[hamdani yusuf (OP)]

Here is a Venn diagram of real number:

Reference: https://www.physicsforums.com/threads/is-my-classification-of-transcendental-correct.921555

Here's a pattern.
There are infinitely many more integers than natural numbers.
There are infinitely many more rational numbers than integers.
There are infinitely many more algebraic numbers than rational numbers.
There are infinitely many more real numbers than algebraic numbers.

But somehow the state of countability changes only between real numbers and algebraic numbers.

Interestingly, all the examples for non-algebraic numbers shown in the diagram above are analytic numbers.
Non-analytic numbers can be constructed by filling ai and bi in a generalized continued fraction with numbers in an irregular pattern, e.g. by putting random numbers for each of them. This makes most of real numbers are non-analytic numbers, i.e. cannot by represented using a finite amount of information.

[paul cotter]

How many numbers exist?-answer: as many as you want. Rumours suggesting a shortage of numbers have been thoroughly debunked.

[alancalverd]

Indeed you can prove by induction that there are more than you could ever need, and they are all free, but it doesn't stop people panicking!

[hamdani yusuf (OP)]

This way, the examples for non-algebraic numbers in the Venn diagram below can be classified as analytic numbers, which means that they can be represented as a regular pattern in a generalized continued fraction. It means that they can be stated using a finite number of bits of information.

This new class of real numbers can also be called compressible number or algorithmic number. They can be produced by a finite algorithm using only natural numbers. 
So,  the number onion diagram would contain
natural > integer > rational > algebraic > algorithmic > real.

[Eternal Student]

Hi.

   I quite like a bit of mathematics, so It's nice to see anyone thinking about some Maths on this forum.

Here's a pattern.
There are infinitely many more integers than natural numbers.
There are infinitely many more rational numbers than integers.
......
But somehow the state of countability changes only between real numbers and algebraic numbers.

       Sounds OK.   Your use of "somehow" makes it sound like it shouldn't happen.   Isn't it possible that one of those increases just was much, much bigger than another?  The phrase "infinitely many more" just isn't a very restrictive description of an increase in size.   There are many increases you can make which are infinite increases in size.
     Exactly as you have identified, some of those increases leave the end set with the same cardinality as the starting set - but some increases can and do change the cardinality.
   
   

But somehow the state of countability changes only between real numbers and algebraic numbers.

     You may be over-emphasizing exactly where this change takes place.    You can add a finite set of transcendental numbers to the algebraic numbers and that's obviously no problem -  the final set is still countable.   Moreover you can add an infinite BUT countable set of transcendentals to the algebraic numbers and you would still have a final set that is countable.    You can keep on adding more infinite (but countable) sets of transcendentals and the resulting set will still be countable.    Nothing changes or breaks until you try to add an uncountable set of transcendentals,   for example if you tried to add ALL of the transcendental numbers.
     To say this a different way,  it is not as if the algebraic numbers are the final, last or maximal countable set you could ever build.   You can certainly build bigger sets that are still countable.   Mathematicians just tend to stop at the algebraic numbers because that's an interesting set of numbers in its own right and not because it was as big as they could get while still keeping the set countable.
     You can also play the game from the other end and try to work downwards....    starting from the Reals you can easily strip out an infinite (but countable) set of numbers and you'd still have a set which is uncountable.    You can be even more devious and strip out an uncountable set of numbers and yet still have failed to reduce the cardinality (you can still end up with a set this is uncountable).
     Somewhere between the algebraics and the Reals,  the cardinality jumps from that of N  to that of ℜ,   but there is no set of numbers that you can identify that describes the maximal countable set or the minimal uncountable set.

    Infinity and cardinality, it's an interesting topic.

Best Wishes.

[hamdani yusuf (OP)]

You can add a finite set of transcendental numbers to the algebraic numbers and that's obviously no problem -  the final set is still countable.   Moreover you can add an infinite BUT countable set of transcendentals to the algebraic numbers and you would still have a final set that is countable.    You can keep on adding more infinite (but countable) sets of transcendentals and the resulting set will still be countable.    Nothing changes or breaks until you try to add an uncountable set of transcendentals,   for example if you tried to add ALL of the transcendental numbers.

I'd like to borrow a concept from biology in grouping organisms called monophylecy. A group of numbers worth naming if it's monophyletic.
In your example, they are either polyphiletic or paraphyletic.

[Eternal Student]

Hi again.

I'd like to borrow a concept from biology in grouping organisms....

     I expect you can do that.

In your example, they are either polyphiletic or paraphyletic.

    Maybe but probably only because Human beings tend to start with rational numbers (fractions).   Constructing the Real numbers is something mankind and especially mathematicians have been interested in for hundreds if not thousands of years.  It's very interesting stuff but we do need to recognise that how things have been done historically does not need to be taken as anything resembling a natural order or evolution.   Speaking objectively, Numbers did not come about in just one way, instead human preference has just lead us to develop them one way but other ways are available.

     Quite often mankind starts from the counting numbers (the Natural numbers) and historically it was even assumed that these were god-given and could not be constructed from anything simpler.   More recently it has been considered that set theory and formal mathematical logic should be the foundation of all mathematics and it is possible to construct the Naturals from these more elementary mathematical objects if you try (see, for example Peano's axioms:   http://www2.hawaii.edu/~robertop/Courses/TMP/7_Peano_Axioms.pdf    which provides what looks like a fairly nice and short document describing the construction of the Natural numbers.   You could also find some details elsewhere like Wikipedia if you wanted but their page(s) are not set up as well for learning).   If you wanted an entirely set theoretic approach then you just note that an empty set  ∅ should exist in your set theory system and then define the element 0, "zero",  (the first Natural number that Peano's axioms requires) to be the empty set ∅.    The successor, S(n) can then be defined as the power set of n.   For example, the number   1  is defined to be  S(0)  =^def  P(∅) =^def {∅}  where I have written   =^def   to indicate that this is a definition and not to be mistaken as the equality operation = that is defined in the Peano Axioms themselves.   Note that although  ∅   and {∅}  may superficially look to be the same set, they are not.    ∅, the empty set, has no elements   but  {∅}  has one element which is the empty set.     By continuing to take successors we obtain  the number 2 =^def  {∅, {∅} }   which has 2 distinct elements;     3 =^def  { ∅, {∅}, {{∅}},  {∅, {∅}} } which has 4 distinct elements....    etc.... (it gets messy but you just keep taking power sets).
    One way or another, whether god-given or otherwise, we tend to arrive at the Natural numbers as something that is fairly basic and elementary.    Where we go from there seems to be just a human preference.   Historically, we have tended to construct the rational numbers,  "the fractions"     a/b  and indeed the ancient Greeks were only happy with these.    They were greatly troubled about the apparent existence of numbers like √2   which seemed to come out of their knowledge of geometry   (e.g. Pythagoras' theorem applied to determine the length of the hypotenuse for a right angled triangle with the two other sides of length 1).   However, if you now accept that irrational numbers like √2 are required then there is no reason why you couldn't start to extend the Natural numbers by adding some irrational numbers first rather than by adding some rational numbers.
     You can easily have a mathematical structure called a Group where the elements are the Natural numbers plus one or more irrational numbers.   Those elements can be combined under the usual operation of addition +.
    What I'm trying to say is that it is only human preference that has lead to the development of rational numbers before the development of irrational numbers.    In your classification of different sets of numbers as being a "monophylectic" or "paraphylectic",   there may not be any objective reason to consider   "this place" or "this set of numbers"  as being the natural ancestor of some subsequent sets of numbers that we might construct.   
    To say it another way,  when presented with an assorted collection of sets of numbers (e.g. some with Naturals, some with transcendentals, some with a mixture) there may not be any objective way of deciding where their common ancestor might have been and therefore if you have a monophyletic or paraphylectic collection of sets.   It's been mainly just human preference that has lead us to think that the natural path of development should be 
     Naturals --->  Rationals  ---> Irrationals  ---> Transcendentals  ---> Reals   
...but all sorts of other development routes are possible, with various branches, forks, re-convergence to a common set and then splitting off again  etc. etc.

 Naturals  --->   Naturals plus √2    --->  Rationals plus √2   ---->   Rationals plus √2 and π   --->  Reals   ----> Complex
                                                                      \                                                                                   /\    \
                                                                        \                                                                                 /      \
                                                                        \/                                                                               /         \
                                                                    Algebraics   -------------->   Algebraics plus e and Ln 10          \/
                                                                                                                                                                     Hyperreal

I'm not a philosopher but I'll suggest that Nature didn't evolve the number system down one set route.... it has just always been there and it's only human beings that try to understand it by developing the full system of the Real numbers from simpler structures.  There are many routes to get from the Naturals to the Reals and no reason to objectively assert that one is the right way.

Best Wishes.

LATE EDITING:   Sorry, the branch diagram for the construction of some sets of numbers only works when viewed on a widescreen monitor with the forum maximised to take up the whole screen.   Hopefully you get the idea, even if your small mobile phone device is splitting up the diagram onto many lines.   I should have done a proper diagram as a JPEG image  or something, sorry.

[Zer0]

It's sort of the point here. Physics breaks down below this length.

Reminds me of the concept of Singularities.

P.S. - Perhaps you are Gone, Never to Return...so long & Peace!
✌️

[evan_au]

there are more (numbers) than you could ever need, and they are all free

In AI (Artificial Intelligence) work, numbers are not free. It takes serious electrical energy to train a large AI model, and even more to run it.
- The more resolution in your calculations (eg using 64 bit floating point instead of 32 point floating point), the faster it converges, but the more electricity it takes to train the model.
- An AI model which is to be used by millions of people (eg a self-driving car algorithm) might use far more electricity to execute than it took to train, so researchers are looking at bringing AI execution down to 16 bits, 8 bits or perhaps even less?
- Some of the recent gaming video chips turned into supercomputers (eg NVIDEA*) have partitioned their computation hardware so it can do 1 x 32-bit operation or 4 x 8-bit operations at once.

..which leads to a debate about "how many numbers do you need?". It's a bit simplisitic to say that the nerves in our brains only have 2 states: firing or not firing.
- There is a complex modulation with rates of firing
- and a poorly-understood overall modulation by brain waves (alpha, delta, etc).
- but rough calculations suggest that neurons on the brain have a marvellous energy economy in terms of energy per transition, which is the envy of engineers designing computers out of silicon...

* other gaming chips are available, and other supercomputers are available, but not many that are both...

[evan_au]

When I was a kid, the concept of infinity did my brain in.

But even finite numbers can do your brain in. "Graham's Number" is so mind-bogglingly huge that if you tried to hold all the digits of it in your head, your head would collapse into a black hole.
- The proof of this comes from the observation that information has a certain entropy, and the surface of a black hole is the smallest area with a certain entropy
- Apparently, the Schwarzschild radius of Graham's number is bigger than your head... (Note: I haven't tested this calculation myself)
https://en.wikipedia.org/wiki/Graham%27s_number

Podcast (1 hour:15 minutes): https://www.preposterousuniverse.com/podcast/2022/10/17/214-antonio-padilla-on-large-numbers-and-the-scope-of-the-universe/

[Eternal Student]

Hi again,

I've enjoyed reading  @evan_au 's comments and might get around to listening to that podcast.  It has also reminded me of something else that was relevant from an older comment by @hamdani yusuf .

It means that they (...certain types of numbers...) can be stated using a finite number of bits of information....(and going on to suggest that other types of numbers can not be represented with a finite amount of information)....

       
   Any number can be represented with a finite amount of information or digital bits of information.   It's just that you can't have this ability for all numbers simultaneously.   Let's try and explain this simply and without recourse to much mathematics:
    Let's take Graham's number as our example, that can be taken as your standard unit for quantity.   So you just set one bit for a quantity equal to Graham's number.     This means you can also represent some other numbers (like two lots of Graham's number) quite easily.    However, a small number like 1,  then needs to be expressed as a fraction of  Grahams number and that is now going to be extremely information dense.
   To re-phrase the concept in more conventional mathematical language,  we can always perform a change of basis so that any number can be easily represented.
   Anyway, it's just my opinion but I would say that as human beings we are all too quick to assume that some numbers intrinsically have weird or interesting properties as if that is a property of the number.   Quite often it is just a property of the representation we are using.

Best Wishes.

[hamdani yusuf (OP)]

What I'm trying to say is that it is only human preference that has lead to the development of rational numbers before the development of irrational numbers.    In your classification of different sets of numbers as being a "monophylectic" or "paraphylectic",   there may not be any objective reason to consider   "this place" or "this set of numbers"  as being the natural ancestor of some subsequent sets of numbers that we might construct.   

In the Venn diagram shown below, smaller ellipses are subsets of the larger ellipses. Algorithm to precisely express the numbers in smaller sets are more restricted compared to larger sets.
Algorithm allowed to express natural numbers only involve addition and multiplication of smaller natural numbers.
Algorithm to express integer numbers requires subtraction operator, besides addition and multiplication of natural numbers.
Algorithm to express rational numbers requires division operator, besides what's needed to express integer numbers.
Algorithm to express algebraic numbers requires root operator, besides what's needed to express rational numbers.
Algorithm to express algorithmic numbers requires finite algorithm, besides what's needed to express algebraic numbers.
The non-algorithmic real numbers can't be expressed using finite algorithm.

You can't jump to use root operator for the first iteration of number classification because it requires exponentiation by rational number.

This way, the examples for non-algebraic numbers in the Venn diagram below can be classified as analytic numbers, which means that they can be represented as a regular pattern in a generalized continued fraction. It means that they can be stated using a finite number of bits of information.

This new class of real numbers can also be called compressible number or algorithmic number. They can be produced by a finite algorithm using only natural numbers. 
So,  the number onion diagram would contain
natural > integer > rational > algebraic > algorithmic > real.

As I mentioned earlier, all examples above are algorithmic numbers. Non-algorithmic numbers can't be accurately expressed using finite algorithmic symbols.

[Eternal Student]

Hi.

You can't jump to use root operator for the first iteration of number classification because it requires exponentiation by rational number.

  Not necessarily.
   For example,  the Taylor series for √(1+x)  is just a series involving only exponentiation by natural number powers (which is just repeated multiplication if you didn't even want to use that exponentiation).   Admittedly, I would like some use of division.   Anyway set x = 1 and then you have a series  which can be the motivation for them wanting to build a bigger algebraic structure that includes the limit of that series.    (Just to be clear, they don't need to know anything about Taylor series - they can just observe that there is a series which seems to be leading somewhere and be motivated to build a bigger algebraic structure that would include the limit of such a series).
   
Best Wishes.

[evan_au]

You can't jump to use root operator for the first iteration of number classification because it requires exponentiation by rational number.

From another perspective, there is a binary square root algorithm that operates very similar to binary division - just compare, subtract and shift.
- It requires no exponentiation.
- However, since it requires an infinite number of shifts and subtractions to calculate √2, you can't call it a finite algorithm.
- ...even if you did have a computer that could store and compare numbers with an infinite number of bits.

[hamdani yusuf (OP)]

Hi.

You can't jump to use root operator for the first iteration of number classification because it requires exponentiation by rational number.

  Not necessarily.
   For example,  the Taylor series for √(1+x)  is just a series involving only exponentiation by natural number powers (which is just repeated multiplication if you didn't even want to use that exponentiation).   Admittedly, I would like some use of division.   Anyway set x = 1 and then you have a series  which can be the motivation for them wanting to build a bigger algebraic structure that includes the limit of that series.    (Just to be clear, they don't need to know anything about Taylor series - they can just observe that there is a series which seems to be leading somewhere and be motivated to build a bigger algebraic structure that would include the limit of such a series).
   
Best Wishes.

Thanks for the correction.
But still, the main point remains. You can't jump number classifications.

Admittedly, I would like some use of division.

Which is multiplication by rational numbers.

[hamdani yusuf (OP)]

You can't jump to use root operator for the first iteration of number classification because it requires exponentiation by rational number.

From another perspective, there is a binary square root algorithm that operates very similar to binary division - just compare, subtract and shift.
- It requires no exponentiation.
- However, since it requires an infinite number of shifts and subtractions to calculate √2, you can't call it a finite algorithm.
- ...even if you did have a computer that could store and compare numbers with an infinite number of bits.

Shift left is related to multiplication, while shift right is related to division.

Infinite sum can be expressed using finite algorithm, as long as it follows a regular pattern.

[Eternal Student]

Hi.

But still, the main point remains. You can't jump number classifications.

    We can.   Since we already know the Reals exist or can be constructed by some route,   we can just introduce an element √2 and describe how it combines with the Naturals under addition as an axiom.   We don't have to tell them (or ourselves, or whoever it was constructing the Reals by some other route) how or why they might generate that number or want to build a bigger algebraic structure that includes that element just from stuff they notice from the numbers they already have.
    If the axioms are consistent with how we already know that √2 behaves in the Reals  (e.g.   3 + (2+√2)  + (4+√2) =   (9 + √2) + √2   etc. etc. - you just tell them exactly how the new element should combine and behave)   then the structure they build will be a copy of what we would describe as the Group generated by N and √2.     (Specifically, since those axioms were consistent with and necessarily follow from what we know already about the Reals, we do no harm by adding those axioms.  We don't produce any new inconsistencies and we don't give them access to any logical statements that we don't have access to ourselves).
     I appreciate that this may not be your (or many people's) idea of a natural route for the evolution or development of the Reals from the Naturals.   It only seems "natural" if there was some reason or motivation to build a bigger algebraic structure based on what you can see emerging or generate from the numbers you already have.   However, from a pure Mathematics point of view, it is possible to create a structure like this regardless of whether you could see any reason to incorporate such an element that behaves √2.    Our own historical development of the number system progressed in various ways, sometimes there were clear reasons for wanting to include a new element and it was very much as if you could almost generate that number from what you already had.  However, sometimes you couldn't generate anything like that element from what you already have but there was still some reason to want to include a new thing you could call a number that had a particular property.  Here are some examples:
   Example 1:    The need for irrational numbers like √2  was first suggested by some results from geometry.   It wasn't until much later that results like Taylor series were known and you can then see that we can almost generate the new number from what you already have.   
    Example 2:    The complex numbers can be developed literally just by taking as an axiom that we want a new element i and the properties that this should have under the field operations of + and x.   We take as an axiom that this element has an interesting property under multiplication, i²  = -1   even though no other element we already had in the Reals has any ability resembling this.   We didn't (and can't) obtain anything like the element i  just by taking some sequence or series of real numbers and considering limits.   I haven't tried but I would say that we can't obtain the element i by anything that looks like a simple extension of some algorithm or process applied to elements of ℜ.  Whether we can or not, I can assert that we just didn't do it that way.   We identified such an element i as being desirable and wished to construct an algebraic structure that behaved like a number system (a field) with that element for a different reason.   We just wanted a field where equations like  x²  +1  = 0  had a solution and the earliest development of the Complex numbers was entirely based on just assuming the existence of a number which will do the job.     Where Complex numbers are still studied at school  (which I think is now only on some further maths A-level courses) this is still exactly how they present the Complex numbers -  you just start by assuming the existence of a number i   with the property i² = -1.
      Now, as it happens, we certainly don't just have to take the existence of i as an axiom.   We can construct a field based on the structure ℜ X ℜ   with  field operations  + and x  defined a certain way.   You don't generally see that being presented until University level studies to the best of my knowledge.  However, it's just worth noting that it can be done.   It may seem odd if historically someone had just started the other way round but it is entirely possible that they could have done so....  "Look here's an algebraic structure we can build".  People would have said "...and why would we want that?  I can build a garden spade that also has a drinking trough for camels but we will never want it".   However, some years later the original person can reply  "ah ha...  the thirsty camel has finally come while you were gardening... or in my case  I have an element (0,1)  such that  (0,1)² = (1,0)  and that's precisely what we need for the element i".

   Anyway... this post is getting too long.   The gist of it is... crumbs, I can't remember, it was a sort of ramble about number systems and camels... oh yes, vaguely the development of the number system progresses in all sorts of ways.  You don't always need to identify how you might generate a new number from what you already have,  sometimes you will just identify a need for such a new number (like i ) and chose to extend your algebraic structure, your number system, to include it.   A set of space aliens may have developed a new element √2 that they wanted to include with their Natural numbers perhaps because an equation like  x² - 2 = 0  was very important to them.

Best Wishes.

[hamdani yusuf (OP)]

Starting from natural numbers,
Integers are naturally formed by inverse operator of addition.
Rational numbers are naturally formed by inverse operator of multiplication.
Algebraic numbers are naturally formed by inverse operator of exponentiation.
Following the pattern, it seems possible to construct higher class of numbers using higher level of operations, and use their inverse.

[hamdani yusuf (OP)]

https://en.m.wikipedia.org/wiki/Hyperoperation

In mathematics, the hyperoperation sequence [nb 1] is an infinite sequence of arithmetic operations (called hyperoperations in this context)[1][11][13] that starts with a unary operation (the successor function with n = 0). The sequence continues with the binary operations of addition (n = 1), multiplication (n = 2), and exponentiation (n = 3).

After that, the sequence proceeds with further binary operations extending beyond exponentiation, using right-associativity. For the operations beyond exponentiation, the nth member of this sequence is named by Reuben Goodstein after the Greek prefix of n suffixed with -ation (such as tetration (n = 4), pentation (n = 5), hexation (n = 6), etc.) [5] and can be written as using n − 2 arrows in Knuth's up-arrow notation.

Hyperoperations allow us to exactly express a non-algebraic real number.
For example, x^x=2.
According to Wolframalpha, x=e^(W(log(2)). It involves inverse of tetration.
More real numbers can be constructed using inverse of pentation, hexation, and so on.

[hamdani yusuf (OP)]

The real numbers can be constructed using inverse of Hyperoperations as expansion of natural numbers.
Start from H1, which is addition, we can expand natural numbers to integers by adding negative integers. Here's an example.
x+10 = 1
In Hyperoperation notation,
x (H1)  10 = 1
x = -9, which is not a natural number.

Next is H2, which is multiplication. We can expand integers  to rational numbers by adding fractions. Here's an example.
x * 10 = 1
In Hyperoperation notation,
x (H2)  10 = 1
x = 0.1 which is not an integer.