# Uniqueness of zero

## Main Question or Discussion Point

We know that there are several different infinities, and there appears to be some kind of duality between infinity and zero. So how do we know that zero is unique? There as several distinct concepts of "nothing" in the english language that are often confused, as exemplified in the statement "nothing is better than eternal happiness, and a ham sandwich is better than nothing, so a ham sandwich is better than eternal happiness". The concept of nothing and its relation to zero and the empty set seems quite subtle. So to restate the question, how do we know that zero is unique?

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arildno
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We may first limit ourselves to the real numbers, rather than to set theory.

In this case, infinity is not a real number, so it falls outside discussion.

As for the uniqueness of the real number 0, that can be proved by means of the axioms governing the real numbers:

1. Definition of 0:
There exists a real number "0", so that for every real number "a", we have a+0=0
2. Definition of additive inverse (-a):
For every number "a", there exists a number (-a) so that a+(-a)=0

This is basically what we need, in addition to the "rearrangement" laws that addition commutes&associates, along with with the properties of the equivalence relation "=", i.e, including the symmetry, reflexivity and transitivity properties.

The proof goes as follows.

Suppose there exists a number "z" so that for any particular number "a", we have a+z=a.

Thus, we may construct the identity:

a+z=a+0

Adding (-a) to each side, we get:

(a+z)+(-a)=(a+0)+(-a)
or, by rearrangement:

z+(a+(-a))=0+(a+(-a))
i.e,
z+0=0+0
i.e
z=0

Thus, the uniqueness of "0" has been proved.

apeiron
Gold Member
The elaborateness of the definition nicely brings attention to the fact of how constructed and limited a notion of nothing we are dealing with in zero.

This is "nothing" from the point of view of a 1D number line, a highly (indeed maximally) constrained space of numbers that turns out to have a bunch of crisp arithmetical properties, such as commutative relationships.

To generalise our idea of zero - to move towards a truer view of nothingness - we would have to start removing those constraints represented by numbering restricted to a single dimension.

So how would "zero" look as the origin of an infinite-D "number-line"? A properly unbounded kind of zero-ness.

Infinity is also here being defined in the usual way - as an operation on a 1D number-line with unbounded extension. Again, what does infinity look like in the 2D complex number plane or the 16D sedonion space?

In different numerical systems, "0" or "zero" or "nothingness" may have different meanings, I don't think it is reasonable to try to discuss "the ultimate zero."

There exists a real number "0", so that for every real number "a", we have a+0=0
That's quite unlikely.

apeiron
Gold Member
In different numerical systems, "0" or "zero" or "nothingness" may have different meanings, I don't think it is reasonable to try to discuss "the ultimate zero."
.
Well, do we know if they do have different meanings? Then why? Then how we can move beyond that to arrive at a more general zero?

As I say, Arildno's go at a proof is a good illustration that the meaning of zero is the product of a whole system of necessary constraints. And the way to generalise in philosophy (and maths) is to systematically remove constraints to discover what remains.

So geometry generalises to topology once you remove the constraint of quantitative measurements - moving from actual paths to possible paths.

Algebra is also exploring its own generalisation with octonions, sedonions, and so on to infinite dimensional number.

It may not seem reasonable, but it is happening already.

Zero is just a concept... When you put your mind to the concept of zero you are not making a zero... No the zero already exists and you are just glancing at it from a distance in a new unique way. Same is true for infinity and the reason we can grasp a view of it is because we understand its function from our own point of view.

The above proof for the uniqueness of zero as it applies to the real numbers works like this - he took the definitions that are relevent to it and some usually understood axioms and showed that these lead to a unique additive identity. If we were to look at a different system, different rules and definitions apply, maybe they will also lead to a unique zero, maybe they won't. The zero of that system is a different entity to the zero of the real numbers, different rules apply to it, it works in a different way, referring to both of them as "zero" or "0" is a convenience.

BobG
Homework Helper
We know that there are several different infinities, and there appears to be some kind of duality between infinity and zero. So how do we know that zero is unique? There as several distinct concepts of "nothing" in the english language that are often confused, as exemplified in the statement "nothing is better than eternal happiness, and a ham sandwich is better than nothing, so a ham sandwich is better than eternal happiness". The concept of nothing and its relation to zero and the empty set seems quite subtle. So to restate the question, how do we know that zero is unique?
Your analogy is similar to asking if 5 seconds is greater than 5 kilograms. Even Arildno's proof is going to run into problems if we're adding time and mass together.

Zero is zero. It's the domain that's changed in the analogy.

apeiron
Gold Member
Your analogy is similar to asking if 5 seconds is greater than 5 kilograms. Even Arildno's proof is going to run into problems if we're adding time and mass together.
.
This may sound a smart arse reply, but seriously, the planckscale does give us a yardstick to judge such issues. 5 lightseconds does defined an expanse of spacetime. We can then judge how full of energy density that expanse is. An argument could be made in this direction - and it would also help show why we would believe our reality, the one we can observe, is in fact "all connected". All properties can be reduced to a single geometry (hopefully).

Zero is zero. It's the domain that's changed in the analogy.
Domain is another way of saying context, or global constraints. So what does a generalised domain look like once we have abstracted over all the particular domains we can think of?

BobG
Homework Helper
This may sound a smart arse reply, but seriously, the planckscale does give us a yardstick to judge such issues. 5 lightseconds does defined an expanse of spacetime. We can then judge how full of energy density that expanse is. An argument could be made in this direction - and it would also help show why we would believe our reality, the one we can observe, is in fact "all connected". All properties can be reduced to a single geometry (hopefully).
No, it doesn't sound like a smart arse reply. In fact, that's why I edited my reply to compare time and mass. Putting time and distance was too easy for you guys to shoot a hole in. :rofl:

The original post still just uses wordplay to create an absurdity that doesn't actaully exist. Both sets of "nothing" are empty. That doesn't mean you can compare ham sandwiches to eternal happiness. The problem has nothing to do with the uniqueness of zero.

All properties can be reduced to a single geometry? Hopefully, maybe, but probably not.

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arildno
Homework Helper
Gold Member
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The elaborateness of the definition nicely brings attention to the fact of how constructed and limited a notion of nothing we are dealing with in zero.

This is "nothing" from the point of view of a 1D number line, a highly (indeed maximally) constrained space of numbers that turns out to have a bunch of crisp arithmetical properties, such as commutative relationships.

To generalise our idea of zero - to move towards a truer view of nothingness - we would have to start removing those constraints represented by numbering restricted to a single dimension.

So how would "zero" look as the origin of an infinite-D "number-line"? A properly unbounded kind of zero-ness.

Infinity is also here being defined in the usual way - as an operation on a 1D number-line with unbounded extension. Again, what does infinity look like in the 2D complex number plane or the 16D sedonion space?
Nonsensical, vague blatherboy-"definitions" in philosophy are ALL WORTHLESS, and certainly not "truer"

arildno
Homework Helper
Gold Member
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The above proof for the uniqueness of zero as it applies to the real numbers works like this - he took the definitions that are relevent to it and some usually understood axioms and showed that these lead to a unique additive identity. If we were to look at a different system, different rules and definitions apply, maybe they will also lead to a unique zero, maybe they won't. The zero of that system is a different entity to the zero of the real numbers, different rules apply to it, it works in a different way, referring to both of them as "zero" or "0" is a convenience.
Quite so.

I played the "real numbers game", and by following those rules, I came to the true conclusion that "0" is, indeed, unique within that game.

By the fallacious process imagined by apeiron, by removing "constraints" as he calls the axioms, he'll end up with..nothing, and might as well shut up. Zero may be thought of as an example of an identity element of a group - it is not nothing - I don't know what you mean by nothing.

An identity element of a group is always unique.

Infinite numbers are never elements of groups. They are mathematically unrelated to zero.

The idea of a duality between zero and infinity is meaningless to me. There are many infinities that are non-commensurate, in fact there is no largest one, so which one is dual to zero - whatever dual means?For instance,is the infinity of the continuum dual to zero - or how about the set of all sbsets of the continuum?

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Your analogy is similar to asking if 5 seconds is greater than 5 kilograms. Even Arildno's proof is going to run into problems if we're adding time and mass together.

Zero is zero. It's the domain that's changed in the analogy.
I assume by my analogy you mean the part I put in quotation marks. In that case, my point is that the two uses of the word nothing have fundamentally different meanings - they do not simply refer to different contexts. The first nothing means "there does not exist a thing" whereas the second means "the absence of any thing". A third meaning (possibly a confused meaning) is the idea of nothing as a thing or state.

I don't know what the duality between 0 and infinity is. There are certainly many different zeros, 0, (0, 0), (0, 0, 0) to name but a few. Indeed, if for every infinity there is a vector space with that many dimensions, then we can assign a distinct zero to each infinity.

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apeiron
Gold Member
The idea of a duality between zero and infinity is meaningless to me. There are many infinities that are non-commensurate, in fact there is no largest one, so which one is dual to zero - whatever dual means? For instance,is the infinity of the continuum dual to zero - or how about the set of all sbsets of the continuum?
This is the thought I have: fundamentals always come in complementary pairs. This is a fact of logic. So uniqueness in a monadic sense is impossible. Candidate fundamentals like zero must have a matching concept that is asymmetric, so that together they are a broken symmetry resulting from the breaking of a deeper symmetry.

Now one answer might be that the duality is of the form of the binary code - counting reduced to its actual simplest form. The 1 and the 0. A something and a nothing. The many, all other number and pattern, can then be constructed from these atomistic components.

So forget numberlines, this is a dimensionless approach. All you need is pure absence/pure presence, and dimensionality itself can be constructed in the information theoretic view.

But binary code is a symmetric kind of "symmetry-breaking" and so unnatural. We really should prefer a stable asymmety.

A second way of finding the duality here would be to contrast the continuity of the numberline with the discreteness of its points. This is a classic dichotomy - discrete~continuous, familiar from Zeno's paradoxes.

So this would mean we could treat any chosen number as being dual to the line on which it lives. The relationship is a broken symmetry, asymmetric. But I don't see that zero, as just another point in this system, would be uniquely dual in that sense. Rather all points would be dual to the whole of the line.

What I was arguing before was that zero might be properly dual in an infinite dimensional number realm. On a 1D line, points just sort of slide around as nothing intersects at a point to single it out as unique. But add further dimensions and yes, an origin is being fixed at a place. The zero becomes the only point dual to the whole of the space arguably.

This is the thought I have: fundamentals always come in complementary pairs. This is a fact of logic.
What is the complement of a photon? Where is your proof that the complement of a thing is different from the thing itself? In addition, the complement of zero is zero.

apeiron
Gold Member
What is the complement of a photon? Where is your proof that the complement of a thing is different from the thing itself? In addition, the complement of zero is zero.
This is a claim about what we eventually find to be fundamental. So things that lack clear and unambiguous asymmetry/complementarity could not, by this definition, be fundamental.

So elephants, dewdrops, the number 16, would all be things that are not obviously dual - at the level of description we are using - and so are not themselves fundamental.

Is a photon fundamental then? No, its just another of the zoo of particles or couplings. So we would have to dig down deeper (or rise up a level of abstraction, same thing) to seek what was fundamental.

So in QM, a photon would be granted these kinds of more general dualities. A photon would be particle~wave. It would have position~momentum. It would have energy~time. We would be viewing the photon in terms of the fundamental symmetries of nature.

Of course, these QM and Noether dualities may not themselves be the most basic. But we would seem to be on the right path here.

Other complementaries that would appear to apply to photons would be event~context. Or at least the exchange of a photon is an event (of all the things that could have potentially happened within the context of this light cone, it was this event that took place, and equally, none of the others that had existed as possibilities).

So you see how basic dichotomisation is to modern physics as well as ancient philosophy. It is the logical step that starts any idea building.

Is a photon fundamental then? No ...
Yes.

What's more, it has a complement. What is the complement of a photon?

apeiron
Gold Member
Yes.

What's more, it has a complement. What is the complement of a photon?
Ohmygawd! You mean they found the photino!!:surprised

Evo
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Closed pending moderation decision.