The natural numbers and logical consequences of them

[mr3000]

TL;DR

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

What does this mean/imply?

 

[anuttarasammyak]

What do you mean by slicing numbers? Could you give us some examples?

 

[mr3000]

Dividing them. Such as by 10, 25, 38, and so on.

 

[PeroK]

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

What does this mean/imply?

The natural numbers form a countably infinite set. You can partition the natural numbers into a finite collection of infinite sets. E.g. into odd and even numbers. Or, into sets depending on the remainder when divided by 10.

 

[FactChecker]

TL;DR: The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

The definition of infinity is that it is how many natural numbers there are.

That is where "infinity" begins. It is the size of the smallest infinite set. It's size is denoted by $\aleph_0$. There are larger infinite sets of size $\aleph_1$, $\aleph_2$, ... etc.

You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

Yes.

What does this mean/imply?

Good question. The subject is interesting. You might be interested in the work of Georg Cantor in the late 1800s. He formalized the study of "infinity" and proved that there were sets with sizes larger than the size of the set of natural numbers.

 

[haruspex]

What does this mean/imply?

Operations like addition and multiplication are only defined for numbers, which are necessarily finite. What if we try to extend those to include $\aleph_0$? Can you deduce any formulae?

 

[Hornbein]

TL;DR: The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

The definition of infinity is that it is how many natural numbers there are. You can take those infinite natural numbers and slice them into an infinite number of infinite sets, each of which can then be sliced the same way ad infinitum.

What does this mean/imply?

I don't know that it means anything. It's just an interesting fact. That's what "pure mathematics" is all about. If you make any reference to meaning they don't like it. They want no reference to the real world contaminating their math.

 

[Hornbein]

The definition of infinity is that it is how many natural numbers there are.

Infinity literally means "without end." It doesn't have anything in particular to do with the natural numbers. Things can be considered infinitely small because they are endlessly small. No matter how small it is (but not zero) there is something that is smaller.

 

[weirdoguy]

They want no reference to the real world contaminating their math.

Who are "they"? Not mathematicians obviously.

 

[FactChecker]

I don't know that it means anything. It's just an interesting fact. That's what "pure mathematics" is all about. If you make any reference to meaning they don't like it. They want no reference to the real world contaminating their math.

The development of physics and mathematics have gone hand-in-hand for hundreds of years.
"Pure" math is pure until it is applied to something. The key parts of the mathematics used in Special and General Relativity existed as "pure" math until it was used in Einstein's theory and atomic bombs started exploding.

 

[anuttarasammyak]

The natural numbers form a countably infinite set. You can partition the natural numbers into a finite collection of infinite sets. E.g. into odd and even numbers. Or, into sets depending on the remainder when divided by 10

An example of infinite collection of infinite sets is $\{p^n\}$ where p is prime number. We can do similar on n and so on to get infinite sequence of infinity.

What does this mean/imply?

It is .. as it is.

 

[mathwonk]

the fact that the natural numbers contain an infinite number of disjoint infinite subsets, implies for the arithmetic of infinite cardinals, that if A is the cardinal number measuring the size of the natural numbers, then A.A = A. On the other hand if 2^A denotes the cardinal number of all subsets (finite and infinite) of the natural numbers, one can show that 2^A > A.

 

[haruspex]

the fact that the natural numbers contain an infinite number of disjoint infinite subsets, implies for the arithmetic of infinite cardinals, that if A is the cardinal number measuring the size of the natural numbers, then A.A = A.

That’s what I was trying to lead @mr3000 to in post #6, but got no response.

 

[mr3000]

Is there any basis for this outside of set theory?

 

[FactChecker]

Is there any basis for this outside of set theory?

You asked about the number of items in the set of natural numbers and how they can be partitioned, so I don't think any other considerations are relevant. They would just confuse things.

 

[haruspex]

Is there any basis for this outside of set theory?

Infinity is to do with things. How you can discuss things rigorously without developing a theory of sets eludes me.

 

[FactChecker]

Is there any basis for this outside of set theory?

You probably want to say it is a (?? thing ??) that is greater than any natural number. But what is (?? thing ??)? What are its other properties?

Cantor's formal study of "infinity" involves mappings between sets in addition to plain set theory.
IMHO, this discussion can not make progress until you show that you have at least read about Cantor's approach.

 

[mr3000]

What do you mean by slicing numbers? Could you give us some examples?

I mean multiplication, division, addition, subtraction of each of the natural numbers

 

[anuttarasammyak]

These are called arithmetic operations. I do not find a direct relation to partitioning or decomposing a set into disjoint subsets which I suspect your main subject is.

Results of arithmetic operations to natural number except division by 0 which is not defined belong to 0 and plus-minus rational number
$$\pm\frac{n}{m}$$
whose cardinal is same with natural number.