# Infinity question about known numbers?

#### kris kaczmarczyk

1)
Can I say that humans in history named, recognized countable set of numbers (1,2..Pi,sqrt(2) ... etc)
the count of this set is named aleph-0 ?
2)
Do mathematicians investigate the "other" continuum set of numbers which is infinity bigger ?
Do we suspect, knows anything about that set like that .
For example I am naming pi_2019 number: pi-to-the power of pi ,to-the power of pi .... (times 2019 power it) (new number? I have named;)
3)
Did I move one number from one set to the other (named digits) ?

#### fresh_42

Mentor
2018 Award
1)
Can I say that humans in history named, recognized countable set of numbers (1,2..Pi,sqrt(2) ... etc)
the count of this set is named aleph-0 ?
No. At least not without telling us what's actually in this set and what not.
$\aleph_0$ is the number of elements in $\mathbb{N}=\{\,1,2,3,4,\ldots\,\}$ without $\pi$ and alike.
However, if you only add finitely many elements to this set, then it still has $\aleph_0$ many numbers.
2)
Do mathematicians investigate the "other" continuum set of numbers which is infinity bigger ?
Yes.
Do we suspect, knows anything about that set like that .
There are many more than just one set bigger than $\mathbb{N}$. $\mathbb{R}$ is just one of them.
For example I am naming pi_2019 number: pi-to-the power of pi ,to-the power of pi .... (times 2019 power it) (new number? I have named;)
See above. As long as you only "name" as many numbers as $\mathbb{N}$ has, or less, as long will your set have $\aleph_0$ many elements.
3)
-Did I move one number from one set to the other (named digits) ?
No, not really. But you cannot say it this way. It makes no sense. I suggest to read https://en.wikipedia.org/wiki/Cardinality

#### kris kaczmarczyk

1)
ℕ number of people in history each named , recognized ℕ numbers.
Because ℕ x ℕ is still ℕ and count of ℕ is Aleph-0 , I dare to say that my question still stands?

#### fresh_42

Mentor
2018 Award
There have only been finitely many people who ever lived or still live.

You haven't defined your sets, so it is impossible to say which element is in them and which is not, and even less can be said which cardinality they have.

#### WWGD

Gold Member
1)
ℕ number of people in history each named , recognized ℕ numbers.
Because ℕ x ℕ is still ℕ and count of ℕ is Aleph-0 , I dare to say that my question still stands?
Actually number of people is not the whole of N; there have lived only infinitely-many people ( Some 80 billion IIRC).

#### kris kaczmarczyk

I rest my case it is other way around , basically it is infinitely less than I thought and ascribed to people knowledge.

The number we named , recognized , get from any equation ever written is < than aleph-0 , it is finite number,
oh well so much still to discover.

At last the number of possible pencil drownings is greater than number of all functions and number of all possible functions is greater than continuum ( all subsets of continuum lines ) and circle is not the Riemann function, so we still go a chance ?

#### jbriggs444

Homework Helper
At last the number of possible pencil drownings is greater than number of all functions and number of all possible functions is greater than continuum ( all subsets of continuum lines ) and circle is not the Riemann function, so we still go a chance ?
The usual abstraction of a "pencil drawing" is a continuous real-valued (or 2-dimensional vector-valued) function over some bounded interval in the reals. Since it is continuous, it suffices to specify the result at all rational values in the domain. Unless I am mistaken, the cardinality of the set of such functions is only the cardinality of the continuum.

You can get the same information content by carefully sawing off the pencil to a real-valued length. If you want to break through to a higher order of infinity, you need to lose the continuity restriction.

An engineer might point out that a real pencil drawing involves only finitely many specks of graphite with only finitely many distinguishable positions each. So only finitely many drawings are possible with any given pencil on any given sheet of paper.

#### Mark44

Mentor
Actually number of people is not the whole of N; there have lived only infinitely-many people ( Some 80 billion IIRC).
Only a finite number of people have ever lived. Is that what you were trying to say?

#### WWGD

Gold Member
Only a finite number of people have ever lived. Is that what you were trying to say?
Yes, sorry, pretty roundabout way of saying it.

#### Mark44

Mentor
Not just roundabout -- it's actually incorrect as written.
there have lived only infinitely-many people

#### phyzguy

1)
Can I say that humans in history named, recognized countable set of numbers (1,2..Pi,sqrt(2) ... etc)
the count of this set is named aleph-0 ?
I would say that people have recognized an uncountably infinite set of numbers. The set:
{All real numbers between zero and one} is uncountably infinite and I recognize it to be so.

#### WWGD

Gold Member
Not just roundabout -- it's actually incorrect as written.
Yes, it is nonsensical as written. I meant finitely-many but somehow wrote infinitely.

#### Mark44

Mentor
Yes, it is nonsensical as written. I meant finitely-many but somehow wrote infinitely.
That's what I thought.

#### kris kaczmarczyk

The usual abstraction of a "pencil drawing" is a continuous real-valued (or 2-dimensional vector-valued) function over some bounded interval in the reals. Since it is continuous, it suffices to specify the result at all rational values in the domain. Unless I am mistaken, the carnality of the set of such functions is only the cardinality of the continuum.

You can get the same information content by carefully sawing off the pencil to a real-valued length. If you want to break through to a higher order of infinity, you need to lose the continuity restriction.

An engineer might point out that a real pencil drawing involves only finitely many specks of graphite with only finitely many distinguishable positions each. So only finitely many drawings are possible with any given pencil on any given sheet of paper.
I am living in the Platonic world. If you believe Cantor any set of all subset of the set has next-step aleph-n carnality.
Segment has carnality of space if we count as my profesor at high-school defined: to buckets we remove elements for each if they are both empty they equal (Lebasgue ?).
But becasue all Reimann function represents all possible subsets of R-real number combination. Their set is greater than ℂ - eual to numbers of point on the segment.
What I was trying to say that art has its own beauty, and circle , which is not a Riemann function - x is associated with 2 y pints sometimes expands the set of all proper lines (functions) artists could draw.

#### kris kaczmarczyk

There have only been finitely many people who ever lived or still live.

You haven't defined your sets, so it is impossible to say which element is in them and which is not, and even less can be said which cardinality they have.
Let me rephrase my question: If we resale all numbers into segment [0,1] (like 0.314156535..., 0.2, 0.3, 0.17)
Remove all numbers from the set: defined as [p/q] p,q ∈ N and q≠ 0 (as far as I remember it has ℂ cardinality, (all primes are gone, for example) only 0.314155635 will stay from my example. There is infinite difference between ℂ and ℝ (R-C=>R). So what is left of my segment looks like nothing was changed (although 0 and 1 are gone, so it is open segment but it still has ℝ components ); Is there any pattern of "gaps" we can observe zooming out the (0,1) segment now looking for missing C ; also do we see islands of numbers there which we never thought about - named before; I just wonder what do we missing ? is there some special square root etc...

#### phyzguy

Let me rephrase my question: If we resale all numbers into segment [0,1] (like 0.314156535..., 0.2, 0.3, 0.17)
Remove all numbers from the set: defined as [p/q] p,q ∈ N and q≠ 0 (as far as I remember it has ℂ cardinality, (all primes are gone, for example) only 0.314155635 will stay from my example. There is infinite difference between ℂ and ℝ (R-C=>R). So what is left of my segment looks like nothing was changed (although 0 and 1 are gone, so it is open segment but it still has ℝ components ); Is there any pattern of "gaps" we can observe zooming out the (0,1) segment now looking for missing C ; also do we see islands of numbers there which we never thought about - named before; I just wonder what do we missing ? is there some special square root etc...
One property of this set. Between any two rational numbers, no matter how close, there are an infinite number of irrational numbers, and between any two irrational numbers, no matter how close, there are an infinite number of rational numbers. So there are no "gaps".

#### Mark44

Mentor
Segment has carnality of space if we count as my profesor at high-school defined: to buckets we remove elements for each if they are both empty they equal (Lebasgue ?).
First off, carnality and cardinality are words with very different meanings. The one you mean is cardinality.

Second, two sets have the same cardinality if there is a one-to-one pairing between the two sets. For example, the cardinalities of the real intervals [0, 1] and [0, 100] are the same -- the pairing is y = 100x, with $x \in [0, 1]$ and $y \in [0, 100]$. The definition from your professor omits some important details about the actual pairing.

#### kris kaczmarczyk

I guess I mean: https://en.wikipedia.org/wiki/Lebesgue_measure , but i remember geometric explanation: the cross of segment and ℝ. If you select point in space below the height of segment cross , you can draw straight line crossing segment and infinite line, and then you have one to one relation (Euclidean space) between segment and infinite line representing ℝ
So we have "one-to-one pairing" - my high schooling professor explanation.

#### kris kaczmarczyk

One property of this set. Between any two rational numbers, no matter how close, there are an infinite number of irrational numbers, and between any two irrational numbers, no matter how close, there are an infinite number of rational numbers. So there are no "gaps".
Thank you - I miss it, it is getting worst and worst for us humans, we all are finite set and what we recognized you event cannot see and "touch" - if you point with "Platonic" needle , you will never touch "known numbers" (0 probability) no matter how small the segment and how much it was zoomed out?

#### PeterDonis

Mentor
If you believe Cantor any set of all subset of the set has next-step aleph-n cardinality.
That's not quite what Cantor's Theorem says. The theorem says that the power set (set of all subsets) of any set $S$ has a higher cardinality than $S$. But it does not say that it is the next aleph after the cardinality of $S$.

#### mfb

Mentor
The set of numbers discussed individually in the history of humanity is finite. So what?

Similarly: The set of stars that has been discussed individually is a tiny subset of the total number of stars in the observable universe. That doesn't make the other stars less real.

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