[hamdani yusuf (OP)]

To be clear,

Looks like another never ending, merry-go-round thread by Hamdani to put on ignore.  Bye and have fun.

No one forced you to write in my threads. If you prefer to only read things that already have obvious answers, that's up to you.

[hamdani yusuf (OP)]

   I honestly don't know.    It depends what you consider to be an "expansion procedure like that".

Adding more members to a set of number by finite amount of additional information.

[hamdani yusuf (OP)]

For example, would you allow taking limits?     If   {Xn}  is a sequence of numbers you already admit in your set,  then  why not also include the Real number X whenever the sequence Xn → X.   That seems like a reasonable expansion procedure.    Anyway, that expansion procedure gets you straight from Q  to  R   in one step.

I haven't found a reason to deny it.
AFAIK, integers are part of rational numbers, which in turn are part of algebraic numbers, which are part of real numbers. Is there a set of numbers which contains more than algebraic numbers but less than real numbers?

[Eternal Student]

Hi.

Is there a set of numbers which contains more than algebraic numbers but less than real numbers?

   If you just want a SET of numbers,  yes,  many of them.
There's an infinite set of distinct transcendental numbers,   T.   These are real numbers that are not algebraic.   Actually MOST of the real numbers are these,   the algebraic numbers are countable so they are a tiny drop in the ocean compared to the transcendental numbers.
   Anyway, you could just keep adding the transcendental numbers one at a time until you got bored.

   You can cut down some of those intermediate sets of numbers if you put more of a restriction on your set of numbers.   For example, instead of just asking for a set of numbers, insist that the new set of numbers is always a proper Field in its own right.   These would be called Field Extensions.   (Example:   If you try to add π,  then the field operations automatically generate  π², π³, ....,  1/π, ...., -π, .... (705 + 3/π), ....   so  to ensure the set really is closed under field operations you'd need all of these numbers added in one go).
  None the less, I still think there's an infinite set of simple field extensions you could find.  (I haven't spent too long thinking about it but I reckon you could try a simple field extension by adding π and show that √π is still missing,  so extend again with √π and show  the √(√π) is still missing..... etc.....    the root of what you've just extended with should always be missing).

Best Wishes.

[evan_au]

∞^∞ = ∞

However, 2^∞ > ∞
(At least for the counting numbers), so this one seems suspect.

∞−∞ and ∞/∞) are not generally well-defined.

In practice, this means that you need more information to determine the answer.

For example, L'Hopital's rule allows you to calculate
Limit of a/b as a→∞  and b→∞
in those scenarios where you know the derivative of a and b (and both aren't infinite).

[hamdani yusuf (OP)]

Here are what I  got after go down the rabbit hole.

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

In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are π and e.[1][2]

Though only a few classes of transcendental numbers are known — partly because it can be extremely difficult to show that a given number is transcendental — transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers comprise a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic.[3][4][5][6] The converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, algebraic non-rational and transcendental real numbers.[3] For example, the square root of 2 is an irrational number, but it is not a transcendental number as it is a root of the polynomial equation x2 − 2 = 0. The golden ratio (denoted {\displaystyle \varphi }\varphi  or {\displaystyle \phi }\phi ) is another irrational number that is not transcendental, as it is a root of the polynomial equation x2 − x − 1 = 0. The quality of a number being transcendental is called transcendence.

https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry)
The periods are intended to bridge the gap between the algebraic numbers and the transcendental numbers. The class of algebraic numbers is too narrow to include many common mathematical constants, while the set of transcendental numbers is not countable, and its members are not generally computable.

The set of all periods is countable, and all periods are computable,[6] and in particular definable.
Many of the constants known to be periods are also given by integrals of transcendental functions. Kontsevich and Zagier note that there "seems to be no universal rule explaining why certain infinite sums or integrals of transcendental functions are periods".

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

In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.[1][2] In other words, a transcendental function "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, subtraction, multiplication, division, raising to a power, and root extraction.[3]

https://en.m.wikipedia.org/wiki/Computable_number
In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers[1] or the computable reals or recursive reals.

[hamdani yusuf (OP)]

There's also continued fraction to classify subsets of real numbers.

https://en.m.wikipedia.org/wiki/Continued_fraction
In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on.[1] In a finite continued fraction (or terminated continued fraction), the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers {\displaystyle a_{i}}a_{i} are called the coefficients or terms of the continued fraction.[2]

Rational numbers have finite continued fraction. Irrational numbers, including transcendental numbers have infinite continued fraction.

[hamdani yusuf (OP)]

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

In 1900, David Hilbert posed an influential question about transcendental numbers, Hilbert's seventh problem: If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is a^b necessarily transcendental? The affirmative answer was provided in 1934 by the Gelfond–Schneider theorem. This work was extended by Alan Baker in the 1960s in his work on lower bounds for linear forms in any number of logarithms (of algebraic numbers).[18]

It seems like we only need one more step to go from power numbers to transcendental numbers. But not all transcendental numbers can be constructed this way.

[Eternal Student]

Hi.

If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is ab necessarily transcendental?

That has a significant copy-and-paste error.    ab should be   a^b =  a^b

Best Wishes.

[Eternal Student]

Hi.

Here are what I  got after go down the rabbit hole.

    Seems reasonable.   For whatever it's worth I didn't know all of the classifications you've managed to find.  I have no doubt that there's more ways you could divide up the real numbers.   There should be loads of sets of numbers that someone has given a name to and had some passing reason to want to study them.

    If you were studying the Real numbers as part of a structured course in Real Analysis,  then at about this point people would push your attention to the remarkable facts that do seem to appear:
1.    It's very difficult to construct the Real numbers by algebraic methods.   (Conventional wisdom is that you can't).
2.    However techniques from Real Analysis allow the construction of an object (a complete, totally ordered field extension of the Rationals) that we can call the real numbers.
3.     Furthermore, accepting certain axioms, we have a uniqueness property:  Any complete totally ordered field extension of the rationals is the same field (up-to isomorphism).

   Item 3,  the uniqueness property, is oddly reassuring but not necessarily important.   It's psychologically reassuring:   When Mr. Smith thinks of the real numbers they are using the same thing as Monsieur Smithe  except that one says    "one", "two", "three"   while the other calls it  "un", "deux", "trois"   etc.    It's philosophically reassuring in that perhaps the Real Numbers aren't completely arbitrary or abstract things - there is actually only one thing (up-to isomorphism) that they can be.    The uniqueness of the Real Numbers is often the final result or crowning glory presented at the end of an introductory course on Real Analysis,  however, I'm not sure it really matters for the purposes of Mathematics.   (Maybe it just prevents squabbles among mathematicians claiming that their Real Numbers are better than your Real numbers).
   

However, 2^∞ > ∞
(At least for the counting numbers), so this one seems suspect.

   Cardinal arithmetic is what they what they sometimes call this.  Ummm... I have no idea what else to say but just didn't want you to think your post had gone un-noticed.  Thanks @evan_au.

Best Wishes.

[alancalverd]

If you could divide infinitely, we might ask how would anything get anywhere at a fundamental length

Dangerous use of "fundamental".

The Planck length is the base unit of length in Planck units, just as the meter is in SI units, and it has no special physical attributes.

Unlike the fundamental constants such as e, ε₀, μ₀ etc which do determine the measured behavior of the universe.

[hamdani yusuf (OP)]

Hi.

If a is an algebraic number that is not zero or one, and b is an irrational algebraic number, is ab necessarily transcendental?

That has a significant copy-and-paste error.    ab should be   a^b =  a^b

Best Wishes.

Thanks for the correction.

Now here's an opinion.
If the numerator and denominator are allowed to be infinite, then any real number can be expressed as ratio between two integers with infinite precision. The same thing can't be done for complex numbers or hypercomplex numbers.

[hamdani yusuf (OP)]

∞^∞ = ∞

However, 2^∞ > ∞
(At least for the counting numbers), so this one seems suspect.

∞−∞ and ∞/∞) are not generally well-defined.

In practice, this means that you need more information to determine the answer.

For example, L'Hopital's rule allows you to calculate
Limit of a/b as a→∞  and b→∞
in those scenarios where you know the derivative of a and b (and both aren't infinite).

I've been feeling the same since the beginning of this thread. So did some critiques of Cantor. Particularly, I'm not convinced by diagonal argument. Eternal Student has shown a loop hole in it. I think it's similar to the cases described in the video below. 

Three false proofs, and what lessons they teach.

Time stamps:
0:00 - Fake sphere proof
1:39 - Fake pi = 4 proof
5:16 - Fake proof that all triangles are isosceles
9:54 - Sphere "proof" explanation
15:09 - pi = 4 "proof" explanation
16:57 - Triangle "proof" explanation and conclusion

[hamdani yusuf (OP)]

If the numerator and denominator are allowed to be infinite, then any real number can be expressed as ratio between two integers with infinite precision. The same thing can't be done for complex numbers or hypercomplex numbers.

I've been thinking through this for quite a while now, but I still can't find a reasonable objection / credible refutation against it. I started to think that the problem is part of information theory, particularly in data compressability. Perhaps it is related to the concept of entropy.
For a start, we can think of integers as part of real numbers which have high compressability, or low entropy. They can be expressed in a finite bits of information, as a multiple of an identity number, which is 1.

[hamdani yusuf (OP)]

For a start,

The next step is to fill up the gaps between those integers. The simplest way is by dividing an integer with another integer. We get a rational number.
To express a non-integer rational number, we can use some bits of information as the numerator, some others as denominator, and a few bits for the operator, which is division.

When the numerator and denominator are arranged vertically, the division operator is written as a horizontal line between them. When the numerator and denominator are arranged horizontally, the division operator is written as a diagonal line between them.

For convenience, the denominator can be set as an integer exponent of a base number, usually 10, which is called decimal system. The fractional part is separated from the whole part by a small character. Some regions use comma, while some others use period or dot.

The decimal expression can simplify arithmetic operations like addition and subtraction. But it sacrifices precision, especially if the denominator of the rational number has prime factors other than 2 or 5. But for most practical cases, we don't need infinite precision, hence the rounding error doesn't often cause problems.

[hamdani yusuf (OP)]

When expressed as ratio, 1/2, 1/3, and 1/4 don't seem to need much different amount of information. But when expressed in decimal, 1/3 needs infinite number of digits. Although the decimal digits are repetitive, which make the information compressible.

It turns out that we can compress the data required to express some irrational numbers if they are algebraic, which is a root of a non-zero polynomial in one variable with integer coefficients. The amount of information needed depends on the coefficients of the polynomial equation.

[Bored chemist]

When expressed as ratio, 1/2, 1/3, and 1/4 don't seem to need much different amount of information. But when expressed in decimal, 1/3 needs infinite number of digits. Although the decimal digits are repetitive, which make the information compressible.

Use base 12 rather than 10.
The problem goes away.

[hamdani yusuf (OP)]

If the numerator and denominator are allowed to be infinite, then any real number can be expressed as ratio between two integers with infinite precision.

Like wise, if the degree of polynomial is allowed to be infinite, then any real number can be expressed as a root of polynomial.

[hamdani yusuf (OP)]

When expressed as ratio, 1/2, 1/3, and 1/4 don't seem to need much different amount of information. But when expressed in decimal, 1/3 needs infinite number of digits. Although the decimal digits are repetitive, which make the information compressible.

Use base 12 rather than 10.
The problem goes away.

But this solution is not general, since it depends on the chosen base number, as well as the denominator. If the denominator has a prime factor not shared with the base number, the number of digits will be infinite.

A more general solution is using continued fraction, which is independent from base number selection. 

[Eternal Student]

Hi.

Particularly, I'm not convinced by diagonal argument. Eternal Student has shown a loop hole in it.

    That was an error in the method shown in that particular text (which was a PopSci version of Cantor's proof that ℜ is uncountable ).  However, it's possible to fix that and most texts describing the diagonalisation method do exactly that.   Just to be clear, the argument is basically sound, it's just that the PopSci article wasn't a perfect version of it.
     However, there's no reason why you should go along with it.  A lot of people won't like the Axiom of Choice, for example.  Earlier posts have mentioned that something like a weak version of this, the axiom of countable choice, does seem to be required.   So you can reject the Axiom of Choice and have a perfectly mainstream set theory, or system of mathematics, where Cantor's diagonalisation method would not hold, if you want.

Now here's an opinion.
If the numerator and denominator are allowed to be infinite, then any real number can be expressed as ratio between two integers with infinite precision........  [AND subsequent posts discussing the idea]...

     A / B     has  A as the numerator and B as the denominator.
I have no rules to know what to do, or what it means when A or B is infinite.
Presumably you meant A and B are arbitrary Real numbers (and B≠0), they can be big but they are real numbers, so they are never infinite.

   Also what do you then mean by "with an infinite precision"? 
   Does that mean that you can give me the expression A/B to represent the real number √2  and then I can't find any small positive number ξ>0  so that     A/B   is a distance ξ (or more) away from  √2 ?     

Best Wishes.