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Axiom

A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B

is counter-intuitiveintuitive.

Correct; since ancient time [see Euclid's Elements] it was assumed :

Common notion 5.

The whole is greater than the part.

But is also quite ancient the discovery that this principle may lead to problems in some cases; see the so-called Galileo's paradox.

Thus, Cantor's intuition to define a new way of "counting" the elements of a collection : the notion of one-to-one correspondence is prior to (and independent of) the concept of counting number.

The benefit of this approach was its applicability to every type of collections: finite and infinite.

In the case of finite collection*, the new approach was cocnsistent with the usual one, based on counting numbers.


Having said that, tehre are some alternative approaches to mathematical infinity; see e.g.

and

See also Finitism.

Axiom

A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B

is counter-intuitive.

Correct; since ancient time [see Euclid's Elements] it was assumed :

Common notion 5.

The whole is greater than the part.

But is also quite ancient the discovery that this principle may lead to problems in some cases; see the so-called Galileo's paradox.

Thus, Cantor's intuition to define a new way of "counting" the elements of a collection : the notion of one-to-one correspondence is prior to (and independent of) the concept of counting number.

The benefit of this approach was its applicability to every type of collections: finite and infinite.

In the case of finite collection*, the new approach was cocnsistent with the usual one, based on counting numbers.


Having said that, tehre are some alternative approaches to mathematical infinity; see e.g.

and

See also Finitism.

Axiom

A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B

is intuitive.

Correct; since ancient time [see Euclid's Elements] it was assumed :

Common notion 5.

The whole is greater than the part.

But is also quite ancient the discovery that this principle may lead to problems in some cases; see the so-called Galileo's paradox.

Thus, Cantor's intuition to define a new way of "counting" the elements of a collection : the notion of one-to-one correspondence is prior to (and independent of) the concept of counting number.

The benefit of this approach was its applicability to every type of collections: finite and infinite.

In the case of finite collection*, the new approach was cocnsistent with the usual one, based on counting numbers.


Having said that, tehre are some alternative approaches to mathematical infinity; see e.g.

and

See also Finitism.

1
source | link

Axiom

A2: If set A is a proper subset of set B, then A has a smaller cardinality than set B

is counter-intuitive.

Correct; since ancient time [see Euclid's Elements] it was assumed :

Common notion 5.

The whole is greater than the part.

But is also quite ancient the discovery that this principle may lead to problems in some cases; see the so-called Galileo's paradox.

Thus, Cantor's intuition to define a new way of "counting" the elements of a collection : the notion of one-to-one correspondence is prior to (and independent of) the concept of counting number.

The benefit of this approach was its applicability to every type of collections: finite and infinite.

In the case of finite collection*, the new approach was cocnsistent with the usual one, based on counting numbers.


Having said that, tehre are some alternative approaches to mathematical infinity; see e.g.

and

See also Finitism.