## defining a number as "real"

**Moderators:** gmalivuk, Moderators General, Prelates

- LE4dGOLEM
- is unique......wait, no!!!!
**Posts:**5972**Joined:**Thu Oct 12, 2006 7:10 pm UTC**Location:**:uoıʇɐɔol

### defining a number as "real"

I had a semi-interesting discussion last night, in which it was claimed that 1 was not a number. I replied that it is a real number. Plied for a definition, I stated that a "real number" is any number that can be used in a mathematical equation that makes sense, for example, one plus one is two, because having a quantity of one thing and another quantity of one of the same thing means that you have a quantity of two of that thing. Thus, infinity is not a "proper" number, because it cannot be used in a "proper" maths equation - it is not subject to the normal laws of + - / and *. The retort followed that pi (or any other trancendental (sp?) number) cannot be a real number as they continue on infinitely, and as infinity is not real, any number that uses an aspect of infinity cannot be real.

So how do you define a "real" number?

So how do you define a "real" number?

Une See Fights - crayon super-ish hero webcomic!

doogly wrote:It would just be much better if it were not shitty.

- bitwiseshiftleft
**Posts:**295**Joined:**Tue Jan 09, 2007 9:07 am UTC**Location:**Stanford-
**Contact:**

I stated that a "real number" is any number that can be used in a mathematical equation that makes sense

Well, that doesn't quite work, since complex numbers can be easily used in equations that make sense, not to mention quaternions, surreal numbers, matrices, etc.

To define R (the set of reals), we first of all assume the rationals are all members of R. This ensures R is non-empty.

Secondly, we have something called the Least Upper Bound axiom. This states that every subset S of R that is bounded above has a least upper bound in R. Or, to put it another way, if every member of S is less than some real number b (an upper bound), then there is some real number s such that every member of S is smaller than s (s is an upper bound of S), and for every number x that is smaller than s, there is a member of S that is larger than x (there is no upper bound of S smaller than s - i.e. s is a least upper bound). s is generally called the supremum of S.

For example, its fairly easy to show that there is no number in the rationals that squares to 2. However, by looking at the set of numbers that square to less than 2 (which exists in the rationals), we can use the least upper bound axiom to show that the supremum of this set exists and squares to 2.

### Re: defining a number as "real"

LE4dGOLEM wrote:Thus, infinity is not a "proper" number, because it cannot be used in a "proper" maths equation - it is not subject to the normal laws of + - / and *.

Ya, this isn't quite true.....but that's ok. In set theory, addition and multiplication of infinite cardinals is an extension of addition and multiplication of finite cardinals (i.e. - the naturals).

el sjaako wrote:A defenition that is close to or equivilent to what token said is that every real number is a limit of some series of rational numbers. For example, pi is the limit of a series that starts with 3, then 31/10, then 314/100, then 3141/1000, and so on (aka 3,3.1,3.14,3.141).

I think what you are thinking of is the Fundamental Axiom of Analysis - that every non-decreasing sequence with an upper bound has a real limit - which is, as you say, equivalent to the Least Upper Bound axiom.

Token wrote:I think what you are thinking of is the Fundamental Axiom of Analysis - that every non-decreasing sequence with an upper bound has a real limit - which is, as you say, equivalent to the Least Upper Bound axiom.

I don't know if I'm referring to that. I went to this university thing, and it was all about real nubers. The guy explained (and I'm guessing at the translation of dutch terms here) that the collection of real numbers is the collection of all equivalence-classes of Cauchy series. I just don't think the original poster would be very happy with that anwser without a day or two of explanations to understand what it actually means.

Looking more closely at what you wrote there that is basicly what I'm saying, but I didn't know it was an axiom, much less a fundamental one.

- EradicateIV
**Posts:**361**Joined:**Mon Mar 19, 2007 7:33 pm UTC**Location:**Brownsville, PA-
**Contact:**

I just finished my intro to theory class, we learned there are 7 axioms/property's that describe real numbers.

Your 5 axioms of alegebra (Working within fields)

(communitivity, associativity, inverse, identity, and distributive)

Your 2 order properties (Making these fields ordered) which knocks out imaginary and mod p fields.

I. An element of a field must either be an element of p ("positive numbers"), the inverse of that element is an element of p, or that number is 0

II. If you have two elements that belong to p ("Positive numbers") then x+y is an element of p and so is x*y.

then atleast, to seperate rationals from reals.

Your completeness property (see also that lowest upper bound axiom above).

Use a set of sqrt(2) decimal representations, converges to sqrt(2), but oh no, sqrt(2) is not rational.

Yay Reals!

This proves I deserved my A+ in there

Your 5 axioms of alegebra (Working within fields)

(communitivity, associativity, inverse, identity, and distributive)

Your 2 order properties (Making these fields ordered) which knocks out imaginary and mod p fields.

I. An element of a field must either be an element of p ("positive numbers"), the inverse of that element is an element of p, or that number is 0

II. If you have two elements that belong to p ("Positive numbers") then x+y is an element of p and so is x*y.

then atleast, to seperate rationals from reals.

Your completeness property (see also that lowest upper bound axiom above).

Use a set of sqrt(2) decimal representations, converges to sqrt(2), but oh no, sqrt(2) is not rational.

Yay Reals!

This proves I deserved my A+ in there

1010011010

Our truth is only as good as our assumptions.

Our truth is only as good as our assumptions.

Looking more closely at what you wrote there that is basicly what I'm saying, but I didn't know it was an axiom, much less a fundamental one.

Fundamental in the sense that without it (or an equivalent axiom), pretty much all of Analysis after about the second lecture of the most basic course would go out the window.

Token wrote:Looking more closely at what you wrote there that is basicly what I'm saying, but I didn't know it was an axiom, much less a fundamental one.

Fundamental in the sense that without it (or an equivalent axiom), pretty much all of Analysis after about the second lecture of the most basic course would go out the window.

It's only necessary if you define the reals as a (the) complete ordered field (which I personally think defeats the main point of Analysis, which is mathematical rigor.) If you construct them as Dedekind cuts or equivalence classes of Cauchy sequences then the least upper bound property (and equivalents) aren't axioms but theorems.

-M

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

You can't define the reals as the only complete ordered field, because it's not. It is, however, the only complete archimedean ordered field, so you could define it that way. I like defining that way, and using dedekind cuts to prove that such an object actually exists. Others like to claim that the real numbers are dedekind cuts, which I like less because for me, a real number is intuitively just a real number, and is not a subset of the rationals. That's just crazy talk!

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

"With math, all things are possible." —Rebecca Watson

tetoru wrote:Token wrote:Looking more closely at what you wrote there that is basicly what I'm saying, but I didn't know it was an axiom, much less a fundamental one.

Fundamental in the sense that without it (or an equivalent axiom), pretty much all of Analysis after about the second lecture of the most basic course would go out the window.

It's only necessary if you define the reals as a (the) complete ordered field (which I personally think defeats the main point of Analysis, which is mathematical rigor.) If you construct them as Dedekind cuts or equivalence classes of Cauchy sequences then the least upper bound property (and equivalents) aren't axioms but theorems.

I never said that it was necessary, merely that it is fundamental. Whether you take it as axiom or prove it otherwise, it's still damn important.

- Thunderbird4!
- It's ketchup, I swear!
**Posts:**307**Joined:**Sun May 06, 2007 3:40 am UTC**Location:**Out of my mind-
**Contact:**

A simple definition of a real number is one that can be found on an infinitely long number line. Slightly less simple it can be given by an infinite decimal representation. That covers your 42, -23/129, pi etc.

The imaginary numbers can't be found on that number line because absolutely no point on the line is that number. Take i for example. It is the (sqrt)-1. Now as I'm sure you know there is no real number (number found on your number line) that when multiplied by itself will equal -1. (Hell if you want to be technical there is no number, just a fancy letter to represent the value people created to represent this "number")

The best real-world example I can give is that you can have actual parts of lets say an apple that are real numbers (the missing pieces can be negative for this) you can be missing 1 piece which would be a -1, you could have 4 pieces left, thats 4, you could have 22/7 pieces left, but none the less you still have some or an absence of some. However if I offered to give you i apples it wouldn't be possible. It isn't possible to get i parts of a real object like an apple.

The imaginary numbers can't be found on that number line because absolutely no point on the line is that number. Take i for example. It is the (sqrt)-1. Now as I'm sure you know there is no real number (number found on your number line) that when multiplied by itself will equal -1. (Hell if you want to be technical there is no number, just a fancy letter to represent the value people created to represent this "number")

The best real-world example I can give is that you can have actual parts of lets say an apple that are real numbers (the missing pieces can be negative for this) you can be missing 1 piece which would be a -1, you could have 4 pieces left, thats 4, you could have 22/7 pieces left, but none the less you still have some or an absence of some. However if I offered to give you i apples it wouldn't be possible. It isn't possible to get i parts of a real object like an apple.

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

Thunderbird, the problem with your definition is that it leads to the whole .999...=?=1 problem, and to people thinking that every point on the number line has another point "right next to it". I've heard stories of a professor proving that between any two distinct real numbers there is another, and getting the objection, "But what if the numbers are right next to each other?" (The professor responded by saying, "Well in that case..." and then doing the proof again.)

Intuitive definitions are nice as far as they go, but they lead to problems when people think that if there intuition comes into conflict with rigor, their intuition is right, which unfortunately is all too common.

Intuitive definitions are nice as far as they go, but they lead to problems when people think that if there intuition comes into conflict with rigor, their intuition is right, which unfortunately is all too common.

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

"With math, all things are possible." —Rebecca Watson

- Thunderbird4!
- It's ketchup, I swear!
**Posts:**307**Joined:**Sun May 06, 2007 3:40 am UTC**Location:**Out of my mind-
**Contact:**

- Yakk
- Poster with most posts but no title.
**Posts:**11087**Joined:**Sat Jan 27, 2007 7:27 pm UTC**Location:**E pur si muove

The problem with using axioms to define the real numbers is you hit Godel right in the face.

You cannot restrict the models your axioms describe to a single thing with a single set of properties. There are multiple different models which are described by any set of axioms of the real numbers that have different properties.

You can, however, restrict the type of thing that is a real number line using axioms. :)

You cannot restrict the models your axioms describe to a single thing with a single set of properties. There are multiple different models which are described by any set of axioms of the real numbers that have different properties.

You can, however, restrict the type of thing that is a real number line using axioms. :)

Yakk wrote:The problem with using axioms to define the real numbers is you hit Godel right in the face.

You cannot restrict the models your axioms describe to a single thing with a single set of properties. There are multiple different models which are described by any set of axioms of the real numbers that have different properties.

You can, however, restrict the type of thing that is a real number line using axioms.

Sorta. You can't categorically define the reals using first-order axioms, but you can using second-order logic. And even if you construct the reals in first-order ZFC, well, by Lowenheim-Skolem there are still countable models of ZFC, so this doesn't help you any: you can still get a countable uncountable set of reals out of the process. (Think about that for a while if you really want your brain to bleed.)

-M

Formal definition:

http://en.wikipedia.org/wiki/Real_number

Layman's definition:

Any sequence of digits, with potentially infinite digits after the '.' but a finite amount before it, is a real number. Why the hell would they think 1 isn't a number? What number do you subtract from three to get two?

http://en.wikipedia.org/wiki/Real_number

Layman's definition:

Any sequence of digits, with potentially infinite digits after the '.' but a finite amount before it, is a real number. Why the hell would they think 1 isn't a number? What number do you subtract from three to get two?

Don't pay attention to this signature, it's contradictory.

- Thunderbird4!
- It's ketchup, I swear!
**Posts:**307**Joined:**Sun May 06, 2007 3:40 am UTC**Location:**Out of my mind-
**Contact:**

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

Alky wrote:Formal definition:

http://en.wikipedia.org/wiki/Real_number

Layman's definition:

Any sequence of digits, with potentially infinite digits after the '.' but a finite amount before it, is a real number. Why the hell would they think 1 isn't a number? What number do you subtract from three to get two?

So 0.999... and 1 are different as real numbers because they are different as sequences of digits?

I'm looking forward to the day when the SNES emulator on my computer works by emulating the elementary particles in an actual, physical box with Nintendo stamped on the side.

"With math, all things are possible." —Rebecca Watson

"With math, all things are possible." —Rebecca Watson

He could have said an "actual" number to mean more or less the same, I guess. As opposed to "nonsensical", maybe.

His wording is not very clear to me, but doesn't outright exclude i, for instance. Because he talks about "numbers" (And not "element of an algebraically closed field" or whatever) I'd be tempted to say the complex numbers. (which is, for me, precisely the set of things that can be qualified as "numbers")

But then he talks about problems to describe pi.

So maybe we need to limit ourselves to "the smallest algebraically closed extension of the natural numbers", i.e. the algebraic numbers.

LE4d, give us more precise requirements?

- Yakk
- Poster with most posts but no title.
**Posts:**11087**Joined:**Sat Jan 27, 2007 7:27 pm UTC**Location:**E pur si muove

tetoru wrote:Yakk wrote:The problem with using axioms to define the real numbers is you hit Godel right in the face.

You cannot restrict the models your axioms describe to a single thing with a single set of properties. There are multiple different models which are described by any set of axioms of the real numbers that have different properties.

You can, however, restrict the type of thing that is a real number line using axioms. :)

Sorta. You can't categorically define the reals using first-order axioms, but you can using second-order logic. And even if you construct the reals in first-order ZFC, well, by Lowenheim-Skolem there are still countable models of ZFC, so this doesn't help you any: you can still get a countable uncountable set of reals out of the process. (Think about that for a while if you really want your brain to bleed.)

Mate, second-order logic itself admits multiple "implementations". So you can show that there is only one model of real numbers per model of second-order logic. Godel to the n!

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

Yeah, but don't constructions have the same problem? After all, the collection of dedekind cuts depends on your model. They're provably the same up to isomorphism anyways.

"With math, all things are possible." —Rebecca Watson

skeptical scientist wrote:Alky wrote:Formal definition:

http://en.wikipedia.org/wiki/Real_number

Layman's definition:

Any sequence of digits, with potentially infinite digits after the '.' but a finite amount before it, is a real number. Why the hell would they think 1 isn't a number? What number do you subtract from three to get two?

So 0.999... and 1 are different as real numbers because they are different as sequences of digits?

I didn't say that they were unique real numbers. But people are going to believe that no matter what you tell them.

Don't pay attention to this signature, it's contradictory.

- Yakk
- Poster with most posts but no title.
**Posts:**11087**Joined:**Sat Jan 27, 2007 7:27 pm UTC**Location:**E pur si muove

skeptical scientist wrote:Yeah, but don't constructions have the same problem? After all, the collection of dedekind cuts depends on your model. They're provably the same up to isomorphism anyways.

No, there are statements in some models of (second order logic) or (the real numbers) that are true, yet are false in a different model, as far as I know.

Take the godel-statement S for a theory T. Let U := T & S, and V := T & ~S.

Then U has a statement that is true that is false in V, namely S. Both are as consistent as the original theory T. And both are examples of models of T.

I am not sure what you mean by isomorphism, when one theory has an additional number G within a theory of the natural numbers that the other one lacks. Are you saying that the real numbers defined by the two distinct second-order logics are isomorphic across logics, provably?

- EradicateIV
**Posts:**361**Joined:**Mon Mar 19, 2007 7:33 pm UTC**Location:**Brownsville, PA-
**Contact:**

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

Yakk wrote:skeptical scientist wrote:Yeah, but don't constructions have the same problem? After all, the collection of dedekind cuts depends on your model. They're provably the same up to isomorphism anyways.

No, there are statements in some models of (second order logic) or (the real numbers) that are true, yet are false in a different model, as far as I know.

This is only true of incomplete theories. If you have a complete theory, every statement (in the language) that is true in one model will be true in every other model. But the models themselves will be different.

Take the godel-statement S for a theory T. Let U := T & S, and V := T & ~S.

Then U has a statement that is true that is false in V, namely S. Both are as consistent as the original theory T. And both are examples of models of T.

Well, they are not themselves models, but they have models which are also models of T. But the only reason this works is because your original theory was incomplete.

I am not sure what you mean by isomorphism, when one theory has an additional number G within a theory of the natural numbers that the other one lacks. Are you saying that the real numbers defined by the two distinct second-order logics are isomorphic across logics, provably?

No, I mean that you can prove in a theory T that all objects satisfying some requirements are isomorphic. Then in any model of T, if X and Y are objects satisfying the same requirements, X and Y will be isomorphic. However, if M and M' are different models of T, and X and X' are objects of the models M and M' which satisfy the requirements, we may have that M and M' are not isomorphic from our outside perspective. For example, let T be the theory of natural numbers with set variables, and let A and B be two sets such that 0 is in both A and B, and if n is in A or B, n+1 is as well. Then A and B will be provably equal, by a simple induction argument. However, there are secretly nonstandard models of the natural numbers, and in these models there are sets that look just like A and B as far as the theory is concerned, but are not the natural numbers as far as we're concerned with our outside perspective.

In particular, I meant that you can prove that any ordered archimedean field with the lub property is isomorphic to the reals, so you get the same object up to isomorphism (in your theory) by letting R be the unique such object, or by constructing it with dedekind cuts. But what you get still depends on your model.

Last edited by skeptical scientist on Mon May 07, 2007 6:32 pm UTC, edited 1 time in total.

"With math, all things are possible." —Rebecca Watson

- jestingrabbit
- Factoids are just Datas that haven't grown up yet
**Posts:**5967**Joined:**Tue Nov 28, 2006 9:50 pm UTC**Location:**Sydney

tendays wrote:Guys, read the OP - he isn't asking for a definition of a real number, but the definition of a "real" number. <- quotes.

He could have said an "actual" number to mean more or less the same, I guess. As opposed to "nonsensical", maybe.

His wording is not very clear to me, but doesn't outright exclude i, for instance. Because he talks about "numbers" (And not "element of an algebraically closed field" or whatever) I'd be tempted to say the complex numbers. (which is, for me, precisely the set of things that can be qualified as "numbers")

But then he talks about problems to describe pi.

So maybe we need to limit ourselves to "the smallest algebraically closed extension of the natural numbers", i.e. the algebraic numbers.

LE4d, give us more precise requirements?

Exactly what I thought about the first post and every subsequent one in the thread but your's.

I tend to think that the question of what a number is doesn't have a hard and fast answer and I suspect that whether a number is real or not isn't expressed by a dichotomy but by a range of realness.

If you want an infinity that obeys all the regular rules (so long as those are a subset of the field axioms) you need look no further than the hyperreals. Whats more, I'd say that the quaternions were probably a little more real than the hyperreals despite the fact that they have a noncommutative multiplication, so I don't think that the way the number works in equations decides realness. Perhaps how well it connects with an aspect of observed physical reality does it for me.

otoh, if someone told me 1 wasn't real I'd probably have a bit of a chuckle and change subjects or leave the room.

- Yakk
- Poster with most posts but no title.
**Posts:**11087**Joined:**Sat Jan 27, 2007 7:27 pm UTC**Location:**E pur si muove

skeptical scientist wrote:Yakk wrote:skeptical scientist wrote:Yeah, but don't constructions have the same problem? After all, the collection of dedekind cuts depends on your model. They're provably the same up to isomorphism anyways.

No, there are statements in some models of (second order logic) or (the real numbers) that are true, yet are false in a different model, as far as I know.

This is only true of incomplete theories. If you have a complete theory, every statement (in the language) that is true in one model will be true in every other model. But the models themselves will be different.

There are no complete logic models that can model the reals, however. Well, there are, but they are inconsistent (and you can prove everything) and/or you cannot tell if something is proven or not (which makes calling it "logic" a bit of a stretch).

Take the godel-statement S for a theory T. Let U := T & S, and V := T & ~S.

Then U has a statement that is true that is false in V, namely S. Both are as consistent as the original theory T. And both are examples of models of T.

Well, they are not themselves models, but they have models which are also models of T. But the only reason this works is because your original theory was incomplete.

Sure sure. But if you have a theoretical framework describing an axiomitization of the reals, your theoretical framework is guaranteed to be incomplete. Right?

I am not sure what you mean by isomorphism, when one theory has an additional number G within a theory of the natural numbers that the other one lacks. Are you saying that the real numbers defined by the two distinct second-order logics are isomorphic across logics, provably?

No, I mean that you can prove in a theory T that all objects satisfying some requirements are isomorphic. Then in any model of T, if X and Y are objects satisfying the same requirements, X and Y will be isomorphic. However, if M and M' are different models of T, and X and X' are objects of the models M and M' which satisfy the requirements, we may have that M and M' are not isomorphic from our outside perspective. For example, let T be the theory of natural numbers with set variables, and let A and B be two sets such that 0 is in both A and B, and if n is in A or B, n+1 is as well. Then A and B will be provably equal, by a simple induction argument.

Depending on your axiomitization of the natural numbers, it might not even require induction. :)

However, there are secretly nonstandard models of the natural numbers, and in these models there are sets that look just like A and B as far as the theory is concerned, but are not the natural numbers as far as we're concerned with our outside perspective.

That I don't get. How are they not natural numbers?

Or, more accurately, why do you say they are not natural numbers?

What are these "natural numbers with addition and multiplication" you are talking about -- if there is no way to describe their properties precicely enough to distinguish them from things you aren't talking about?

In particular, I meant that you can prove that any ordered archimedean field with the lub property is isomorphic to the reals, so you get the same object up to isomorphism (in your theory) by letting R be the unique such object, or by constructing it with dedekind cuts. But what you get still depends on your model.

So, for a given model of the theory of second order logic, it is talking about one particular set R which uniquely fullfills the properties of the reals. But the particular unique set R that is being talked about varies by your choice of model of the theory of second order logic varies.

Ie, there are sets R and R` in models M and M` respectively of theories T and T` respectively. Both T and T` contain the axioms & proof-rules of second order logic (and additional axioms) and are as consistent as second order logic was, so M and M` are models of the theory of second order logic.

But the set R in model M of theory T contains an element with properties that no element of the set R` in model M` of theory T` contains, expressed in the theory-symbols of second order logic (which exist in both T and T`).

Whew. I think that works. Less certain now that I wrote it all down. And I've got a headache!

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

Yakk wrote:There are no complete logic models that can model the reals, however. Well, there are, but they are inconsistent (and you can prove everything) and/or you cannot tell if something is proven or not (which makes calling it "logic" a bit of a stretch).

This is of course true, but I still think the existence of such logic models has philosophical implications. There exist theories for which the truth of every statement is determined, but even in such theories, there are nonisomorphic models. So there are models of, say, the reals such that all the true statements are exactly the same, but the reals themselves are not. This will still be a problem whether you define them as dedekind cuts or as the unique X satisfying YZW.

Take the godel-statement S for a theory T. Let U := T & S, and V := T & ~S.

Then U has a statement that is true that is false in V, namely S. Both are as consistent as the original theory T. And both are examples of models of T.

Well, they are not themselves models, but they have models which are also models of T. But the only reason this works is because your original theory was incomplete.

Sure sure. But if you have a theoretical framework describing an axiomitization of the reals, your theoretical framework is guaranteed to be incomplete. Right?

If you just want a theoretical framework describing an axiomatization of the reals, I suppose it depends exactly what you mean by "axiomatization of the reals". For example, the theory of real closed fields is a complete decideable theory, and is in some sense an axiomatization of the reals (although as with every consistent theory with infinite models, there will be multiple nonisomorphic models). But if you want your theory to be strong enough to describe more than just the reals (for example, if you want it to be strong enough to do all of mathematics) then what you said is of course true.

However, there are secretly nonstandard models of the natural numbers, and in these models there are sets that look just like A and B as far as the theory is concerned, but are not the natural numbers as far as we're concerned with our outside perspective.

That I don't get. How are they not natural numbers?

Or, more accurately, why do you say they are not natural numbers?

What are these "natural numbers with addition and multiplication" you are talking about -- if there is no way to describe their properties precicely enough to distinguish them from things you aren't talking about?

There are numbers which are not 0, S0, SS0, SSS0, SSSS0, etc., for any amount of applications of the successor function. In other words, there are numbers which are bigger than every natural number in some nonstandard models.

To see this, we observe that (crossing our fingers and hoping) Peano Arithmetic is a consistent theory. Let L be the language of Peano Arithmetic, and let L' be L together with an additional constant symbol c. Let PA' be the theory you get starting with the axioms of PA in the language L'. Now the natural numbers are a model of PA' under any interpretation of c, since we don't have any axioms that give c any properties. Therefore, the natural numbers are a model of PA'+"c!=0"+"c!=S0"+...+"c!=SSS...S0" for any long string of Ss, since we can interpret c to be a sufficiently large number so that this is true; hence these theories are all consistent. By compactness, the theory PA'+"c!=0"+"c!=S0"+... is also consistent, and by completeness it has a model, which is a model of PA, since this theory extends PA, and it has an element modelling the constant c, with the additional property that this element is neither 0 nor 1 nor 2 nor 3 nor 4 nor any other number you can name. That's what I call "not a natural number".

Whew. I think that works. Less certain now that I wrote it all down. And I've got a headache!

It's no worse than countable models of ZFC.

"With math, all things are possible." —Rebecca Watson

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

adlaiff6 wrote:The product of the binary interpretation of any string in {'.'}o{0,1}* and 2 to any integer power.

There are two problems with this definition.

1) What the hell is a binary interpretation anyways?

2) Apparently all real numbers are non-negative

Also, I thought {0,1}* was the set of finite strings of 0s and 1s, and if you wanted infinite strings you denoted it as {0,1}^N or {0,1}^ω.

Actually, if you're a set theorist, a real number is a set whose elements are all of the form {{{a},{a,b}}:{{a},{a,b}}~{{c},{c,d}}} (an equivalence class of "ordered pairs" interpreted as representing products of natural numbers with their formal inverses) where b is a set of the form {{}}, {{},{{}}}, {{{}}, {{},{{}}}} etc. (a "natural number" which is not {}) and a is a set of the form {{s},{s,a'}} (an "integer") where s is either 0 or 1 (interpreted as showing the sign of a) and a' is a natural number. Needless to say, verifying that 1*1=1 could be extremely time consuming with these definitions, which is why I usually advise forgetting the whole thing.

In other words, a real number is an infinite set consisting of infinite sets of two element sets, whose elements are themselves finite sets of sets of sets of sets, many of which are empty.

"With math, all things are possible." —Rebecca Watson

- adlaiff6
**Posts:**274**Joined:**Fri Nov 10, 2006 6:08 am UTC**Location:**Wouldn't you rather know how fast I'm going?-
**Contact:**

skeptical scientist wrote:1) What the hell is a binary interpretation anyways?

I'll let you know once they invent computers.

skeptical scientist wrote:2) Apparently all real numbers are non-negative

D'oh! Got me there.

Umm...count the first bit as the sign? I guess?

Well, I suppose I can do that since I never defined binary interpretation.

skeptical scientist wrote:Also, I thought {0,1}* was the set of finite strings of 0s and 1s, and if you wanted infinite strings you denoted it as {0,1}^N or {0,1}^ω.

I could really go either way on this one. I've never tried to write down an infinite string. You're probably correct, but you knew what I meant, which is the whole purpose of communication, isn't it?

I'm not.skeptical scientist wrote:Actually, if you're a set theorist

skeptical scientist wrote:, a real number is a set whose elements are all of the form {{{a},{a,b}}:{{a},{a,b}}~{{c},{c,d}}} (an equivalence class of "ordered pairs" interpreted as representing products of natural numbers with their formal inverses) where b is a set of the form {{}}, {{},{{}}}, {{{}}, {{},{{}}}} etc. (a "natural number" which is not {}) and a is a set of the form {{s},{s,a'}} (an "integer") where s is either 0 or 1 (interpreted as showing the sign of a) and a' is a natural number. Needless to say, verifying that 1*1=1 could be extremely time consuming with these definitions, which is why I usually advise forgetting the whole thing.

In other words, a real number is an infinite set consisting of infinite sets of two element sets, whose elements are themselves finite sets of sets of sets of sets, many of which are empty.

I blindly agree with you, claiming the desire not to read all those squiggly brackets. Maybe when it's not almost 4AM here.

3.14159265... wrote:What about quantization? we DO live in a integer world?

crp wrote:oh, i thought you meant the entire funtion was f(n) = (-1)^n

i's like girls u crazy

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

adlaiff6 wrote:skeptical scientist wrote:1) What the hell is a binary interpretation anyways?

I'll let you know once they invent computers.

Something that purports to be a definition had better, y'know, define stuff. Assuming I know what a binary interpretation is just because I live in a world with computers is silly; even if I know what a binary integer like 101001 is, how the hell am I supposed to interpret an infinite decimal like .001011101001... or worse .0011111... if you don't tell me how?

"With math, all things are possible." —Rebecca Watson

- adlaiff6
**Posts:**274**Joined:**Fri Nov 10, 2006 6:08 am UTC**Location:**Wouldn't you rather know how fast I'm going?-
**Contact:**

skeptical scientist wrote:adlaiff6 wrote:skeptical scientist wrote:1) What the hell is a binary interpretation anyways?

I'll let you know once they invent computers.

Something that purports to be a definition had better, y'know, define stuff. Assuming I know what a binary interpretation is just because I live in a world with computers is silly; even if I know what a binary integer like 101001 is, how the hell am I supposed to interpret an infinite decimal like .001011101001... or worse .0011111... if you don't tell me how?

It's pretty much the same way you interpret decimal.

3.14159265... wrote:What about quantization? we DO live in a integer world?

crp wrote:oh, i thought you meant the entire funtion was f(n) = (-1)^n

i's like girls u crazy

- skeptical scientist
- closed-minded spiritualist
**Posts:**6142**Joined:**Tue Nov 28, 2006 6:09 am UTC**Location:**San Francisco

adlaiff6 wrote:skeptical scientist wrote:adlaiff6 wrote:skeptical scientist wrote:1) What the hell is a binary interpretation anyways?

I'll let you know once they invent computers.

Something that purports to be a definition had better, y'know, define stuff. Assuming I know what a binary interpretation is just because I live in a world with computers is silly; even if I know what a binary integer like 101001 is, how the hell am I supposed to interpret an infinite decimal like .001011101001... or worse .0011111... if you don't tell me how?

It's pretty much the same way you interpret decimal.

And everyone has inherent knowledge of how to interpret infinite decimal expansions without ever seeing a definition? We know that never leads to confusion. After all, who could think something silly like that .999...≠1?

"With math, all things are possible." —Rebecca Watson

- TheLordOfPhysics
**Posts:**8**Joined:**Wed May 09, 2007 9:52 pm UTC

if you can physically represent it it is real. like 1 being i have one bradpeice, two bradpeices and so on and so forth because i can physiclly represent it. were as negatives are representations of debt and owning "bradpeices's" if you will.

Friend - "Hey what would you rather do logarithms or a girl?"

Me - "Obvo logarithms, because they dont complain if your done in 10 seconds"

^^^True Story^^^

Me - "Obvo logarithms, because they dont complain if your done in 10 seconds"

^^^True Story^^^

### Who is online

Users browsing this forum: No registered users and 4 guests