Higher Infinity and the Foundations of Mathematics

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and the foundations of mathematics

Joel David Hamkins

The City University of New York College of Staten Island & The CUNY Graduate Center

Mathematics, Philosophy, Computer Science

American Association for the Advancement of Science Paciﬁc Division 95th annual meeting Riverside, CA, June 18-20, 2014

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and the foundations of mathematics

The story of inﬁnity and what is going on in the foundations of mathematics

Different kinds of inﬁnity, reaching up to the higher inﬁnite

Pervasive independence phenomenon, meaning numerous fundamental questions in set theory and mathematics are neither provable nor refutable

Troubling philosophical questions at the core of mathematics concerning the nature and meaning of truth

Higher inﬁnity and foundations of math Joel David Hamkins, New York 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know?

Higher inﬁnity and foundations of math Joel David Hamkins, New York 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0

Higher inﬁnity and foundations of math Joel David Hamkins, New York 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10

Higher inﬁnity and foundations of math Joel David Hamkins, New York million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10 100

Higher inﬁnity and foundations of math Joel David Hamkins, New York 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10 100 million

Higher inﬁnity and foundations of math Joel David Hamkins, New York 100 zeros z }| { = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10 100 million

googol

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol zeros 100 z }| { = 10googol = 1010 = 1000000000 ······ 0000

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00

googol plex

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large ﬁnite numbers

Large ﬁnite numbers as a metaphor for inﬁnity. What is the largest number you know? 0 1 10 100 million 100 zeros z }| { googol = 10100 = 100000 ··· 00 googol zeros 100 z }| { googol plex = 10googol = 1010 = 1000000000 ······ 0000

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex = 10(googol)! googol plex bang = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

Higher inﬁnity and foundations of math Joel David Hamkins, New York = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex = 10(googol)! googol plex bang = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol bang plex = 10(googol)! googol plex bang = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

Higher inﬁnity and foundations of math Joel David Hamkins, New York = 10(googol)! googol plex bang = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol plex bang = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex = 10(googol)!

Higher inﬁnity and foundations of math Joel David Hamkins, New York = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex = 10(googol)! googol plex bang

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol bang plex bang plex plex googol bang bang plex bang bang

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex = 10(googol)! googol plex bang = 10googol!

Which is larger?

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Getting larger

googol plex = 10googol

googol bang = (googol)! = 1 · 2 · 3 ··· googol

Which is larger?

googol bang plex = 10(googol)! googol plex bang = 10googol!

Which is larger?

googol bang plex bang plex plex googol bang bang plex bang bang

Higher inﬁnity and foundations of math Joel David Hamkins, New York 10 ·· 10· } googol = 1010 Consider all expressions in the googol plex bang stack language. googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex

Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number?

This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Truly vast numbers

googol stack

Higher inﬁnity and foundations of math Joel David Hamkins, New York Consider all expressions in the googol plex bang stack language. googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex

Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number?

This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Truly vast numbers 10 ·· 10· } googol googol stack = 1010

Higher inﬁnity and foundations of math Joel David Hamkins, New York googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex

Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number?

This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Truly vast numbers 10 ·· 10· } googol googol stack = 1010 Consider all expressions in the googol plex bang stack language.

Higher inﬁnity and foundations of math Joel David Hamkins, New York Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number?

This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Truly vast numbers 10 ·· 10· } googol googol stack = 1010 Consider all expressions in the googol plex bang stack language. googol stack bang plex plex bang googol plex stack bang plex bang googol bang stack plex

Higher inﬁnity and foundations of math Joel David Hamkins, New York This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number?

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Question Is there a feasible computable procedure to say which of any two such expressions describes the larger number?

This is an open mathematical question. However, we do know how to compare expressions having at most a googol terms.

Higher inﬁnity and foundations of math Joel David Hamkins, New York . .

6 It has inﬁnitely many rooms. 5 Each room has a guest. The hotel is completely full.

4 A new guest shows up. What is the manager to do? He announces: everybody move to the next higher room. 3 Guest in Room n 7→ Room n + 1 2 Room 1 is now vacant, and so the new guest can check in. 1

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s Grand Hotel

A new high-rise hotel just went up.

Higher inﬁnity and foundations of math Joel David Hamkins, New York Each room has a guest. The hotel is completely full. A new guest shows up. What is the manager to do? He announces: everybody move to the next higher room.

Guest in Room n 7→ Room n + 1

Room 1 is now vacant, and so the new guest can check in.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s Grand Hotel . .

A new high-rise hotel just went up. 6 It has inﬁnitely many rooms. 5

Higher inﬁnity and foundations of math Joel David Hamkins, New York A new guest shows up. What is the manager to do? He announces: everybody move to the next higher room.

Guest in Room n 7→ Room n + 1

Room 1 is now vacant, and so the new guest can check in.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s Grand Hotel . .

A new high-rise hotel just went up. 6 It has inﬁnitely many rooms. 5 Each room has a guest. The hotel is completely full.

Higher inﬁnity and foundations of math Joel David Hamkins, New York He announces: everybody move to the next higher room.

Guest in Room n 7→ Room n + 1

Room 1 is now vacant, and so the new guest can check in.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s Grand Hotel . .

A new high-rise hotel just went up. 6 It has inﬁnitely many rooms. 5 Each room has a guest. The hotel is completely full.

4 A new guest shows up. What is the manager to do?

Higher inﬁnity and foundations of math Joel David Hamkins, New York Room 1 is now vacant, and so the new guest can check in.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s Grand Hotel . .

A new high-rise hotel just went up. 6 It has inﬁnitely many rooms. 5 Each room has a guest. The hotel is completely full.

4 A new guest shows up. What is the manager to do? He announces: everybody move to the next higher room. 3 Guest in Room n 7→ Room n + 1 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s Grand Hotel . .

A new high-rise hotel just went up. 6 It has inﬁnitely many rooms. 5 Each room has a guest. The hotel is completely full.

4 A new guest shows up. What is the manager to do? He announces: everybody move to the next higher room. 3 Guest in Room n 7→ Room n + 1 2 Room 1 is now vacant, and so the new guest can check in. 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York But now, Hilbert’s Bus pulls up. The bus has inﬁnitely many seats, each occupied by a new guest. What can the manager do? He announces:

Room n 7→ Room 2n

Now only the even-numbered rooms are occupied. The odd rooms are free.

Seat k 7→ Room . . . 2k + 1

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

More guests arrive . . New guests are thus accommodated as they arrive.

Higher inﬁnity and foundations of math Joel David Hamkins, New York He announces:

Room n 7→ Room 2n

Now only the even-numbered rooms are occupied. The odd rooms are free.

Seat k 7→ Room . . . 2k + 1

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and foundations of math Joel David Hamkins, New York Seat k 7→ Room . . . 2k + 1

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

More guests arrive . . New guests are thus accommodated as they arrive. But now, Hilbert’s Bus pulls up. 6 The bus has inﬁnitely many seats, each occupied by a new guest. 5 What can the manager do? He announces: 4 Room n 7→ Room 2n 3 Now only the even-numbered rooms are occupied. The odd rooms are free. 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York 2k + 1

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and foundations of math Joel David Hamkins, New York Room n 7→ Room 2n

Car c, Seat s 7→ Room ... 3c5s

Everyone is accommodated.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s train arrives . . Now, Hilbert’s train arrives. Infinitely many cars, each with infinitely many seats. 6 A person in every car c, seat s. 5 Can they fit into the hotel?

Higher inﬁnity and foundations of math Joel David Hamkins, New York Car c, Seat s 7→ Room ... 3c5s

Everyone is accommodated.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s train arrives . . Now, Hilbert’s train arrives. Infinitely many cars, each with infinitely many seats. 6 A person in every car c, seat s. 5 Can they fit into the hotel?

4 Room n 7→ Room 2n 3

Higher inﬁnity and foundations of math Joel David Hamkins, New York 3c5s

Everyone is accommodated.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s train arrives . . Now, Hilbert’s train arrives. Infinitely many cars, each with infinitely many seats. 6 A person in every car c, seat s. 5 Can they fit into the hotel?

4 Room n 7→ Room 2n 3 Car c, Seat s 7→ Room ... 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York Everyone is accommodated.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s train arrives . . Now, Hilbert’s train arrives. Infinitely many cars, each with infinitely many seats. 6 A person in every car c, seat s. 5 Can they fit into the hotel?

4 Room n 7→ Room 2n 3 Car c, Seat s 7→ Room ... 3c5s 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Hilbert’s train arrives . . Now, Hilbert’s train arrives. Infinitely many cars, each with infinitely many seats. 6 A person in every car c, seat s. 5 Can they fit into the hotel?

4 Room n 7→ Room 2n 3 Car c, Seat s 7→ Room ... 3c5s 2 Everyone is accommodated. 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Countability

The underlying concept here is countability.

A set is countable if it ﬁts in Hilbert’s Hotel, if it can be placed in one-to-one correspondence with a set of natural numbers.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Equinumerosity

A more fundamental concept is equinumerosity.

Two sets A and B are equinumerous if they can be placed into one-to-one correspondence with each other.

Without counting, we can see that the set of people in this room is equinumerous with the set of noses, simply because we each have a nose and nobody has two.

Higher inﬁnity and foundations of math Joel David Hamkins, New York Yes! The set of fractions is countable; it ﬁts into Hilbert’s hotel.

p fraction 7→ Room 3p5q q

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Rational numbers

Next comes Hilbert’s Inﬁnite Parade, densely packed in itself, with a person for every rational number.

1 3 5 Rational numbers = fractions, like 2 , 8 , 16 ,...

Can they all ﬁt into Hilbert’s Hotel?

Higher inﬁnity and foundations of math Joel David Hamkins, New York 3p5q

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Rational numbers

Next comes Hilbert’s Inﬁnite Parade, densely packed in itself, with a person for every rational number.

1 3 5 Rational numbers = fractions, like 2 , 8 , 16 ,...

Can they all ﬁt into Hilbert’s Hotel?

Yes! The set of fractions is countable; it ﬁts into Hilbert’s hotel.

p fraction 7→ Room q

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Rational numbers

Next comes Hilbert’s Inﬁnite Parade, densely packed in itself, with a person for every rational number.

1 3 5 Rational numbers = fractions, like 2 , 8 , 16 ,...

Can they all ﬁt into Hilbert’s Hotel?

Yes! The set of fractions is countable; it ﬁts into Hilbert’s hotel.

p fraction 7→ Room 3p5q q

Higher infinity and foundations of math Joel David Hamkins, New York No. Suppose towards contradiction that we could. Room 1 ← 2.718281828 ··· Room 2 ← 3.14159265 ··· Room 3 ← 5.3851648 ··· . . . .. Define a number d = 0.d1d2d3 ··· th so that dk is different from k digit of number in Room k. In particular, d is different from the number in Room k, for any k. So d is not in any room! Thus, R is uncountable, and does not fit into Hilbert’s Hotel.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

The set of real numbers R Cantor’s Cruise Ship pulls in, with a passenger for every real number. Can we ﬁt them into Hilbert’s Hotel?

Higher inﬁnity and foundations of math Joel David Hamkins, New York In particular, d is different from the number in Room k, for any k. So d is not in any room! Thus, R is uncountable, and does not ﬁt into Hilbert’s Hotel.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

The set of real numbers R Cantor’s Cruise Ship pulls in, with a passenger for every real number. Can we fit them into Hilbert’s Hotel? No. Suppose towards contradiction that we could. Room 1 ← 2.718281828 ··· Room 2 ← 3.14159265 ··· Room 3 ← 5.3851648 ··· . . . .. Define a number d = 0.d1d2d3 ··· th so that dk is different from k digit of number in Room k. In particular, d is different from the number in Room k, for any k. So d is not in any room! Thus, R is uncountable, and does not fit into Hilbert’s Hotel. Higher infinity and foundations of math Joel David Hamkins, New York The Higher Infinite The Central Discovery of Set Theory Mathematical Truth and Existence

Different sizes of inﬁnity

Conclusion There are different sizes of inﬁnity.

The inﬁnity of the natural numbers is smaller than the inﬁnity of the real number line.

How many sizes of inﬁnity are there?

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Different sizes of inﬁnity

Conclusion There are different sizes of inﬁnity.

The inﬁnity of the natural numbers is smaller than the inﬁnity of the real number line.

How many sizes of inﬁnity are there?

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Reaching higher More generally, Cantor proved that the power set of any set is a strictly larger set.

P(X) = set of all subsets of X = { Y | Y ⊆ X } To see why this is true, suppose that X and P(X) have the same size. So there is a one-to-one correspondence

i 7→ Yi

with i ∈ X and Yi ⊆ X.

The trick: let Y = { i ∈ X | i ∈/ Yi }.

So i ∈ Y just in case i ∈/ Yi , and thus Y 6= Yi for any i. So the correspondence misses the set Y , a contradiction! Therefore, the power set P(X) is strictly larger than X in size.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Reaching higher More generally, Cantor proved that the power set of any set is a strictly larger set.

P(X) = set of all subsets of X = { Y | Y ⊆ X } To see why this is true, suppose that X and P(X) have the same size. So there is a one-to-one correspondence

i 7→ Yi

with i ∈ X and Yi ⊆ X.

The trick: let Y = { i ∈ X | i ∈/ Yi }.

So i ∈ Y just in case i ∈/ Yi , and thus Y 6= Yi for any i. So the correspondence misses the set Y , a contradiction! Therefore, the power set P(X) is strictly larger than X in size.

Higher inﬁnity and foundations of math Joel David Hamkins, New York < P(N) < P(P(N)) < P(P(P(N))) < ···

For any given infinity X, the power set P(X) is larger. For any set A of infinite sets, there is an infinity larger than any set in A.

Conclusion The number of different sizes of inﬁnity is greater than any given inﬁnity.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Infinitely many different infinities So we may construct larger sizes of infinity by iterating the power set.

Higher inﬁnity and foundations of math Joel David Hamkins, New York < P(P(N)) < P(P(P(N))) < ···

For any given infinity X, the power set P(X) is larger. For any set A of infinite sets, there is an infinity larger than any set in A.

Conclusion The number of different sizes of inﬁnity is greater than any given inﬁnity.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Infinitely many different infinities So we may construct larger sizes of infinity by iterating the power set.

N < P(N)

Higher inﬁnity and foundations of math Joel David Hamkins, New York < P(P(P(N))) < ···

For any given infinity X, the power set P(X) is larger. For any set A of infinite sets, there is an infinity larger than any set in A.

Conclusion The number of different sizes of inﬁnity is greater than any given inﬁnity.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Infinitely many different infinities So we may construct larger sizes of infinity by iterating the power set.

N < P(N) < P(P(N))

Higher infinity and foundations of math Joel David Hamkins, New York For any given infinity X, the power set P(X) is larger. For any set A of infinite sets, there is an infinity larger than any set in A.

Conclusion The number of different sizes of inﬁnity is greater than any given inﬁnity.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Infinitely many different infinities So we may construct larger sizes of infinity by iterating the power set.

N < P(N) < P(P(N)) < P(P(P(N))) < ···

Higher infinity and foundations of math Joel David Hamkins, New York Conclusion The number of different sizes of infinity is greater than any given infinity.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Infinitely many different infinities So we may construct larger sizes of infinity by iterating the power set.

N < P(N) < P(P(N)) < P(P(P(N))) < ···

For any given infinity X, the power set P(X) is larger. For any set A of infinite sets, there is an infinity larger than any set in A.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Infinitely many different infinities So we may construct larger sizes of infinity by iterating the power set.

N < P(N) < P(P(N)) < P(P(P(N))) < ···

For any given infinity X, the power set P(X) is larger. For any set A of infinite sets, there is an infinity larger than any set in A.

Conclusion The number of different sizes of inﬁnity is greater than any given inﬁnity.

Higher inﬁnity and foundations of math Joel David Hamkins, New York 0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

Higher inﬁnity and foundations of math Joel David Hamkins, New York 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

Higher inﬁnity and foundations of math Joel David Hamkins, New York 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2

Higher inﬁnity and foundations of math Joel David Hamkins, New York . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . .

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York . . ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω2 ··· Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . .

Higher inﬁnity and foundations of math Joel David Hamkins, New York Counting to ω2 is just like counting to 100.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

How to count

Anyone can learn to count in the ordinal numbers

0 1 2 3 ··· ω ω + 1 ω + 2 ω + 3 ··· ω · 2 ω · 2 + 1 ω · 2 + 2 ω · 2 + 3 ··· ω · 3 ω · 3 + 1 ω · 3 + 2 ω · 3 + 3 ··· . . ω · n ω · n + 1 ··· ω · n + k ··· . . ω2 ··· Counting to ω2 is just like counting to 100.

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1

Higher inﬁnity and foundations of math Joel David Hamkins, New York ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ···

Higher inﬁnity and foundations of math Joel David Hamkins, New York ω ωω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

ω ωω

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

ω ωω ωω ··· ωω

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0

Higher inﬁnity and foundations of math Joel David Hamkins, New York Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York CK ··· ω1 ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· ω1 ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· ω2 ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York ··· α ········· The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

But of course the ordinals continue far beyond... 0 1 ··· ω ω + 1 ··· ω · 2 ··· ω · n + k ··· ω2 ω2 +1 ··· ω2 +ω ··· ω2 +ω·n+k ··· ω2 ·r +ω·n+k ··· ω3 ω3 + 1 ··· ω4 ··· ωm ··· ωω ··· ω ωω ωω ω ··· ω ··· 0 ··· 1 ··· Ch CK ω1 ··· ω1 ··· ··· ω1 ··· ω2

Higher inﬁnity and foundations of math Joel David Hamkins, New York The ordinals continue on and on without end, falling like grains of sand through the transﬁnite hourglass.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting past inﬁnity

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

The sizes of inﬁnity

Ordinals can be used to index successive instances of any transﬁnitely iterated process.

Indeed, Cantor had invented them speciﬁcally for such a purpose, to keep track of his “derived sets,” which are obtained by successively casting out isolated points from a set of real numbers.

Higher inﬁnity and foundations of math Joel David Hamkins, New York . . Vγ

Vβ

Vα

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Cumulative universe of all sets The cumulative universe of all sets is constructed by iteratively computing the power set operation.

Vω

We start with the empty set V0 = ∅. At each stage, add all subsets of what has been created so far.

Vα consists of the sets created by stage α. The entire set-theoretic universe V consists of all sets added during the course of this iterative procedure. Higher inﬁnity and foundations of math Joel David Hamkins, New York . . Vγ

Vβ

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Cumulative universe of all sets The cumulative universe of all sets is constructed by iteratively computing the power set operation.

Vα

Vω

We start with the empty set V0 = ∅. At each stage, add all subsets of what has been created so far.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Cumulative universe of all sets The cumulative universe of all sets is constructed by iteratively computing the power set operation.

Vβ

Vα

Vω

We start with the empty set V0 = ∅. At each stage, add all subsets of what has been created so far.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Cumulative universe of all sets The cumulative universe of all sets is constructed by iteratively computing the power set operation.

Vγ

Vβ

Vα

Vω

We start with the empty set V0 = ∅. At each stage, add all subsets of what has been created so far.

Cumulative universe of all sets The cumulative universe of all sets is constructed by iteratively computing the power set operation. . . Vγ

Vβ

Vα

Vω

We start with the empty set V0 = ∅. At each stage, add all subsets of what has been created so far.

The ZFC axioms of set theory The Zermelo-Fraenkel (ZFC) axioms of set theory express the fundamental truths of this cumulative picture. Zermelo introduced them in order to formalize his proof of the well-ordering principle (later augmented by Fraenkel). The set-theoretic universe can serve as a mathematical foundation for all mathematics. On this view, every mathematical object as a set, and to be precise in mathematics means to specify something set-theoretically. Historically important for unifying mathematics. Set theory is a Grand Uniﬁed Theory of Everything. There are also alternative foundations, particularly from category theory, including some exciting current foundations-in-development.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Higher inﬁnity and foundations of math Joel David Hamkins, New York ℵ1 is the next inﬁnite cardinal

ℵ2 and so on. . .

ℵω, the supremum of ℵn for ﬁnite n. The ﬁrst singular cardinal.

ℵω+1

ℵα And so on

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting the sizes of inﬁnity

ℵ0 is the smallest inﬁnite cardinal

Higher inﬁnity and foundations of math Joel David Hamkins, New York ℵ2 and so on. . .

ℵω, the supremum of ℵn for ﬁnite n. The ﬁrst singular cardinal.

ℵω+1

ℵα And so on

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting the sizes of inﬁnity

ℵ0 is the smallest inﬁnite cardinal

ℵ1 is the next inﬁnite cardinal

Higher inﬁnity and foundations of math Joel David Hamkins, New York . .

ℵω, the supremum of ℵn for ﬁnite n. The ﬁrst singular cardinal.

ℵω+1

ℵα And so on

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting the sizes of inﬁnity

ℵ0 is the smallest inﬁnite cardinal

ℵ1 is the next inﬁnite cardinal

ℵ2 and so on.

Higher infinity and foundations of math Joel David Hamkins, New York ℵω, the supremum of ℵn for finite n. The first singular cardinal.

ℵω+1

ℵα And so on

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting the sizes of inﬁnity

ℵ0 is the smallest inﬁnite cardinal

ℵ1 is the next inﬁnite cardinal

ℵ2 and so on. . .

Higher inﬁnity and foundations of math Joel David Hamkins, New York ℵω+1

ℵα And so on

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting the sizes of inﬁnity

ℵ0 is the smallest inﬁnite cardinal

ℵ1 is the next inﬁnite cardinal

ℵ2 and so on. . .

ℵω, the supremum of ℵn for ﬁnite n. The ﬁrst singular cardinal.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Counting the sizes of inﬁnity

ℵ0 is the smallest inﬁnite cardinal

ℵ1 is the next inﬁnite cardinal

ℵ2 and so on. . .

ℵω, the supremum of ℵn for ﬁnite n. The ﬁrst singular cardinal.

ℵω+1

ℵα And so on

Higher inﬁnity and foundations of math Joel David Hamkins, New York Yes! Let λ be the supremum of

ℵ0 ℵℵ0 ℵℵℵ ℵℵℵ ··· 0 ℵ0

It follows that ℵλ is also the supremum of this sequence, and so ℵλ = λ, an ℵ-ﬁxed point. We catch our tail.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Aleph ﬁxed points

th For every ordinal α, we have ℵα, the α inﬁnite cardinal.

Initially, we often have α < ℵα.

1 < ℵ1

5 < ℵ5

Could there be an ordinal λ with λ = ℵλ? An ℵ-ﬁxed point.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Aleph ﬁxed points

th For every ordinal α, we have ℵα, the α inﬁnite cardinal.

Initially, we often have α < ℵα.

1 < ℵ1

5 < ℵ5

Could there be an ordinal λ with λ = ℵλ? An ℵ-ﬁxed point. Yes! Let λ be the supremum of

ℵ0 ℵℵ0 ℵℵℵ ℵℵℵ ··· 0 ℵ0

It follows that ℵλ is also the supremum of this sequence, and so ℵλ = λ, an ℵ-ﬁxed point. We catch our tail.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The answer is extremely interesting...

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Inaccessible cardinals

Let’s ask for more.

A cardinal κ is inaccessible, if it cannot be reached from below κ is not the limit of fewer than κ many cardinals of size less than κ. (regular) If a set X has size less than κ, then P(X) is also less than κ. (strong limit)

Are there any such cardinals?

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Inaccessible cardinals

Let’s ask for more.

Are there any such cardinals? The answer is extremely interesting...

Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals tall cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals strong cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals supercompact cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York almost-huge cardinals

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts

supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

A few large cardinal concepts almost-huge cardinals supercompact cardinals strong cardinals tall cardinals measurable cardinals subtle cardinals weakly superstrong cardinals unfoldable cardinals totally indescribable cardinals weakly compact cardinals Mahlo cardinals uplifting cardinals inaccessible cardinals worldly cardinals Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

The large cardinal hierarchy near Ramsey cardinals From Victoria Gitman:

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large cardinal concepts Each large cardinal concept expresses a precise, robust, technical concept of inﬁnity. often generalize properties of ω to higher cardinals intensely studied, fundamental concepts in set theory arise largely from natural questions in inﬁnite combinatorics and logic large cardinal hierarchy has a roughly linear nature large cardinals in many cases have surprising regularity consequences down low these consequences provide a very welcome unifying structure theory for the nature of the set-theoretic universe Thus, large cardinals provide a coherent unifying foundation for mathematics

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Transcendent hierarchy

A principal feature of the large cardinal hierarchy is that each level of the hierarchy completely transcends those below it.

For example, from the existence of a smaller large cardinal, we cannot one cannot prove the existence of any larger cardinals. Indeed, from a smaller cardinal, we cannot even prove that the existence of the larger cardinals is even consistent with the axioms of set theory.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Are there any inaccessible cardinals? Let me illustrate in the case of inaccessible cardinals. Observation

If κ is inaccessible, then Vκ is a model of the set theory axioms.

In particular, if κ is the smallest inaccessible cardinal, then Vκ has no inaccessible cardinals inside it. It is therefore consistent with the axioms of set theory that there are no inaccessible cardinals. Similarly, we cannot prove from one hundred inaccessible cardinals, that there is another one, because if κ is the 101st inaccessible cardinal, then Vκ will be a model of the axioms of set theory and have precisely 100 inaccessible cardinals. This phenomenon is pervasive in the large cardinal hierarchy.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Large cardinals as a pinnacle of achievement

The discovery of the large cardinal hierarchy is an amazing achievement of mathematics.

For thousands of years, mathematicians and philosophers have struggled with the concept of inﬁnity.

In the past century, we have achieved a deep, informed, technically precise, robust understanding of the higher inﬁnite, and furthermore one which we’ve found to provide a unifying vision for the foundations of mathematics, providing answers to many fundamental questions.

It is a remarkable achievement.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Sets of real numbers

Cantor noticed from the earliest days that all the naturally arising sets of real numbers are either countable or have the same size as R.

set of all integers Z...... countable set of all rational numbers Q...... countable set of algebraic numbers A ...... countable intervals [0, 1]...... size continuum Cantor set C ...... size continuum closed sets ...... either countable or size continuum

No set A ⊆ R of intermediate size was ever found.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Continuum Hypothesis

So Cantor asked:

Question (Cantor)

Is every set of reals A ⊆ R either countable or equinumerous with R?

The Continuum Hypothesis (CH) is the assertion that yes, indeed, every set A ⊆ R is either countable or has size continuum; there are no inﬁnities between Z and R.

In other words, CH asserts that there are no inﬁnities between the integers Z and the reals R.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Continuum Hypothesis

Cantor struggled to settle the CH for the rest of his life.

Hilbert gave a stirring speech at the dawn of the 20th century, identifying the mathematical problems that would guide mathematics for the next century.

Hilbert’s Problem Number 1: Cantor’s Continuum Hypothesis.

Higher inﬁnity and foundations of math Joel David Hamkins, New York In 1938, Kurt Godel¨ constructed an artiﬁcial mathematical universe, called the constructible universe L, and proved that all the ZFC axioms of set theory are true in L, and furthermore that the continuum hypothesis CH is also true there.

Therefore, it is consistent with ZFC that the CH holds.

In other words, if ZFC is consistent, then we cannot refute CH.

The constructible universe is ingenious, containing only the bare minimum of sets that absolutely must be present on the basis of the axioms of set theory, given that one will include all the ordinals.

The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

An incredible partial answer in 1938 The CH question remained open for decades.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

An incredible partial answer in 1938 The CH question remained open for decades.

In 1938, Kurt Godel¨ constructed an artiﬁcial mathematical universe, called the constructible universe L, and proved that all the ZFC axioms of set theory are true in L, and furthermore that the continuum hypothesis CH is also true there.

Therefore, it is consistent with ZFC that the CH holds.

In other words, if ZFC is consistent, then we cannot refute CH.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Another incredible partial answer in 1962

In 1962, Paul Cohen provided a complementary partial answer. Cohen constructed another kind of artiﬁcial mathematical universe, inventing the method of forcing to do so, in which the axioms of set theory are all true, but CH fails.

In other words, it is consistent with the axioms of set theory that CH is false.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Independence of CH

Conclusion If the ZFC axioms of set theory are consistent, then we cannot settle the CH question on the basis of those axioms. 1 We cannot prove CH, since it fails in Cohen’s universe. 2 we cannot refute CH, since it holds in Godel’s¨ universe.

Thus, the continuum hypothesis is independent of the axioms of set theory, neither provable nor refutable.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Forcing

Cohen provided not only a universe where CH fails, but also an extremely powerful tool, a model-construction method known as forcing.

Forcing allows us to construct diverse extensions of any given set-theoretic universe, and this has been undertaken in thousands of arguments.

With forcing, we are often able to construct models of set theory exhibiting speciﬁc truths or having speciﬁc relations to other models of set theory.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

An abundance of models

With forcing, set theorists have thus discovered 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.

ℵn Would you like CH + ¬♦? Do you want 2 = ℵn+2 for all n? Suslin trees? Would you like to separate the bounding number from the dominating number? Do you want an indestructible strongly compact cardinal? Would you prefer that it is also the least measurable cardinal?

We build models to order.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Pervasive Independence Phenomenon

The pervasive independence phenomenon, to my way of thinking, is the central discovery of set theory in the past 50 years.

Largely or even completely unexpected by earlier experts, this phenomenon is now everywhere to be found and fundamental to our understanding of the subject.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Philosophical issues concerning truth and existence

Fine, we can neither prove nor refute CH.

Still, we want to know: is it true?

Although this question may seem innocent, interpreting it sensibly leads directly to difﬁcult philosophical issues.

The main troubles arise from the fact that we don’t actually have an adequate account of what it means to say that a mathematical statement is ‘true’.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

The debate on pluralism Differing philosophical approaches to mathematical truth. Universe view (singularist). There is an ambient, real, set-theoretic universe consisting of all sets, and we aim to discover what is true in it. The independence phenomenon is a distraction about the weakness of our current theories.

Multiverse view (pluralist). There is a huge variety of concepts of set, each giving rise to its own set-theoretic world, with different mathematical truths. We aim to discover these possibilities and how the various worlds are connected.

This is an active, ongoing debate in set theory and the philosophy of set theory.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Universe perspective Set theorists with the universe perspective often point to the unifying consequences of large cardinals as evidence that they are on the right track towards the ultimate theory.

Consequentialism. We judge mathematical theories by their consequences. Scientiﬁc criteria in mathematics. The theories make testable predictions. Unifying elegance. Theories are proposed that exhibit elegant unifying features, which no complementary theory can exhibit.

Ultimately, the universists must provide non-mathematical reasons for their favored truths.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Multiverse perspective

On the multiverse perspective Pervasize independence phenomenon is evidence of distinct incompatible concepts of set Diversity of models of set theory is evidence of actual distinct set-theoretic worlds Points to a new goal: to investigate how these models are connected. Analogy with Euclidean and non-Euclidean geometry: a singular concept splinters.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Summary

Set theorists have discovered the large cardinal hierarchy, an amazing clariﬁcation of the higher inﬁnite. Set theorists have also discovered the pervasive independence phenomenon, by which many statements in mathematics are neither provable nor refutable. These technical developments have led to a philosophical crises concerning the nature of mathematical truth and existence. As a result, there is an ongoing philosophical debate in set theory and the philosophy of set theory concerning how we are to overcome those issues.

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Summary

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Summary

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Summary

Higher inﬁnity and foundations of math Joel David Hamkins, New York The Higher Inﬁnite The Central Discovery of Set Theory Mathematical Truth and Existence

Thank you.

Slides and articles available on http://jdh.hamkins.org.

Joel David Hamkins The City University of New York

Higher inﬁnity and foundations of math Joel David Hamkins, New York