I "Define" ℕ from ℤ or ℝ from ℂ, etc...

RaamGeneral

We can easily define, for instance, complex numbers starting from (as a couple of) real number: z = (a,b) ∈ ℂ with a,b ∈ ℝ and the property (0,1)^2 = (-1,0)
We can define integers in a similar manner starting from natural numbers: https://en.wikipedia.org/wiki/Integer#Construction
And rational from integers.

We agree that integers are generalization of natural numbers, as complex are generalization of reals.
So, shouldn't it be possible and easier to do the opposite? (Shouldn't it be easier to define a subset?)

I mean, it should be possible to define real numbers starting from complex numbers, but I can't think of any way. Sure, I can say (a,0) are real numbers, but in this way I already know and I'm already assuming that a,b∈ℝ in (a,b).

I hope my question is clear. Thanks.

fresh_42

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2018 Award
Please define a complex number without the usage of reals. If you want to go down, then you have to solve this.

fbs7

Please define a complex number without the usage of reals. If you want to go down, then you have to solve this.
Hmm... can't you define

z ∈ ℂ iff z = $a + bi$,

$a = \lim_{n \rightarrow \infty} \sum_{u=-n}^n a_u 10^{u}$
$b = \lim_{n \rightarrow \infty} \sum_{v=-n}^n b_v 10^{v}$

i2 = -1

$a_u, b_v$∈ ℕ
∃N such that $a_u, b_v$ = 0 for all u>N, v>N (otherwise the series will diverge)

Btw, I'm just poking my feet in turbulent waters, I'm no mathematician

fbs7

Hmm.. a bit of re-reading of the OP... if my previous post is somewhat correct (and most probably what I wrote is very incorrect), that would allow one to define ℝ in terms of ℂ by setting $b_u = 0$, but it uses ℕ, ℤ and ℚ for that... so it doesn't really meet the OP's expectations

So, how to define ℂ without using ℕ... holy choochoo, that sounds very difficult!!

fresh_42

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2018 Award
So, how to define ℂ without using ℕ... holy choochoo, that sounds very difficult!!
It had to be something like this: Let $\mathbb{C}$ be the minimal field, which is the algebraic and topological closure of an Archimedean ordered prime field of characteristic zero. However, this simply describes the way bottom up by other means: Take $\mathbb{Q}$ and complete it! However, it can be seen, that in any way top down, there have to be basic ingredients which fix the characteristic and value function, i.e. the rationals. Ergo, there is no way top down.

jbriggs444

How about if we start with the field of complex numbers as a given. And we identify two complex numbers, "0" and "1" thusly:

0 is the unique complex number such that 0*0 = 0 and 0+0 = 0.
1 is the unique complex number distinct from 0 such that 1*1 = 1.

Now we have a basis on which to construct a subset of the complex numbers which is isomorphic to the naturals and a subset which is isomorphic to the integers.
We could easily then find a subset which is isomorphic to the rationals.
It should not be too much trouble to carefully mutter some incantations about limits and find a subset which is isomorphic to the reals.

Is that the sort of thing that you are after?

FactChecker

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2018 Award
Certainly it is easier to define the subset, $R$ of $C$ if we ignore that the definition of $C$ is difficult. $R$ is simply the subset where the imaginary part is zero. Likewise, the natural numbers are simply the positive integers. The arithmetic operations remain valid, with restrictions. For instance, the natural numbers are not closed under subtraction and the operation of square root would need to be restricted in $R$.

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fresh_42

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2018 Award
How about if we start with the field of complex numbers as a given. And we identify two complex numbers, "0" and "1" thusly:

0 is the unique complex number such that 0*0 = 0 and 0+0 = 0.
1 is the unique complex number distinct from 0 such that 1*1 = 1.

Now we have a basis on which to construct a subset of the complex numbers which is isomorphic to the naturals and a subset which is isomorphic to the integers.
We could easily then find a subset which is isomorphic to the rationals.
It should not be too much trouble to carefully mutter some incantations about limits and find a subset which is isomorphic to the reals.

Is that the sort of thing that you are after?
This only defines $\mathbb{Z}[ i ]$ or $\mathbb{Q}[ i ]$ if divisions are implicated.

fbs7

It had to be something like this: Let $\mathbb{C}$ be the minimal field, which is the algebraic and topological closure of an Archimedean ordered prime field of characteristic zero. However, this simply describes the way bottom up by other means: Take $\mathbb{Q}$ and complete it! However, it can be seen, that in any way top down, there have to be basic ingredients which fix the characteristic and value function, i.e. the rationals. Ergo, there is no way top down.
Howly choo-choo.... CHOOOOO!! I have no idea what that means, other than it sounds VERY complicated! :)

How about we make an attempt in binary (actually ternary), to get over the complications of defining ℕ; once again, this is a non-mathematician speaking, so feel free to LOL-a-lot, ROFL-much and LUMBAOL-totally:

Say I define 3 digits
0 = {}, 1 = {{}}, i = {{{}}}
xn ∈ { 0, 1, i }

Now I define ordered pairs
a = [ 0 a ]
[ a b ]
[ a b c ] = [ a [ b c ]] = [ [ a b ] c ]
...etc...

and I define the (binary) addition rules for the pairs (notice 1+i is irreducible)
0+x = x, 1+0 = 1
x+y = y+x
x+y+z = (x+y)+z = x+(y+z)
[ x 1 ] + [ y 1 ] = [ x+y 0 ]
[ x i ] + [ y i ] = [ x+y+i 0 ]

then define subtraction and negatives
x=y-z iff y=x+z
-x = 0-x

then define multiplication
0*x = 0, 1*x = x, i*i = -1
x*y = y*x
x*y*z = (x*y)*z = x*(y*z)

then I define my complex number as
c = limn->∞ [ cn ... c0 ... c-n ]
where ∃m such that cn = 0 if n > m

I didn't have to resort to ℕ, ℤ, ℚ or ℝ; everything here is a definition, for example I don't care what's the meaning of -3 as that's defined as 0-3, and the only interesting thing about that is that 3+(-3) = 3+0-3 = (3+0)-3 = 3-3 which I can prove = 0, given that 3+0 = 3, therefore 3+(-3) = 0

Now, at the very bottom of this there's this very suspicious "lim" thingie... which is undefined... but I have no idea how to define an "infinite set of"... that sounds extremely incredibly arcane and complicated!

fbs7

Trying to get out of this limn->∞ thingie, as that definitely looks like a dependency on ℕ. Problem is how to try and define an infinite ordered set, without resorting to ℕ. Trying to use {}, {{}}, etc... as substitute for ℕ... let's see!!

A is a stand-in for ℕ
B is an ordered sequence of digits 0, 1, i
{} is a stand-in for index 0
{x} is a stand-in for x+1
#(X) is a stand-in for cardinality of the sequence

// Define indexes
{} ∈ A
{n} ∈ A for all n ∈ A
n < m iff ∃k such that {n} = k and ( k < m or k = m )

// Define infinite binary sequences recursively that represent ||z|| < 1, like 0.10i011ii01 = 0.1000110001 + (0.0010001100)*i
x, y, z ∈ { 0, 1, i }
[ x{} ] ∈ ℂ
X, Y, Z ∈ ℂ
#(X) = n iff xn ∈ X and not exists m such that n < m and xm ∈ X
Xn, Ym, Zl are shorthand for X, Y, Z ∈ B and #(X) = n, #(Y) = m, #(Z) = l
[ Xn y{n} ] ∈ ℂ
[ Xn 0{n} ] = [ Xn ]

// Define 1-digit additions and subtractions, and sequence additions (subtractions are the same, I'm not listing to save space)
0+x = x, x+y = y+x, x+y+z = (x+y)+z = x+(y+z), ∃ 1+i
x-y=z iff x=y+z
Xn = Yn + Zn iff for any m < n
xm = ym + zm or
xm = ym + zm + 1 & x{m} = y{m} + zm - 1 - 1

// Define 1-digit multiplications (trivial) and sequence multiplications (not sure how to define in small expressions, someone smarter can probably do that)
...etc...

That should be it, ℂ defined in terms of recursive sequences of 0, 1 and i - although it only defined for ||z|| < 1; it would require some more writing to define for ||z|| >= 1, but I guess it's the same process. No ℕ, ℤ, ℚ or ℝ.

fresh_42

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2018 Award
That should be it, ℂ defined in terms of recursive sequences of 0, 1 and i - although it only defined for ||z|| < 1; it would require some more writing to define for ||z|| >= 1, but I guess it's the same process. No ℕ, ℤ, ℚ or ℝ.
Are you sure you didn't just constructed $\mathbb{Q}$, since I don't see any Cauchy sequences, Dedekind cuts nor any other topological closure, metric or even an Archimedian ordering and value function! And why is $1+1+\ldots 1 \neq 0$ in your construction?

Again, you always focus on $\mathbb{F}[T]/\langle T^2+1 \rangle$ with a field $\mathbb{F}$, however you fail to mention its characteristic, as well as it it isn't complete. $i$ isn't the problem here, that is just a root, $\mathbb{F}$ is.

fbs7

Hmm... it's a conundrun in my mind, indeed. My way of thinking was based in how I would define that in a computer program (I'm a programmer).

My original thought was to try to express c ∈ ℂ as an infinite series of something, and hope that one can create an infinite series without resorting to ℕ. But it's very difficult, oh no! :(

fbs7

I know that for example π = 3.1415926535897... etc... So, I suspect that whatever x ∈ ℝ, then x can be defined as an infinite ordered series of digits, like 3 + [ 1 4 1 5 9 2 6 5 3 5 8 9 7 .... etc... ]. Is that a valid definition of a real number (that's how I learned in high school, that ℝ means infinite series of digits without a repetition to them), or do we need to use the Cauchy or Dedekind stuffy (which I have no idea what they are)?

If that's a valid way to define a real number, then I suspect I'm bringing ℕ and ℚ anyway, because that assumes an "ordered" sequence, so it assumes ℕ, and it uses divisions by 10, therefore it also assumes ℚ.

So the idea of trying to escape using ℚ to define ℝ, by using series of digits is a fool's errand, right? There's no way to avoid using ℕ to define ℚ, no way to avoid using ℚ to define ℝ, and no way to avoid using ℝ to define ℂ?

jbriggs444

I know that for example π = 3.1415926535897... etc... So, I suspect that whatever x ∈ ℝ, then x can be defined as an infinite ordered series of digits, like 3 + [ 1 4 1 5 9 2 6 5 3 5 8 9 7 .... etc... ]. Is that a valid definition of a real number (that's how I learned in high school, that ℝ means infinite series of digits without a repetition to them), or do we need to use the Cauchy or Dedekind stuffy (which I have no idea what they are)?
A course in Real Analysis is something you could make good use of.

In order to flesh out such a definition for pi, you would need something stronger than "etc". You would need an actual prescription for any given digit. [Possibly not algorithmic].

You will have a problem. The number of possible finite prescriptions is, at most, countably infinite. The number of infinite digit strings is larger than that. Google for Cantors proof of uncountability.

You could also Google "Dedekind cut" or "Cauchy sequence". Those are two tools used in classic constructions of the set of Real Numbers. Let me try to take you briefly through the Dedekind cut approach.

Suppose that we have the field of rational numbers already constructed complete with operations for addition, subtraction, multiplication and division. Now you define "cuts" of the rational numbers. A "cut" is a pair of disjoint, non-empty subsets that split the rational numbers into two pieces. The upper cut contains all rationals greater than the dividing line. The lower cut contains all rationals lower than the dividing line. All rationals are a member of one side or the other.

If the split point is exactly at a rational number, we will arbitrarily choose to put that number in the upper cut. The split point might not be at a rational number. We might, for instance, consider the upper cut of all positive rational numbers whose square is greater than 2 and a lower cut of all remaining rationals. The dividing line would be at the "square root of 2" which is, of course, not rational.

The assertion is that the complete set of all Dedekind upper cuts of the rationals amounts to the set of all real numbers. More technically, that the result is a complete ordered archimedean field.

The remainder of the construction is the busy work of dotting the i's, crossing the t's. Showing how addition, subtraction, multiplication and division are determined and how they form a complete ordered archimedean field.

Note that to construct the reals, we did not have to exhibit a construction for each individual real. We had to demonstrate the existence of a set and of some arithmetic operations on its members.

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fbs7

Oh, wow! That's most impressively described!

I'm always amazed on how kindly people offer help here, even for the most absurd-sounding questions! That you all so much for that!

What an impressive thought is this Dedekind-cut thingie! So you take a line, that has an infinite number of numbers on it... then you use your rationals as a selection criteria in these cuts... what means you're selecting an infinite number of numbers based on an infinite number of criterias based on infinite number of rationals... then we just state that this infinite number of numbers is the reals!!! Holy moses, that seems suspiciously like asking the question "if you have an infinite set and then you remove an infinite number of numbers from that set, then what remains in the set?"

What's the Latex for /mind-is-blown-away ?

jbriggs444

What an impressive thought is this Dedekind-cut thingie! So you take a line, that has an infinite number of numbers on it... then you use your rationals as a selection criteria in these cuts... what means you're selecting an infinite number of numbers based on an infinite number of criterias based on infinite number of rationals
I am going to object mildly to this characterization. The rationals are not quite the selection criteria. They are more like the markers that you separate according to some criterion. (Also note the Latin pluralization rules).

The mind-blowing piece is that you have this array with only a countably infinite number of markers on it, all laid out in order. (A dense order, but still an order). The number of places you can metaphorically place a razor's edge to divide the set of markers in two is vastly greater than the number of markers in the set you are slicing between.

The behavior of the infinite has a few ways of surprising you.

fresh_42

Mentor
2018 Award
The behavior of the infinite has a few ways of surprising you.
That is why I find the approach by Cauchy sequences more intuitive than via the Dedekind cuts.

fbs7

I am going to object mildly to this characterization. The rationals are not quite the selection criteria. They are more like the markers that you separate according to some criterion. (Also note the Latin pluralization rules).

The mind-blowing piece is that you have this array with only a countably infinite number of markers on it, all laid out in order. (A dense order, but still an order). The number of places you can metaphorically place a razor's edge to divide the set of markers in two is vastly greater than the number of markers in the set you are slicing between.

The behavior of the infinite has a few ways of surprising you.
Oh.. thank you for the correction.

""Define" ℕ from ℤ or ℝ from ℂ, etc..."

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