studiot

Is zero finite?

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This issue is offtopic in a thread about electrons so I have started a new thread for discussion.

https://www.scienceforums.net/topic/115443-electrons-how-do-they-work/?tab=comments#comment-1062243

 

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1 hour ago, studiot said:

Indeed, which is why I based my figure smack in the middle of the range that can can currently be found.
This range is very large, larger than the range of atomic sizes for instance.

 

Furthermore zero is a finite number.

Finite is often used to mean a non-zero number

https://en.wikipedia.org/wiki/Finite_number

"In mathematical parlance, a value other than infinite or infinitesimal values and distinct from the value 0"

 

I prefer the definitions of

Dedekind : not infinite

Russell : Able to be counted using a terminating sequence of natural numbers.

 

Consider the equation

x2 - 2x + 1= 0

Is the difference between the two roots of this quadratic or infinite?

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It is hard to come up with any generally applicable definition by which zero should not be finite.

Dedekind: 0 (understood as the empty set, so that the definition applies) does not allow an injection into a proper subset. Hence 0 is finite.

Russell: 0 has a bijection to a set {1,2,...,n} for a natural number n, namely for n = 0 and empty bijection. Hence 0 is finite. 

All well-orderings of 0 are isomorphic. This implies 0 is finite.

Every non-empty family of subsets of 0 has an inclusionwise minimal element (Tarski). Etc.

There are strange sets that are Dedekind-finite but Kuratowski-infinite. 0 is not one of them, since 0 is Kuratowski-finite by definition. 

I prefer Russell myself, probably because I come from Combinatorics. For us it is important not only to know whether some things exists or not, but also to count the number of them. An expression like n^m for natural numbers n and m means, by definition, the number of different functions from a set of m elements to a set of n elements. In particular you are quite aware that 0^0 = 1 means that there is the unique function \( \emptyset \) from the empty set to the empty set, so Russell's definition applies nicely.

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The context in which I brought it up was in physics, not mathematics. In that context, finite is often used to mean non-zero.

For example

https://arxiv.org/abs/hep-th/0611052

I suspect its use stems from situations where you could be dividing by the term, which gives rise to infinities.

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Thank you taeto for that comprehensive analysis. +1

 

31 minutes ago, swansont said:

The context in which I brought it up was in physics, not mathematics. In that context, finite is often used to mean non-zero.

For example

https://arxiv.org/abs/hep-th/0611052

I suspect its use stems from situations where you could be dividing by the term, which gives rise to infinities.

Yes and I think, even in Mathematics it is implied in some activity that the finite something is non zero, for example in finite differences and finite elements.
Otherwise you would never move across the net, although in finite differences you carry on until the differences are zero.

But in Engineering the terms Poles and Zeros are often used, here is an extract from an Engineering  book of that name.
Note the definite use of a real finite zero.

So, no offence meant, swansont, nothing personal and all that, but I think the jury is out on technical use of finite zero, but taeto has shown tha mathematically it is finite.

finite1.thumb.jpg.28c1092f73e21f6007e50dc995cb3f65.jpg

 

We also have the term singularity  (removable and permanent).

 

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No offense taken. Physicists have been known to use math in a way that drives mathematicians crazier than they already are.

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Whether or not to call zero "finite" outside of math is just wordplay.  In math, zero is finite.

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1 hour ago, mathematic said:

Whether or not to call zero "finite" outside of math is just wordplay.  In math, zero is finite.

How big is zero if it's finite? 

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1 minute ago, StringJunky said:

How big is zero if it's finite? 

I guess it depends. As for an outside of math answer, I’m not sure if the OP question is a valid one outside of mathematics. 

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Can zero be seen as  a limit ? (any number divided by a number that we let increase without  limit)

 

In that sense is zero infinite  because there is an infinite process involved in its definition?

 

Also,I wonder is the zero used in 10,20 etc the same as the zero used to denote an absence of quantity?

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8 minutes ago, StringJunky said:

How big is zero if it's finite? 

How big is 1.. ?

How big is 100.. ?

How big is any other number.. ?

I don't think so word "big" applies here. It would make more sense, if there would be second reference number to which zero is compared..

"0 is bigger than -1"

 

"0 has just one digit required to write it in any numeral system". (only 0 and 1 fulfill this. To write down 2 in binary system there are needed two digits %10)

"irrational number has infinite number of digits required to write it in any numeric system". 0 is not irrational number.

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16 minutes ago, geordief said:

Can zero be seen as  a limit ?

Yes zero is formally a limit of many sequences.

In fact there is even a special name for them,

A null sequence is a sequence that has zero as its limit.

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39 minutes ago, studiot said:

Yes zero is formally a limit of many sequences.

In fact there is even a special name for them,

A null sequence is a sequence that has zero as its limit.

Is the subject of the question not dependent on the branch of maths?

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Posted (edited)

At first I said, "Of course zero is finite." But people gave good examples of casual usage in which finite means nonzero. I realized that when something has a small but nonzero probability, I'll say it has a "finite probability," meaning it's not zero. Interesting semantic point! 

Edited by wtf

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Posted (edited)
28 minutes ago, wtf said:

At first I said, "Of course zero is finite." But people gave good examples of casual usage in which finite means nonzero. I realized that when something has a small but nonzero probability, I'll say it has a "finite probability," meaning it's not zero. Interesting semantic point! 

Can't it be reasoned that finite is anything but zero, which is empty?

Edited by StringJunky

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Posted (edited)
59 minutes ago, StringJunky said:

Can't it be reasoned that finite is anything but zero, which is empty?

No. The empty set is finite since it's not in bijection with any proper subset of itself. That's because it has no proper subsets. So it's Dedekind-finite. And it's in bijective correspondence with a natural number, namely zero. So it's finite in the classical (non-Dedekind) sense. It's of interest that absent the axiom of choice, there are infinite sets that are Dedekind-finite. That is, they're not bijectable to any natural number, and they're not in bijection with any proper subset of themselves. These are very weird sets.

The only way we can say zero is finite is by abuse of terminology in casual usage. In fact in any written work we shouldn't even use the the incorrect terminology to make sure nobody is confused.

Edited by wtf

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7 hours ago, wtf said:

In fact in any written work we shouldn't even use the the incorrect terminology to make sure nobody is confused.

So what is your take on the third finite difference column in a (finite  difference) table of values of a quadratic function?

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Finite is more commonly used as the opposite of infinite, in which case zero is finite.

It is possible that someone could think of finite as the opposite of infinitesimal.  In this usage, the question of whether zero is finite is more  open to debate.

I prefer to avoid mathematical debates which are more about definitions.

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On 7/29/2018 at 3:33 AM, studiot said:

So what is your take on the third finite difference column in a (finite  difference) table of values of a quadratic function?

Not understanding  the question. The successive differences of a polynomial are eventually all zeros. I can't relate this to the subject of the thread, which is the casual semantics of distinguishing zero from finite.

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1 hour ago, wtf said:

I can't relate this to the subject of the thread, which is the casual semantics of distinguishing zero from finite.

Such as stated by Wikipedia here

Quote

Wikpedia

 

finite

A countable number less than infinity, being the cardinality of a finite set – i.e., some natural number, possibly 0

A real number, such as may result from a measurement (of time, length, area, etc.)

In mathematical parlance, a value other than infinite or infinitesimal values and distinct from the value 0

 

Descriptive informalities

finite

Next to the usual meaning of "not infinite", in another more restrictive meaning that one may encounter, a value being said to be "finite" also excludes infinitesimal values and the value 0. For example, if the variance of a random variable is said to be finite, this implies it is a positive real number.

 

1 hour ago, wtf said:

Not understanding  the question. The successive differences of a polynomial are eventually all zeros.

We both know that. I was just observing that users of finite differences expect the finite difference to be other than zero.

This is a mathematical operation, not a physics or engineering one though it is (was) much used in both.

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1 hour ago, studiot said:

Such as stated by Wikipedia here

 

We both know that. I was just observing that users of finite differences expect the finite difference to be other than zero.

This is a mathematical operation, not a physics or engineering one though it is (was) much used in both.

 

It's by no means a mathematical question. Mathematically, zero is finite. 

This is a semantic question, concerning how people, even mathematicians who know better, use words in casual conversation.

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On 7/28/2018 at 6:50 PM, swansont said:

The context in which I brought it up was in physics, not mathematics. In that context, finite is often used to mean non-zero.

For example

https://arxiv.org/abs/hep-th/0611052

I suspect its use stems from situations where you could be dividing by the term, which gives rise to infinities.

You beat me to it. It just occurred to me how physicists like to say "neutrinoes have finite mass" to mean that they have positive rest mass. Some particle has "finite radius" meaning positive radius. And so on. It is almost like mass ought to be measured in 1/kg and sizes of things in 1/m. Similarly to how the charge of an electron really should be positive instead of negative, given the direction in which electricity actually flows. But as opposed to mathematics, where things intrinsically are allowed to become infinite, that is not usually so in physics. In mathematics, "finite" rightly means the opposite of "infinite", and the dichotomy does not apply to physics, because then it is just a matter of choice of units. 

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On 28/7/2018 at 7:33 PM, studiot said:

But in Engineering

In engineering, nobody cares about rigorous definitions. What matters is whether it works, not whether someone wants to call some trivially useless cases "finite" or not.

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