What do you call a function that extracts an algebraic number from a transcendental number?

[Kharakov]

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Infinite sets of numbers come in nested categories, and people have come up with lots of categories that can be put on a one to one correspondence with natural numbers, and lots of other categories that can't be. Here's a nesting of categories, where each infinite set is a subset of all the later infinite sets:

...
Integral eighth powers
Integral fourth powers
Integral squares
Non-prime numbers
Natural numbers
Integers
Rational numbers
Algebraic numbers
Computable numbers
Describable numbers
Real numbers
Complex numbers
Quaternions
...

All of those sets up through the describable numbers can be put on a one to one correspondence with natural numbers; starting with the real numbers, they cannot be. In addition, you can get an infinite set by deleting one of these sets from one of its supersets. The prime numbers are the natural numbers with the non-primes deleted; the negative numbers are the integers with the natural numbers deleted; the irrational numbers are the real numbers with the rational numbers deleted; and so forth. In particular for our topic, the transcendental numbers are the real numbers with the algebraic numbers deleted. And in general, any time you have a category like that, defined by taking one of the basic categories and deleting one of its smaller subsets, what you have left will have the same number of elements as the larger category you deleted a subset from. So there are exactly as many prime numbers as natural numbers; there are exactly as many negative numbers as integers; there are exactly as many irrational numbers as real numbers; and there are exactly as many transcendental numbers as real numbers.

"Exactly as many" has a defined meaning when we're talking about infinite sets. When set Q has exactly as many elements as set S, it means there exist subsets P and R, where P is a subset of Q and R is a subset of S, such that P can be put on a one to one correspondence with S and Q can be put on a one to one correspondence with R. You don't have to actually put Q and S on a one to one correspondence with each other. That can be technically very difficult for a variety of boring reasons, so we use the subset method. The idea is that when P can be put on a one to one correspondence with S, it means the number of elements in P and S are equal, so since P is a subset of Q, the number of elements in Q must be greater or equal to the number in S. So if we can do that subset matching in both directions, we get |Q| >= |S| and |S| >= |Q|, and from the two inequalities we deduce |Q| = |S|.

Here's an example. Are there the same number of real numbers R with 0 <= R < 1 as there are infinite strings of decimal digits? On first glance, obviously yes -- they're the same set, right? .358 = .358; .4747... = .4747...; sqrt(1/2) = .70710678118...; and so forth. The problem comes in when we remember that .73999... and .74000... are the same real number, but they're different infinite strings of decimal digits.

It's not only the digits that are important, but magnitude & direction trump digits (well, let's just do the positive reals for now, forget direction). Digits don't even really describe magnitude if one is playing fast and loose with bases, and do the manipulations you mention (which have nothing to do with magnitude of the numbers).

They (naturals- kharakov) include every finite permutation of digits.

Ok, I suppose this is where I'm lost, specifically. Why don't the naturals include every possible permutation of digits, or is there a defined cap of the set of naturals in which they sortof maybe approach infinity but don't really approach infinity because limits aren't defined well for naturals (there is no natural that contains unbounded information, even though the set of naturals has no bounds)?

So all the numbers that can be specified with a finite amount of information can be put in a 1-to-1 correspondence with the natural numbers.

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That's not just the algebraics. The algebraic numbers are the numbers that can be specified with a finite amount of information in the language of polynomials. But there are other languages that are more expressive than polynomials. All the familiar transcendental numbers like pi and e can also be specified with a finite amount of information, using notations with summation symbols or integrals or trigonometric functions or whatever. So to put them in a 1-to-1 correspondence with the natural numbers you merely have to encode those operators as digits, the same way we encode polynomial coefficients as digits when we map algebraics to natural numbers.

The transcendental numbers you can't put in a 1-to-1 correspondence with the natural numbers are precisely the ones that can't be specified with a finite amount of information in any language whatsoever -- it's the ones whose digit sequences are random.

Ok, that appears to be complete bullshit to me- random? I can see chaos, but one can generate digit sequences that go up and down at arbitrary points. In fact, one can create whatever one wants (unless one wants something truly random).

So.. what do you mean? Weierstrass's monsters are unpredictable, but... not random.

I don't see the pertinence of the diagonal argument to whether or not transcendent numbers outnumber rationals?

If there were the same number of rationals as transcendentals, or if there were more rationals, then you could put all the transcendentals into a 1-to-1 correspondence with the rationals or with a subset of the rationals.

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(That's how it works with finite sets; that's how it works with ordinary infinite sets like the natural numbers. If |A| >= |B| then there exist a C and an M such that C is a subset of A and M is a 1-to-1 mapping and M(B) = C. For example, if the number of {lion, tiger, bear}=3 is greater or equal to the number of {goat, sheep}=2 then there exist C = {lion, bear} and M = {goat:bear, sheep:lion} such that {lion, bear} is a subset of {lion, tiger, bear} and {goat:bear, sheep:lion} is a 1-to-1 mapping and M({goat, sheep}) = {lion, bear}.)

So the relevance is that since the diagonal argument proves that you can't put all the transcendentals into a 1-to-1 correspondence with the rationals or with a subset of the rationals, it shows that the hypothesis that there are the same number of rationals as transcendentals, or that there are more rationals, implies a conclusion that we know isn't true.

Looks like tunnel vision to me. Infinite is infinite. One doesn't have more elements than the other, it could just depend on how we define them, which we define first (you define rationals first, because they are easier to define, instead of transcendentals which are assumed to be infinitesimally different than the "smallest" rational number, which there isn't, so it's bullshit).

I still think that you might find paradoxically there are more rationals than transcendentals, since you can find there are more transcendentals than rationals... after all we're playing with infinites.

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What I recently noticed with the transcendentals with the function I mentioned? They have cos and sin like functions, that allow one to use the transcendental constant associated with specific x and n (I'm focusing on n=2) to generate various algebraics. So perhaps there are various exponential function like series expansions. The implication, and I haven't gotten there yet, is that the transcendentals are the "Pi" type constants associated with various exponential functions.

 

[Bomb#20]

It's not only the digits that are important, but magnitude & direction trump digits (well, let's just do the positive reals for now, forget direction). Digits don't even really describe magnitude if one is playing fast and loose with bases, and do the manipulations you mention (which have nothing to do with magnitude of the numbers).

Well, the digits have something to do with the magnitude -- for any specified base, different magnitudes imply different digits. That's really all you need when you're playing counting games -- it doesn't matter which real number is bigger than which when all you're doing with them is pairing them off with members of another set.

Ok, I suppose this is where I'm lost, specifically. Why don't the naturals include every possible permutation of digits, or is there a defined cap of the set of naturals in which they sortof maybe approach infinity but don't really approach infinity because limits aren't defined well for naturals (there is no natural that contains unbounded information, even though the set of naturals has no bounds)?

There's no defined cap, but that doesn't mean you can have one single natural number that has infinitely many digits in it. When you count 1, 2, 3, ... googol, ... googolplex, ... and so forth, you will never get to the number ...321321321. That's an infinite permutation of digits. Every natural number has both a first digit and a last digit.

So all the numbers that can be specified with a finite amount of information can be put in a 1-to-1 correspondence with the natural numbers.

Show

That's not just the algebraics. The algebraic numbers are the numbers that can be specified with a finite amount of information in the language of polynomials. But there are other languages that are more expressive than polynomials. All the familiar transcendental numbers like pi and e can also be specified with a finite amount of information, using notations with summation symbols or integrals or trigonometric functions or whatever. So to put them in a 1-to-1 correspondence with the natural numbers you merely have to encode those operators as digits, the same way we encode polynomial coefficients as digits when we map algebraics to natural numbers.

The transcendental numbers you can't put in a 1-to-1 correspondence with the natural numbers are precisely the ones that can't be specified with a finite amount of information in any language whatsoever -- it's the ones whose digit sequences are random.

...random? I can see chaos, but one can generate digit sequences that go up and down at arbitrary points. In fact, one can create whatever one wants (unless one wants something truly random).

So.. what do you mean? Weierstrass's monsters are unpredictable, but... not random.

By random, I mean a number you could get by first writing "0.", and then flipping a coin and writing 1 for heads and 0 for tails and doing that again and again forever. Any number you generate with chaos or a computer's pseudo-random number generator or any algorithm at all isn't what I'm talking about. I mean something you need an infinite number of bits to specify. Weirstrass's monster is (summation[n=0 to infinity] aⁿ cos (bⁿ pi x). That's specified with a finite amount of information. All the numbers you can specify with a finite amount of information can be put into a one-to-one correspondence with the natural numbers.

Looks like tunnel vision to me. Infinite is infinite. One doesn't have more elements than the other, it could just depend on how we define them, which we define first (you define rationals first, because they are easier to define, instead of transcendentals

Well, hey, I can't make you accept the diagonal argument. If it doesn't convince you, feel free to find somebody more convincing than me to explain it -- I've given it my best shot. Likewise, if you think you can construct a one-to-one mapping between the transcendentals and the rationals, go for it.

which are assumed to be infinitesimally different than the "smallest" rational number,

No they aren't. Every transcendental is a finite nonzero amount different from every rational. There's no lower limit on how different that is -- any two reals have infinitely many rationals between them -- but the differences are always finite. Nobody uses infinitesimals for this analysis.

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What I recently noticed with the transcendentals with the function I mentioned? They have cos and sin like functions, that allow one to use the transcendental constant associated with specific x and n (I'm focusing on n=2) to generate various algebraics. So perhaps there are various exponential function like series expansions. The implication, and I haven't gotten there yet, is that the transcendentals are the "Pi" type constants associated with various exponential functions.

You can generate a lot of transcendentals that way but you're never going to hit all of them.

 

[lpetrich]

There do exist transcendental-to-algebraic functions. Functions like f(x) = 0 for all x. But there is no function that returns a separate algebraic number for each transcendental one, because there are more transcendental ones than algebraic ones. That is a consequence of the theory of infinite sets, and Georg Cantor's diagonal argument is a famous proof of that result.

 

[steve_bank]

There do exist transcendental-to-algebraic functions. Functions like f(x) = 0 for all x. But there is no function that returns a separate algebraic number for each transcendental one, because there are more transcendental ones than algebraic ones. That is a consequence of the theory of infinite sets, and Georg Cantor's diagonal argument is a famous proof of that result.

I'll ask the obvious question. f(x) = 0, 3 = 0?

 

[Kharakov]

nm....

 

[Kharakov]

There's no defined cap, but that doesn't mean you can have one single natural number that has infinitely many digits in it. When you count 1, 2, 3, ... googol, ... googolplex, ... and so forth, you will never get to the number ...321321321. That's an infinite permutation of digits. Every natural number has both a first digit and a last digit.

So they are specifically defined as coming from a point = to 1? They aren't just arbitrarily defined relative to other numbers on an infinite continuum? So if you pick a point in the continuum (which is closer to nature than the so called natural numbers), and define some distance unit as 1, you go to the right of that 1 and you have a second number at 2*distance, etc?

It seems to me that the naturals are less natural than the continuum, which is where mathematics should start from since it comes from a continuum. Maybe they should be called the artificials? Is calling them naturals a joke?

By random, I mean a number you could get by first writing "0.", and then flipping a coin and writing 1 for heads and 0 for tails and doing that again and again forever.

Well, defining something unpredictable as random doesn't make it so, it's just determined by unpredictable factors in nature. That's not random by any means. Even if some minor, undetectable spacetime geometry determines the outcome (after all, if gravity travels at c, and all the universe contributes at all times), you don't have anything random going on with your coin flip. But maybe the undetermined initial conditions are what we mean by random.... so you have that.

Any number you generate with chaos or a computer's pseudo-random number generator or any algorithm at all isn't what I'm talking about. I mean something you need an infinite number of bits to specify. Weirstrass's monster is (summation[n=0 to infinity] an cos (bn pi x). That's specified with a finite amount of information. All the numbers you can specify with a finite amount of information can be put into a one-to-one correspondence with the natural numbers.

So your claim is that there are numbers that cannot be described at all? Cantor's diagonalization can be described, and using the fast and loose "I used every rational number before I flipped the diagonal and made a new one" is... fast and loose. There isn't an "every rational number". You know as well as I that there are many "tricks" you can play with infinity, many paradoxes that can be created with logic. So what?

My IPU wants me to make a story about these indescribable numbers, so I'm going to need you to prove their existence in a finite amount of symbols. Quickly now.

Looks like tunnel vision to me. Infinite is infinite. One doesn't have more elements than the other, it could just depend on how we define them, which we define first (you define rationals first, because they are easier to define, instead of transcendentals

Well, hey, I can't make you accept the diagonal argument. If it doesn't convince you, feel free to find somebody more convincing than me to explain it -- I've given it my best shot. Likewise, if you think you can construct a one-to-one mapping between the transcendentals and the rationals, go for it.

I understand the argument- after every rational is made in an infinite list, you make one that isn't in the list by flipping the diagonals- it's just wrong. It assumes a specific order to the infinite list of rationals that does not exist. Of course, if you say "this arbitrarily picked order that looks sort of nice to some people creates a number not in the arbitrarily artificially picked order when you flip the diagonal", well, you're right.

Reimann rearrange it and get something else, since it's out of order anyway, and the whole "sweep this under the rug" posse is working hard to keep it covered up.


To me, if you make a list, and the flipped diagonal is pi? That's cool. Especially if the list is one that can easily be described. Still, it's an artificially imposed order on the continuum of rationals that does not exist. There is no "one rational than the next". Cantor's argument is broken.