Kinds of Numbers

[lpetrich]

We can start with the natural numbers, the numbers that we get from counting: 0, 1, 2, 3, ...

These are the possible cardinalities (numbers of elements) of finite sets, with 0 being that of the empty set.

But one can get them axiomatically, with  Peano axioms. It starts with axioms of equality. For a set S and a function = with arguments and return value
S = S gives a truth value (true or false)
It need not be an exact match. Consider that we accept 1/2 = 2/4 without 1/2 exactly matching 2/4.
we have:

1. Reflexivity: for all a in S, a = a
2. Symmetry: for all a,b in S: (a = b) -> (b = a)
3. Transitivity: for all a,b,c in S: (a = b and b = c) -> (a = c)
4. Closure: for all a in S, if a = b, then b is in S

Now for Peano's axioms of arithmetic. I'll start with 0, since it is easier to define addition and multiplication that way, and because it includes the empty set's cardinality. One needs a successor function S that operates on elements of the set of natural numbers N.
S(0) = 1, S(1) = 2, S(2) = 3, ...

1. First-element presence: 0 is in N
2. Closure: for all x in N, S(x) is also in N
3. Successor equality: for all x and y in N, (x = y) <-> (S(x) = S)
4. No first-element predecessors: for all x in N, S(x) = 0 is false
5. Mathematical induction: for a predicate (function that returns true/false) f, (f(0) and for all x in N, f(x) -> f(S(x))) -> for all x in N, f(x)

The last one may be expressed in words as "If something is true for the first element, and if its truth for each element implies its truth for the next element, then that something is true for all elements."

Addition can be defined axiomatically with Peano's axioms:

1. For all x in N, x + 0 = x
2. For all x, y in N, x + S = S(x+y)

Likewise for multiplication:

1. For all x in N, x*0 = 0
2. For all x,y in N, x*S = x + (x*y)

Also ordering:
For x and y in N, x <= y is defined as there being some z in N such that x + z = y
One can define <, >=, and > in terms of that, just as one can define inequality, !=, in terms of =.

One can get all these operations' familiar properties from Peano's axioms. Properties like addition and multiplication being commutative and associative, and multiplication being distributive over addition.

 

[lpetrich]

One can also get natural numbers out of set theory, as set cardinalities (numbers of elements). I'll called the cardinality function N for number. One gets arithmetic out of set cardinalities, and one can show that finite-set cardinalities obey Peano's axioms.

If sets A and B can be put into one-to-one correspondence, then N(A) = N(B) (equality)

If set A is a proper subset of set B (B has some elements that A doesn't), then N(A) <= N(B)
For finite sets, N(A) < N(B), because N(A) != N(B), while for infinite sets, N(A) can equal N(B)

Addition: for two disjoint sets A and B (their intersection is the empty set), N(A) + N(B) = N(union of A and B)

Multiplication: for two sets A and B, N(A)*N(B) = N(Cartesian product of A and B: every (a,b) with a in A and b in B)

Exponentiation: for two sets A and B, N(A)^N(B) = N(all functions f(b) with a value in A, for all b in B)

For example, the power set of set A, the set of all subsets of A, has cardinality 2^N(A). That's because each subset defines a membership function that is true for an element of A being in that subset, and false otherwise.

It can be shown by an argument much like Cantor's diagonal argument that N(power set of A) > N(A). That means that there is an infinite number of infinite-set cardinalities.

That's how one can do arithmetic on infinite-set cardinalities.

 

[lpetrich]

Now for subtraction and integers. One can define subtraction by the missing-variable approach. x = a - b is equivalent to b + x = a. One can then set x = (a,b) for convenience. For a and b natural numbers, x maps onto the natural numbers only if b <= a.

One can define equality, addition, multiplication, and ordering in parallel to the Peano-defined natural numbers. Here, (a,b) have a and b in N
(a,b) = (c,d) -- some x in N such that (a+x=c and b+x=d) or (a=c+x and b=d+x)
(a,b) + (c,d) -- (a+c, b+d)
(a,b) + (c,d) -- (a*c+b*d, a*d+b*c)
(a,b) <= (c,d) -- some x and y in N such that (a+x+y=c and b+x=d) or (a+y=c+x and b=d+x)

It can be shown that each one of these pair numbers are equivalent to one of these kinds of numbers: (a,0), (0,0), and (0,a) where a is in N and nonzero.

The (a,0) and (0,0) can be interpreted as equal to natural numbers: x = (x,0). Equal in value, though not in form. That leaves the class of numbers (0,a). This class has the property (a,0) + (0,a) = (0,0), meaning that we can define (0,a) in terms of (a,0). That is, (0,a) is "negative a" or "minus a" or "additive inverse of a" or "- a".

Historically, (Asian) Indian mathematicians recognized the mathematical legitimacy of negative numbers much earlier than Western ones did, calling positive and negative numbers asset and debt numbers. Since commercial transactions were likely familiar to Western mathematicians for several centuries, the lack of extrapolation in the West is curious.

 

[lpetrich]

Now for rational numbers. One can define them as ordered pairs of integers with appropriate properties for their arithmetic. The second number of each pair must be nonzero, because dividing by zero gives nonsense.

(a,b) = (c,d) -- (a*d = b*c)
(a,b) + (c,d) = (a*d+b*c, b*d)
(a,b) * (c,d) = (a*c, b*d)
- (a,b) = (-a, b)
Ordering: (a,b) - (c,d) -- does the result (e,f) have e,f the same signs, opposite signs, or e = 0?

Every rational number is thus equivalent to some number with the form (a,b) where b > 0 and a and b are relatively prime. Also, every one with form (a,1) can be identified with integer a.

Writing rational numbers as fractions is essentially another way of writing this definition: a/b = (a,b)

Let's see if (a/b)/(c/d) = (a*d)/(b*c).

(a/b)*(b*c) =? (c/d)*(a*d)
a*c = a*c
Correct.

Rational numbers have a property that integers lack. They are dense. That means that one can choose a rational number arbitrarily close to any other one, something not true of integers.

 

[lpetrich]

But does density mean that every non-divergent sequence of rational numbers converges on a rational number? In general, the answer is no, and the first such "irrational number" discovered was the square root of 2, discovered nearly 2500 years ago.

One can define convergence of a sequence a(k) for k = 1, 2, 3, ...., as follows. It converges to b if
For any e > 0, there is some n such that |a(k) - b| < e for all k >= n

One can avoid using a convergence target with Cauchy's definition:
For any e > 0, there is some n such that |a(k) - a(l)| < e for all k,l >= n

One can define equality, arithmetic, and ordering for such "Cauchy sequences" of rational numbers, and those sequences thus define the real numbers. If one uses Cauchy sequences of real numbers, one also gets real numbers, so the real numbers are Cauchy-closed, unlike rational numbers. Real numbers include the rational numbers, and real numbers that are not rational numbers are irrational numbers.

A simple type of Cauchy sequence is adding more decimal digits, or analogous digits in other place-system representations. Thus, sqrt(2) is the limit of 1, 1.4, 1.41, 1.414, 1.4142, ...

There are other ways of defining real numbers, like using "Dedekind cuts", but Cauchy sequences are closest to how one computes approximations of them.

 

[Juma]

Consider that we accept 1/2 = 2/4 without 1/2 exactly matching 2/4.

? Neither 1/2 nor 2/4 are natural numbers. They are rational numbers. And the rational number 1/2 matches 2/4 exactly.

 

[beero1000]

Nice. I had a similar type post a while back, in the logic subforum, I think.

Some minor comments:

Now for subtraction and integers. One can define subtraction by the missing-variable approach. x = a - b is equivalent to b + x = a. One can then set x = (a,b) for convenience. For a and b natural numbers, x maps onto the natural numbers only if b <= a.

One can define equality, addition, multiplication, and ordering in parallel to the Peano-defined natural numbers. Here, (a,b) have a and b in N
(a,b) = (c,d) -- some x in N such that (a+x=c and b+x=d) or (a=c+x and b=d+x)
(a,b) + (c,d) -- (a+c, b+d)
(a,b) + (c,d) -- (a*c+b*d, a*d+b*c)
(a,b) <= (c,d) -- some x and y in N such that (a+x+y=c and b+x=d) or (a+y=c+x and b=d+x)

Typo here, you want (a,b) * (c,d) -- (a*c+b*d, a*d+b*c)

It can be shown that each one of these pair numbers are equivalent to one of these kinds of numbers: (a,0), (0,0), and (0,a) where a is in N and nonzero.

The (a,0) and (0,0) can be interpreted as equal to natural numbers: x = (x,0). Equal in value, though not in form. That leaves the class of numbers (0,a). This class has the property (a,0) + (0,a) = (0,0), meaning that we can define (0,a) in terms of (a,0). That is, (0,a) is "negative a" or "minus a" or "additive inverse of a" or "- a".

Historically, (Asian) Indian mathematicians recognized the mathematical legitimacy of negative numbers much earlier than Western ones did, calling positive and negative numbers asset and debt numbers. Since commercial transactions were likely familiar to Western mathematicians for several centuries, the lack of extrapolation in the West is curious.

I'm not sure they didn't know about these concepts, but rather they chose to avoid confronting them - it seems for philosophical reasons. You can still do fine without zero or negative numbers - just write "nothing" and have separate credit/debit columns, etc.

Now for rational numbers. One can define them as ordered pairs of integers with appropriate properties for their arithmetic. The second number of each pair must be nonzero, because dividing by zero gives nonsense.

(a,b) = (c,d) -- (a*d = b*c)
(a,b) + (c,d) = (a*d+b*c, b*d)
(a,b) * (c,d) = (a*c, b*d)
- (a,b) = (-a, b)
Ordering: (a,b) - (c,d) -- does the result (e,f) have e,f the same signs, opposite signs, or e = 0?

Every rational number is thus equivalent to some number with the form (a,b) where b > 0 and a and b are relatively prime. Also, every one with form (a,1) can be identified with integer a.

Writing rational numbers as fractions is essentially another way of writing this definition: a/b = (a,b)

Let's see if (a/b)/(c/d) = (a*d)/(b*c).

(a/b)*(b*c) =? (c/d)*(a*d)
a*c = a*c
Correct.

Rational numbers have a property that integers lack. They are dense. That means that one can choose a rational number arbitrarily close to any other one, something not true of integers.

The property you're looking for here is "perfect" not "dense". The rationals are indeed dense in the reals, but I think you are trying to say that the rationals have no isolated points, not that the reals are the limit points of the rationals.

 

[lpetrich]

Now for algebraic numbers. We got integers by solving equations x + b = a for x given natural numbers a and b, and we got rational numbers by solving equations b*x = a for x given integers a and b.

So can we extend to higher-order polynomials? Solving the likes of a*x^n + ... + a(1)*x + a(0) = 0 for x? For rational-number coefficients, only some solutions are rational numbers. So they are a new kind of number, algebraic numbers. Only some algebraic numbers are real numbers, but in general, algebraic numbers have the form
(Algebraic number) = (real algebraic number) + (real algebraic number)*sqrt(-1)

Sqrt(-1) = i = unit of the "imaginary numbers". That number comes from mathematicians being reluctant to accept the legitimacy of that sort of number.

Such numbers are called "complex numbers", and a complex kind of number is (kind of number) + (kind of number)*i. Thus, complex integers are (integer) + (integer)*i. Complex real numbers are often called simply complex numbers. Let's see what finding polynomial-equation solutions gets us.

• Integers -> Algebraic numbers
• Rational numbers -> Algebraic numbers
• Real algebraic numbers -> Algebraic numbers
• Real numbers -> Complex (real) numbers
• Complex numbers -> Complex numbers

So both algebraic numbers and complex numbers are algebraically closed.

Real numbers that are not real algebraic numbers are transcendental numbers. Numbers like e and pi.

 

[lpetrich]

Next up are computable numbers. These are numbers that can be approximated using a finite-sized algorithm in a finite number of steps.

Computable real numbers are a superset of real algebraic numbers, and they include well-known transcendental numbers like e and pi, but they don't include all real numbers.

One kind of real number that they don't include is  Chaitin's constant, a number whose binary trailing digits are whether Turing machines can or cannot halt. Since the halting problem is undecidable in general, Chaitin's constant is not computable.

Another kind of uncomputable number is a  Specker sequence, a Cauchy sequence of computable numbers with a limit that is not computable.

-

After computable numbers are definable numbers, those that satisfy finite-sized conditions. All computable numbers are definable, and also some non-computable ones, like Chaitin's constant and Specker-sequence limits.

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There is a very interesting circumstance about how many of all these numbers there are. I'll leave out the various sorts of complex numbers, since they have the same cardinality as the numbers used to construct them.

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

However, real numbers have cardinality C, a number greater than aleph-0. Meaning that there are more real numbers than those other sorts of numbers, and that most real numbers are impossible to specify with something finite-sized.

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There appears to be no way of deciding whether there are any varieties of infinity between aleph-0 and C. That there are none is the "continuum hypothesis", and both it and its negation are consistent with the commonly-used Zermelo-Fraenkel axioms of set theory with the Axiom of Choice.

With power sets, it's easier. One starts with a countably-infinite set, one with cardinality aleph-0 = beth-0. Find the power set of that set, and repeat that operation. The cardinalities of those sets are beth-1, beth-2, etc.

The real numbers have cardinality C = beth-1, and all real-valued functions of real numbers form a set with cardinality beth-2. However, all *continuous* real-valued functions of real numbers form a set with cardinality beth-1, like the real numbers themselves. I don't know of any set with cardinality beth-3 that has a reasonably simple interpretation.

 

[lpetrich]

Nice. I had a similar type post a while back, in the logic subforum, I think.
Typo here, you want (a,b) * (c,d) -- (a*c+b*d, a*d+b*c)

I concede.

I'm not sure they didn't know about these concepts, but rather they chose to avoid confronting them - it seems for philosophical reasons. You can still do fine without zero or negative numbers - just write "nothing" and have separate credit/debit columns, etc.

Something like doing (a,0) and (0,0) and (0,a).

The property you're looking for here is "perfect" not "dense". The rationals are indeed dense in the reals, but I think you are trying to say that the rationals have no isolated points, not that the reals are the limit points of the rationals.

That is indeed correct -- no isolated ones.

- - - Updated - - -

Consider that we accept 1/2 = 2/4 without 1/2 exactly matching 2/4.

? Neither 1/2 nor 2/4 are natural numbers. They are rational numbers. And the rational number 1/2 matches 2/4 exactly.

They have equal values but not equal forms.

 

[Juma]

I concede.

I'm not sure they didn't know about these concepts, but rather they chose to avoid confronting them - it seems for philosophical reasons. You can still do fine without zero or negative numbers - just write "nothing" and have separate credit/debit columns, etc.

Something like doing (a,0) and (0,0) and (0,a).

The property you're looking for here is "perfect" not "dense". The rationals are indeed dense in the reals, but I think you are trying to say that the rationals have no isolated points, not that the reals are the limit points of the rationals.

That is indeed correct -- no isolated ones.

- - - Updated - - -

Consider that we accept 1/2 = 2/4 without 1/2 exactly matching 2/4.

? Neither 1/2 nor 2/4 are natural numbers. They are rational numbers. And the rational number 1/2 matches 2/4 exactly.

They have equal values but not equal forms.

Then you should have said that. The equality sign is for the value only. Thus it is an total match between the value if 1/2 and the value of 2/4.

 

[Speakpigeon]

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

Progress in the sciences, logic and mathematics has involved re-defining common notions. We're just no longer talking about the same things. No wonder then.
EB

 

[Juma]

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

EB

Take two sets of more than 1E100 apples and count them. Tell me the answer once you're done.

 

[lpetrich]

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

What do you mean by that?

One-to-one correspondence is how one tests whether sets have the same cardinality (number of elements). What is counterintuitive about infinite sets is that they can be placed in 121C with some proper subsets of them, subsets lacking some of the original set's elements. Mathematician David Hilbert had explained this oddity with  Hilbert's paradox of the Grand Hotel.

121C is sometimes called a "bijection", as in  Bijection, injection and surjection. For a function f that maps from elements of set A to elements of set B:

f is an injection (one-to-one) -- for all a,b in A, f(a) = f(b) -> a = b
f is a surjection (onto) -- for all b in B, there is an a in A such that f(a) = b
f is a bijection (a 121C) -- it is both an injection and a surjection

-

Let's get into 121C's involving infinite sets. First, 0-based vs. 1-based natural numbers.

0 -- 1, 1-- 2, 2 -- 3, 3 -- 4, 4 -- 5, ...

In general, n <-> (n+1), thus establishing the 121C.

Now for natural numbers and integers.

0 -- 0, 1 -- -1, 2 -- 1, 3 -- -2, 4 -- 2, 5 -- -3, 6 -- 3, ...

In general, 2n-1 <-> -n and 2n <-> n.

Now for ordered pairs. This is suitable for constructing a superset of the rational numbers.

0 -- (0,0), 1 -- (1,0), 2 -- (0,1), 3 -- (2,0), 4 -- (1,1), 5 -- (0,2), ...

In general, n(n+1)/2 + k <-> (n-k,k) where 0 <= k <= n

One can extend this argument to arbitrary finite-length ordered multiplets, thus handling the algebraic numbers, the computable numbers, the definable numbers, and any other set whose elements are defined with them.

 

[beero1000]

I concede.


Something like doing (a,0) and (0,0) and (0,a).

The property you're looking for here is "perfect" not "dense". The rationals are indeed dense in the reals, but I think you are trying to say that the rationals have no isolated points, not that the reals are the limit points of the rationals.

That is indeed correct -- no isolated ones.

- - - Updated - - -

Consider that we accept 1/2 = 2/4 without 1/2 exactly matching 2/4.

? Neither 1/2 nor 2/4 are natural numbers. They are rational numbers. And the rational number 1/2 matches 2/4 exactly.

They have equal values but not equal forms.

Then you should have said that. The equality sign is for the value only. Thus it is an total match between the value if 1/2 and the value of 2/4.

That is what he said.

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

Progress in the sciences, logic and mathematics has involved re-defining common notions. We're just no longer talking about the same things. No wonder then.
EB

What is the ordinary sense of "have the same number of elements"? Be precise...

 

[Bomb#20]

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

Progress in the sciences, logic and mathematics has involved re-defining common notions. We're just no longer talking about the same things. No wonder then.
EB

What is the ordinary sense of "have the same number of elements"? Be precise...

I think he already told you: start counting them and then come back and tell me the answer once you're done. So the answer to "Do the natural numbers and the rational numbers have the same number of elements?" is exactly as undefined as "Zero goes into one how many times?". There has to be something deeply deranged about mathematicians, to call an infinite set "countable".

 

[beero1000]

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

Progress in the sciences, logic and mathematics has involved re-defining common notions. We're just no longer talking about the same things. No wonder then.
EB

What is the ordinary sense of "have the same number of elements"? Be precise...

I think he already told you: start counting them and then come back and tell me the answer once you're done. So the answer to "Do the natural numbers and the rational numbers have the same number of elements?" is exactly as undefined as "Zero goes into one how many times?". There has to be something deeply deranged about mathematicians, to call an infinite set "countable".

The entire point of mathematics is to get correct answers while being as lazy as possible. Don't want to repeatedly add? Multiply. Don't want to repeatedly multiply? Exponentiate. I don't feel like counting every single element, so here's a bijection that will do it for me - oh, and it extends to more situations that the one I invented it for? Awesome, less work for me.

Why anyone would insist on doing it the long way is beyond me.

 

[Speakpigeon]

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

What do you mean by that?

Nothing too difficult to understand. Just compare counting a reasonable number of things (and how that necessarily informs what we means by "counting") with the abstract operation of bijecting one set to another, none of which has been effectively counted in the original sense. These are two different processes. Whether we choose to accept the second as an acceptable extension of the first is up to each of us for himself but they are effectively different. I hope you understand that.

One-to-one correspondence is how one tests whether sets have the same cardinality (number of elements). What is counterintuitive about infinite sets is that they can be placed in 121C with some proper subsets of them, subsets lacking some of the original set's elements. Mathematician David Hilbert had explained this oddity with  Hilbert's paradox of the Grand Hotel.

121C is sometimes called a "bijection", as in  Bijection, injection and surjection. For a function f that maps from elements of set A to elements of set B:

f is an injection (one-to-one) -- for all a,b in A, f(a) = f(b) -> a = b
f is a surjection (onto) -- for all b in B, there is an a in A such that f(a) = b
f is a bijection (a 121C) -- it is both an injection and a surjection

-

Let's get into 121C's involving infinite sets. First, 0-based vs. 1-based natural numbers.

0 -- 1, 1-- 2, 2 -- 3, 3 -- 4, 4 -- 5, ...

In general, n <-> (n+1), thus establishing the 121C.

Now for natural numbers and integers.

0 -- 0, 1 -- -1, 2 -- 1, 3 -- -2, 4 -- 2, 5 -- -3, 6 -- 3, ...

In general, 2n-1 <-> -n and 2n <-> n.

Now for ordered pairs. This is suitable for constructing a superset of the rational numbers.

0 -- (0,0), 1 -- (1,0), 2 -- (0,1), 3 -- (2,0), 4 -- (1,1), 5 -- (0,2), ...

In general, n(n+1)/2 + k <-> (n-k,k) where 0 <= k <= n

One can extend this argument to arbitrary finite-length ordered multiplets, thus handling the algebraic numbers, the computable numbers, the definable numbers, and any other set whose elements are defined with them.

Thanks, I'm good. I understand all that I think.
EB

 

[Speakpigeon]

Natural numbers, integers, rational numbers, real algebraic numbers, computable real numbers, and definable real numbers all have cardinality aleph-0, the countable cardinality. That means that there is the same number of all of them, however counterintuitive it may seem.

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

Progress in the sciences, logic and mathematics has involved re-defining common notions. We're just no longer talking about the same things. No wonder then.
EB

What is the ordinary sense of "have the same number of elements"? Be precise...

You know, counting on your fingers, that sort of things.
EB

 

[Speakpigeon]

It's counterintuitive because it's not true in the ordinary sense of "have the same number of elements".

Start counting them and then come back and tell me the answer once you're done.

EB

Take two sets of more than 1E100 apples and count them. Tell me the answer once you're done.

There again you are trying to make me waste my time. You're caught red-handed.
EB