0 - Axioms of ZFC

Question 1

What is the name of the following ZFC Axiom?

For all x, for all y (

For all z (

z is in x

iff

z is in y ))

Answer 1

Axiom of Extensionality

For all x, for all y (

For all z (

z is in x

iff

z is in y ))

Question 2

What is the name of the following ZFC Axiom?

For all x (

There exists a y s.t. y is in x

implies

there exists a z (

z is in x

AND

for all w (

w is in x

implies

w is not in z )))

Answer 2

Axiom of Foundation

For all x (

There exists a y : y is in x

implies

there exists a z (

z is in x

AND

for all w (

w is in x

implies

w is not in z )))

Question 3

The two regulative axioms are ______** and **_____\_.

These tell you that all sets have these properties, but not if there are any sets to begin with.

Answer 3

The two regulative axioms are extensionality** and **foundation.

These tell you that all sets have these properties, but not if there are any sets to begin with.

Question 4

What is the name of the following axiom?

For all x, For all y, There exists a z (

x is in z

AND

y is in z )

Answer 4

Axiom of Pairing

For all x, For all y, There exists a z (

x is in z

AND

y is in z )

Question 5

The axiom of ______\_ says that

if you have 2 sets you can make another set that contains both of these.

Answer 5

The axiom of pairing says that

if you have 2 sets you can make another set that contains both of these.

Question 6

What is the name of the following axiom?

For all x, There exists a z, For all y, For all w (

( y is in w

AND

w is in x )

implies

y is in z )

Answer 6

Axiom of Union

For all x, There exists a z, For all y, For all w (

( y is in w

AND

w is in x )

implies

y is in z )

Question 7

The axiom of _____\_ says that you can create a new set z out of the elements of the elements of some set x.

Answer 7

The axiom of union says that you can create a new set z out of the elements of the elements of some set x.

Question 8

These three axioms are known as the generative or combinatorial axioms because they grant you the ability to create new sets:

Answer 8

These three axioms are known as the generative or combinatorial axioms because they grant you the ability to create​ new sets:

Pairing, Union, Power Set

Question 9

What is the name of the following axiom?

For all x, There exists a y, For all z (

For all w

( w is in z

implies

w is in x )

implies

z is in y )

Answer 9

Power Set

For all x, There exists a y, For all z (

For all w

( w is in z

implies

w is in x )

implies

z is in y )

Question 10

What is the name of the following axiom?

For all w, There exists a y, For all z

( z is in y

iff

z is in w

AND

phi(z) )

where phi(z) is an arbitrary formula in the language of relations

Answer 10

Axiom (Schema) of Separation

For all w, There exists a y, For all z

( z is in y

iff

z is in w

AND

phi(z) )

where phi(z) is an arbitrary formula in the language of relations

Question 11

The axiom of ________\_ allows you to create or partition a subset from a set as long as you can declare a property that it has.

Answer 11

The axiom of separation allows you to create or partition a subset from a set as long as you can declare a property that it has.

Question 12

What is the name of the following axiom?

For all x0…x1 (

For all u,

There exists a unique z with the property phi(z),

implies

For all w, There exists a u, For all y

(y is in w

iff

there exists a z

( z is in u

AND

phi (y, z, x0…x1 )))

Answer 12

Axiom of Replacement

For all x0…x1 (

For all u,

There exists a unique z with the property phi(z),

implies

For all w, There exists a u, For all y

(y is in w

iff

there exists a z

( z is in u

AND

phi (y, z, x0…x1 )))

Question 13

This axiom contains the statement that there is only one x that satisfies some uniqueness and existence principle y:

Answer 13

This axiom contains the statement that there is only one x that satisfies some uniqueness and existence principle y:

Replacement

Question 14

What is the name of the following axiom?

There exists an x

( There exists a y s.t. y is in x

AND

For all z

( z is in x

implies

There exists a w

( z is in w

AND w is in x )))

Answer 14

Axiom of Infinity

There exists an x

( There exists a y s.t. y is in x

AND

For all z

( z is in x

implies

There exists a w

( z is in w

AND w is in x )))

Question 15

This is the only axiom that starts with an existential operator:

Answer 15

This is the only axiom that starts with an existential operator:

Axiom of Infinity

Question 16

The axiom of ______\_ states that the elements of elements of a set x can create a new unique set that is composed of the pairwise disjoint elements of each.

Answer 16

The axiom of choice states that the elements of elements of a set x can create a new unique set that is composed of the pairwise disjoint elements of each.

Question 17

For proving that a set with some condition exists you first have to prove ________** and **_______\_.

You usually use some combination of axioms to prove the first, and then follow it up with the axiom of _____\_ to prove the latter.

Answer 17

For proving that a set with some condition exists you first have to prove existence** and **uniqueness.

You usually use some combination of axioms to prove the first, and then follow it up with the axiom of extensionality to prove the latter.

Question 18

What is this axiom?

Answer 18

Extensionality

Question 19

What is this axiom?

Answer 19

Pairing

Question 20

What is this axiom?

Answer 20

Union

Question 21

What is this axiom?

Answer 21

Empty Set

Question 22

What is this axiom?

Answer 22

Infinity

Question 23

What is this axiom?

Answer 23

Power Set

Question 24

What is this axiom?

Answer 24

Replacement

Question 25

What is this axiom?

Answer 25

Regularity

Question 26

What is this axiom?

Answer 26

Choice