How do we can prove that an infinity set exist?

[Vishtasb]

In the set theory we have inductive sets which are those sets like

\( A \)

 which

\( \emptyset \in A\)

and 

\( \forall a \in A (a^{+} \in A. \)

and 

\( a^{+}= a \cup \{a\} \).

Then we have the  infinity axiom which says there exists an inductive set: 

\[ (\exist A) [\emptyset \in A \and (\forall a \in A) a^{+} \in A]. \]

Now my question is how do we can conclude than infinity sets, specially those sets that are uncountable such as 

\( \mathbb{R} \)

exists? 

[fiveworlds]

Quote

How do we can prove that an infinity set exist?

Georg Cantor (the inventor of set theory) thought infinite sets didn't exist. He even made up the word transfinite to describe sets that which were enormous but not necessarily infinite.

[Strange]

1 hour ago, fiveworlds said:

Georg Cantor (the inventor of set theory) thought infinite sets didn't exist. He even made up the word transfinite to describe sets that which were enormous but not necessarily infinite.

Cantor proved that the "size" of the infinite set of reals is (infinitely) larger than the infinite set of integers. He may not have been happy with the concept of infinity, but that isn't really relevant. 

[uncool]

On 7/2/2019 at 1:09 PM, Vishtasb said:

In the set theory we have inductive sets which are those sets like

\( A \)

 which

\( \emptyset \in A\)

and 

\( \forall a \in A (a^{+} \in A. \)

and 

\( a^{+}= a \cup \{a\} \).

Then we have the  infinity axiom which says there exists an inductive set: 

\[ (\exist A) [\emptyset \in A \and (\forall a \in A) a^{+} \in A]. \]

Now my question is how do we can conclude than infinity sets, specially those sets that are uncountable such as 

\( \mathbb{R} \)

exists? 

By constructing it, using the axioms of set theory.

[John Cuthber]

"How do we can prove that an infinite set exist?"
Well, when we are teaching schoolkids it goes something like this:
Teacher " what's the biggest number?"

Kid" a thousand"
 T "what do you get if you add 1 to a thousand?"

K "A thousand and one"
T "Is that bigger than a thousand?

K "Yes"
T "And if you said you thought that a thousand and one was the biggest number and I asked you to add 1 to it... then what would happen?"

K " you would get a bigger number"
T"Now do you understand why there is no biggest number?

K "yes"

T "So the numbers must go on forever, right?"

K "yes"

Well, that's pretty much defined an infinite set

 

[Vishtasb]

21 minutes ago, John Cuthber said:

"How do we can prove that an infinite set exist?"
Well, when we are teaching schoolkids it goes something like this:
Teacher " what's the biggest number?"

Kid" a thousand"
 T "what do you get if you add 1 to a thousand?"

K "A thousand and one"
T "Is that bigger than a thousand?

K "Yes"
T "And if you said you thought that a thousand and one was the biggest number and I asked you to add 1 to it... then what would happen?"

K " you would get a bigger number"
T"Now do you understand why there is no biggest number?

K "yes"

T "So the numbers must go on forever, right?"

K "yes"

Well, that's pretty much defined an infinite set

 

But this method is about countable sets.

Existence of countable sets can be proved by using infinity axiom. Because they're a kind of inductive sets (or actually can be isomorphism by inductive sets).

But My real question is about uncountable sets.

 

2 hours ago, uncool said:

By constructing it, using the axioms of set theory.

As I know there are two ways to make a new mathematical object.

One method is 'constructive method' and the other is 'axiomatic approach'.

For example in constructive method we defone 0 . Which is the empty set. And any other number is set of all before numbers.

But in axiomatic approach we assume that the set of integers 'exist'.

So if we use the constructive method we can not just say that the set of all real numbers exist. But we can construct it by using some ways such as ordered fileds or Dedkind? Cuts.

Quote

 

 

[Strange]

16 minutes ago, Vishtasb said:

But My real question is about uncountable sets.

Are you familiar with Cantor's diagonal proof? I guess you must be, because the only reason we know there are countable and uncountable sets, with different infinite cardinals is because of Cantor.

17 minutes ago, Vishtasb said:

But in axiomatic approach we assume that the set of integers 'exist'.

Are you asking how we prove the properties of the reals?

[uncool]

22 minutes ago, Vishtasb said:

As I know there are two ways to make a new mathematical object.

One method is 'constructive method' and the other is 'axiomatic approach'.

I truly don't know what distinction you are making here; the axioms are what give you the "material" or "structure" to "construct" things, and (usually more importantly) then provide a method to prove theorems about those things

Constructing an uncountable set is easy. Use the axiom of infinity to get an inductive set; use the axiom schema of specification to get a set we think of as the positive numbers, and use the powerset axiom to get the powerset of the natural numbers. This powerset is uncountable, by Cantor's theorem.

[John Cuthber]

27 minutes ago, Vishtasb said:

But My real question is about uncountable sets.

Then why didn't you ask it?

[mathematic]

Uncountable numbers are best described by being, like the real numbers.  There are proofs that they are not countable.