.me/p5M0LV-3I.
Joel David Hamkins. “The Set-theoretic Multiverse : A Natural Context for Set Theory”. Annals of the Japan Association for Philosophy of Science 19 (2011), pp. 37–55. ISSN: 0453-0691. DOI: 10.4288/jafpos.19.0 37. http://jdh.hamkins.org/themultiverseanaturalcontext.
Joel David Hamkins. “The set-theoretic multiverse”. Review of Symbolic Logic 5 (2012), pp. 416–449. DOI: 10.1017/S1755020311000359. arXiv:1108.4223[math.LO]. http://jdh.hamkins.org/themultiverse.
Joel David Hamkins. “A multiverse perspective on the axiom of constructibility”. In: Infinity and Truth. Vol. 25. LNS Math Natl. Univ. Singap. World Sci. Publ., Hackensack, NJ, 2014, pp. 25–45. DOI: 10.1142/9789814571043 0002. arXiv:1210.6541[math.LO]. http://wp.me/p5M0LV-qE.
Joel David Hamkins. “Is the dream solution of the continuum hypothesis attainable?” Notre Dame J. Formal Logic 56.1 (2015), pp. 135–145. ISSN: 0029-4527. DOI: 10.1215/00294527-2835047. arXiv:1203.4026[math.LO]. http://jdh.hamkins.org/dream-solution-of-ch.
Joel David Hamkins. Lectures on the Philosophy of Mathematics. book manuscript, 270 pages, in preparation. 2018.
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford References
David Hilbert.“ Uber¨ das Unendliche”. Math. Ann. 95.1 (1926), pp. 161–190. ISSN: 0025-5831. DOI: 10.1007/BF01206605.
Yiannis Moschovakis. Notes on set theory. Undergraduate Texts in Mathematics. Springer, New York, 2006. ISBN: 978-0387-28722-5.
Bertrand Russell. Introduction to Mathematical Philosophy. Corrected edition 1920; Reprinted, John G. Slater (intro.), Routledge, London, UK, 1993. London, UK: George Allean and Unwin, 1919. Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford References
Thank you. Slides and articles available on http://jdh.hamkins.org. Joel David Hamkins Oxford
Amsterdam Wijsgerig Festival DRIFT 2019 Joel David Hamkins, Oxford