Reasons why infinity hasn't been implemented into modern math

[epkid08]

I don't really understand why it hasn't been numerically added into modern math. I mean, we make all kinds of properties for zero, but we can't make properties for infinity? It gets messy from time to time, but we could define everything strictly so that it still works algebraically.

Example: (quoted from http://mathforum.org/dr.math/faq/faq.divideby0.html)

5/0 = \infty
5 = 0*\infty
Multiplicative property of 0.
5=0
WRONG!

If we defined \infty numerically:
5/0 = 1/0
5 = 0/0
We then, could define 0/0 as truly undefined, or all real numbers, or some other name. I don't think we need to classify something as undefined, or anything for that matter, unless it is 0/0, 0^0, \infty^\infty etc.

As for infinity, it should be implemented carefully into our modern math.

Now it's time for you to post "Reasons why infinity hasn't been implemented into modern math."

[slider142]

Infinity is present everywhere in modern math. From simple implementations such as the extended real numbers (http://en.wikipedia.org/wiki/Extended_real_numbers) to the more abstract cardinal (http://en.wikipedia.org/wiki/Cardinal_numbers) and exotic ordinal numbers (http://en.wikipedia.org/wiki/Ordinal_numbers).

[CRGreathouse]

Don't forget the projective reals!

[matt grime]

Or indeed the projective anything, the compactifications of spaces, the Riemann sphere, the theory of poles and singularities going back hundred+ years, Laurent series, Mobius transformations,....

[epkid08]

I'm talking numerically, 1/0. You won't ever see its notation in a function etc...

[Hurkyl]

I'm talking numerically, 1/0. You won't ever see its notation in a function etc...
Unless you're working with extended real numbers, cardinal numbers, projective reals, projective anything, compactified spaces, the Riemann sphere, meromorphic functions, Laurent series, Möbius transformations....

You don't see an infinite integer simply because the ring of integers doesn't contain such a thing. The ring of integers doesn't contain 1/2 either.

[kts123]

I'm talking numerically, 1/0. You won't ever see its notation in a function etc...

You won't usually see it in a function over the real numbers, in the same sense you won't see 1/2 in functions over the integers. Infinity is a special class of number, just like rational numbers, irrational numbers, transcendental numbers, imaginary numbers, complex numbers, etc. If you're working with functions that have a domain and range over the real numbers you won't see infinity (OR imaginary numbers or complex numbers.)

[CRGreathouse]

Actually, I'm pretty tired of the word "infinity". A good tenth of the posts on these math forums could be avoided if posters instead named the sort of infinite number they were thinking about. The one-point compactification of the reals? Aleph-2? Epsilon-naught? The IEEE +Infinity?

[kts123]

location.reimannsphere(santa) ???

[epkid08]

You won't usually see it in a function over the real numbers, in the same sense you won't see 1/2 in functions over the integers. Infinity is a special class of number, just like rational numbers, irrational numbers, transcendental numbers, imaginary numbers, complex numbers, etc. If you're working with functions that have a domain and range over the real numbers you won't see infinity (OR imaginary numbers or complex numbers.)

My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

[matt grime]

It doesn't 'define infinity as undefined' (which is a contradiction in terms). It merely, and correctly, states that you can't cancel off zeroes in multiplicative expressions.

[arildno]

I don't really understand why it hasn't been numerically added into modern math.
My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so.

So, according to you, low-level algebra is the heart of modern maths?

[Hurkyl]

Well, epkid08, your post has been answered: mathematics does 'implement infinity'. Allow me to suggest that, before you continue ranting, you spend some time studying mathematics so as to gain an understanding both of how mathematics works and how the notion of infinite is treated and used in mathematics.

Incidentally, your comment on zero is unfounded. It is so incredibly useful that natural languages have evolved to give it special treatment -- the difference between 'zero' and 'not zero' is quite deeply ingrained in grammar. (Apparently so deeply that some people have difficulty comprehending how they could be united into a notion of 'quantity') Grammar can also split the latter case into 'one' and 'more than one', but not in such an essential way. (There is some limited direct linguistic support for larger quantities, but going beyond one usually entails using ordinal or cardinal numbers)

[arildno]

. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

I hardly ever use 2.17, but I am really glad it has already been defined! :smile:

[D H]

, your comment on zero is unfounded. It is so incredibly useful that natural languages have evolved to give it special treatment -- the difference between 'zero' and 'not zero' is quite deeply ingrained in grammar. (Apparently so deeply that some people have difficulty comprehending how they could be united into a notion of 'quantity') Grammar can also split the latter case into 'one' and 'more than one', but not in such an essential way. (There is some limited direct linguistic support for larger quantities, but going beyond one usually entails using ordinal or cardinal numbers)
As an aside, one of my jobs is writing software standards. One such is "All magic numbers shall be named and referenced by that name. For example 'circumference=2*3.14159*radius' violates this rule and is wrong to boot. Pi is a magic number. Whether two is a magic number is debatable. A good starting point is that the only non-magical numbers are zero and one. You can use an unnamed small integer if the usage is well-commented." I never mentioned that zero and one, being the root of almost all mathematics, are actually the most magical numbers of all.

[D H]

My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so.

No. It does define infinity in terms of limits. It does not define operations with infinity as undefined. It simply does not define such operations, period, and for good reason. They are a superfluous and confusing distraction in the topic at hand, which is getting students to grasp the main concepts of elementary algebra.

It doesn't 'define infinity as undefined' (which is a contradiction in terms).
That's going a bit too far. Computer science can be viewed as an offshoot of applied logic / applied mathematics. Several computer programming languages explicitly define "undefined behaviors". Defining what is undefined is not inherently a contradiction in terms. It is just a warning about the potential for nasal demons.

[kts123]

My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

We call it undefined for the same reason we tell grade-schoolers they can't sqaure root negative numbers -- it'd confuse the hell out of them and it's off-topic. When their minds are ripe, and the contexts of our discussion is proper, these concepts are introduced. Crawl before you can walk.

Oh, and, surprise, some infinities are bigger than others and we can divide by zero. Hehehe.

[matt grime]

Computer science can be viewed as an offshoot of applied logic / applied mathematics. Several computer programming languages explicitly define "undefined behaviors". Defining what is undefined is not inherently a contradiction in terms. It is just a warning about the potential for nasal demons.

Declaring something to be undefined is not the same as *defining* it to be undefined. It might be a useful bastardisation of the language, but it is technically cobblers.... "coming up at ten: what is a pin head and how many angels can we make dance on it."

[kts123]

To help explain what Mr. Grime has said, contemplate the following:

"My talent is not having any talents." as opposed to "I have no talents." One makes a declaration, wheras the other implies a sentiment (and in such a way that in contradicts itself.)

[D H]

Well OK then. Is it permissible to say that some operations in mathematics are declared to be undefined, or one must simply say that some operations in mathematics are not defined?

To the OP: This is true even for the extended real number line. For example, the values of 0/0 and \infty/\infty are not defined.

[LukeD]

Well OK then. Is it permissible to say that some operations in mathematics are declared to be undefined, or one must simply say that some operations in mathematics are not defined?

What he means is that when something is undefined, there's no object "undefined" or even 0/0 or infinity/infinity or whatever. They simply don't exist. When you write them in a formula, then you don't have a grammatically well formed formula. It's as meaningless as "5x^2*1 + / = 2x +", just a random string of math symbols. What you've written doesn't have any meaning until you actually do define it.

There are a few engineering and computer science things that are similar to "undefined", but I don't know anything in math like that.

[epkid08]

I remember back in the fifth grade (or whenever), when I was first being taught (x,y) coordinates, and linear functions, any vertical line had a slop that was 'undefined'. Is this only because 'their minds are too ripe'? I mean, I remember back then, I questioned the this, as I thought it should have a slope of infinity. I can definitely see how it would be confusing, depending on how you teach it, but you wouldn't have to get so specific as to confuse the children. For example, if you define the equation, x=5, as having a slope of infinity, the kids wouldn't understand why you can't put it into slope-intercept form.

But my point is that, with the right properties, infinity can be implemented into algebra.

[Take_it_Easy]

But my point is that, with the right properties, infinity can be implemented into algebra.

I agree but some problem would occasionally show up.
How much is \frac{\infty}{\infty}? Or {\infty}-{\infty}??????

[epkid08]

I was only talking about infinity. We don't have uses for those, and other, indeterminates anyways. They lie off of any number line.

Edit: If your post was directed at the fact that kids might get into trouble by trying to divide infinity by infinity etc., we can make properties so that kids don't get mixed up and do that.

[Hurkyl]

any vertical line had a slop that was 'undefined'. ... I mean, I remember back then, I questioned the this, as I thought it should have a slope of infinity.
If you generalize the notion of slope, it turns out the projective real numbers are the 'right' number system to use to measure slopes. And then, a vertical line does indeed have generalized slope equal to projective infinity.

Just for emphasis, this is only true for this generalized notion of slope -- it is perfectly correct to say that the ordinary notion of slope is inapplicable to a vertical line.


But my point is that, with the right properties, infinity can be implemented into algebra.
Have you not been listening? Not only can it be 'implemented' in algebra, it has.

I was only talking about infinity.
Which 'infinity' or otherwise infinite number are you talking about? Or do you even know what you're talking about?

We don't have uses for those, and other, indeterminates anyways. They lie off of any number line.
They can't lie off any number line -- they don't even exist. :tongue: (Assuming Take_it_Easy was referring to projective infinity, or the positive extended real infinity)

[Take_it_Easy]

I was only talking about infinity. We don't have uses for those, and other, indeterminates anyways. They lie off of any number line.

Edit: If your post was directed at the fact that kids might get into trouble by trying to divide infinity by infinity etc., we can make properties so that kids don't get mixed up and do that.


I see.
In fact it is a possible and logic approach.

We learn to be careful (that the divisor is 0) when a division occur, so we can just introduce the propertyies for infinity and warn to be careful when doing such an operation.

By the way, we have to be always careful in every elementary operation.
But that's not bad after all!

[epkid08]

Have you not been listening? Not only can it be 'implemented' in algebra, it has.

Please post an example of usage of numerically defined infinity in algebra.

[D H]

There are several examples throughout this thread, starting from the very first two responses.

[LukeD]

Please post an example of usage of numerically defined infinity in algebra.

Extended reals, projective reals, projective complex numbers, ordinal numbers, cardinal numbers, hyperreal numbers, surreal numbers

need any more examples?

[matt grime]

YOu should't leap in with such counter examples before the OP has defined what he thinks 'numerically defined' means. I have no idea what that phrase indicates.

[epkid08]

al·ge·bra [al-juh-bruh]
–noun
1.the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations.

This is the definition I used as algebra. Infinity numerically defined would be n/0.

[n_bourbaki]

In that case, the extended complex plane would be the (or an) example you seek. As has been pointed out several times in this thread.

[epkid08]

In that case, the extended complex plane would be the (or an) example you seek. As has been pointed out several times in this thread.

part of the definition of your suggestion-
On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field.

If it doesn't fallow the rules of algebra, then it's not true algebra.

[n_bourbaki]

Oh, what are the precise definitions of "algebra", whatever that nebulous concept is supposed to be? Not that it matters - it is clear that nothing anyone will say will dislodge your preconceptions.

[D H]

al·ge·bra [al-juh-bruh]
–noun
1.the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations.

This is the definition I used as algebra.
That is a dictionary definition. It has little meaning to a mathematician.

Infinity numerically defined would be n/0.
No. There are many ways in which division by zero will get you in a lot of trouble, which is why division by zero is not defined in any of the systems described many times in this thread that incorporate the concept of "infinity" as a number.

If it doesn't fallow the rules of algebra, then it's not true algebra.
What, pray tell, are "the rules of algebra"?

One big problem with making "infinity" a number is that the result is an algebraic structure with a lot less to it than the reals. The reals form a field. The extended real number line doesn't even form a ring.

[Hurkyl]

There are many ways in which division by zero will get you in a lot of trouble, which is why division by zero is not defined in any of the systems described many times in this thread that incorporate the concept of "infinity" as a number.
Correction -- dividing a nonzero number by zero is defined in the projective complex numbers (and projective 'anything'), with value projective infinity. Wheels even allow 0/0.

Of course, this is only if you interpret the '/' symbol as denoting that structure's division operation. If we instead interpret '/' as being integer division, then 1/0 is still undefined, even if we want to work in the projective complexes.

[okkvlt]

Something ive been thinking is

if 0^-1 is infinity, then infinity^-1=0, right?

It actually makes alot of sense, both intuitively and algebraicly

[HallsofIvy]

No, they don't make sense for exactly the reasons everyone has been giving, didn't you notice?

[throng]

infinity is strange. it destroys alot of simple algebra if given real qualities. Would i be wrong to say infinity can be infinitely large or infinitely small but doesn't take any other value.

[karthikgnv]

hi dude...if you figure out how a division works..u can always sort the mess you have about the infinity
.
.
.lets analyse how a division works.....

for eg: take 10/2.....the answer is 5
now...what the number 5 tells you about???
it tells you that..."If you repeated ly substract number 2 from number 10 until zero comes, you should do it five times"...after all division is a repeated substraction
.
.
applying the same for your doubt......
if you take 5/0 the answer is obviously INFINITY
.
.
.
because you can remove the number 0 from the number 5 countless times and still you cant do that until your number 5 becomes zero...
.
.
.
thats why you cant say that 5/0 is equals to 1/0
.
.
because removing 0 from 5 is different from removing 0 from 1
.
.
.
.
is that clear???
is my explanation convincing???
let me know plz...
Thankyou

[Hurkyl]

Would i be wrong to say
You would be better off to read the comments that mathematicians have made in this thread (and in other threads you could find by searching for 'infinity').

[karthikgnv]

hello....Hurky..is that correct..which I wrote abt how we should understand the notation of infinity????

[throng]

I'm toying with the total quantity of energy. it wasn't created (always was) can't be destroyed (always will be) so seems infinite in terms of "eternity". All things are "made of" energy. if energy is everything is its total joules infinite?

I would really appreciate any advise in this regard.

[HallsofIvy]

1. You are assuming a lot of things that are not obvious. We say that something is "infinite" if it is unbounded right now. That has nothing to do with "eternity". The fact that the total amount of (mass-) energy in the universe doesn't change does not imply that it is "infinite".

2. How does this have anything to do with the topic of this thread?

[Hurkyl]

hello....Hurky..is that correct..which I wrote abt how we should understand the notation of infinity????
No, it is not.

[jmatejka]

I think "Infinity" is a useful human defined concept, NOT a reality.

[Office_Shredder]

If anyone wants to divide 1/0, before proceeding to talk about infinity, give a clear and lucid explanation as to why it isn't negative infinity instead.

[CRGreathouse]

If anyone wants to divide 1/0, before proceeding to talk about infinity, give a clear and lucid explanation as to why it isn't negative infinity instead.

Two systems I can think of where that would make sense are the projective reals (or complexes) and a system defined intuitively over the positive reals, zero, and infinity. Neither supports 0/0 or infty/infty, though.

[Hurkyl]

Wheels (http://en.wikipedia.org/wiki/Wheel_theory) are a mild generalization that allow all divisions. (Although, they're not in common use, AFAIK)

[csprof2000]

For my money, zero is a lot like infinity, and all finite numbers are a lot like 1.

[HallsofIvy]

Well, considering that 0 IS a finite number that pretty much means that you consider all numbers as "a lot alike"!

[csprof2000]

How can you call 0 finite?

Have you ever given it any thought, or are you just parroting conventional wisdom?

[Hurkyl]

How can you call 0 finite?

Have you ever given it any thought, or are you just parroting conventional wisdom?
How about "because it satisfies the definition of the word 'finite' "? :tongue:

[csprof2000]

http://www.thefreedictionary.com/finite

a. Being neither infinite nor infinitesimal.
b. Having a positive or negative numerical value; not zero.
c. Possible to reach or exceed by counting. Used of a number.
d. Having a limited number of elements. Used of a set.

I would sooner say that zero is neither finite nor infinite than to say it's finite.

Just like zero isn't really positive or negative, you know... it's not a matter of conventional definitions, or even mathematics, per se, but of philosophy. I think that there are various ways one can interpret the number zero, and under some interpretations it's just as finite as 1 or 2 while under others it's just as infinite as... well... infinity.

[Hurkyl]

Ah switching definitions in the middle of a discussion, always a sneaky move to play in a game of semantics. :tongue:

The only time I have ever seen "finite" actually used in a way to exclude zero when one needs a convenient way to speak of things that are "not infinitessimal", in which case one makes an abuse of notation by repurposing 'finite' to refer to things that are finite, but not infinitessimal.

And some prefer to avoid such abuse of notation, using words like 'nonvanishing' or 'nonnegligible' instead of changing the meaning of 'finite'.

[csprof2000]

To be fair, you pretended you knew what *I* was saying when I said what I did, and I never made any representation that I believed in your version of reality. Plus, given the definition I provided from a (possibly less than reputable) third party, it seems a fair enough issue to give me the benefit of the doubt.

How would you define "finite", exactly? How do rigorous texts on various subjects define these things? Zero is certainly a peculiar finite number, if you consider it to be such. No sign, no multiplicative inverse, and semantically meaning the absence of quantity. Circles of radius zero are degenerate, triangles with one side's length equal to zero are degenerate, etc.

It also shares with infinity the property that multiplying by it gives itself, and it is in a very abusive and intuitive way a sort of 1/infinity.

You could do with zero much the same things as has been done with infinity in conventional, everday, run-of-the-mill math, where infinity isn't a number, etc. It would require a little reformulation, but you'd have to lack any imagination to see that it isn't so fundamental as, say, 1, or any other finite number for that matter (why would there be a "hole" anywhere if not at some symmetric or absolute place, like zero of infinity?)

I would imagine that one could also go off into fantastical reveries (much like has been done with infinity) and explore various orders of zero, and start talking about how some zeroes are smaller than others, etc. What are infinitesimals if not a kind of zero?

Anyway, done with this rant. Jeez, you guys take things so personally sometimes. It's not about trying to say you're suckers for seeing things one way.

[Hurkyl]

To be fair, you pretended you knew what *I* was saying when I said what I did
One of the most basic semantic conventions is that when a particular word has an established usage, that anyone using that word unqualified means that usage.

I never made any representation that I believed in your version of reality
Reality? :confused: What does that have to do with anything?

Plus, given the definition I provided from a (possibly less than reputable) third party, it seems a fair enough issue to give me the benefit of the doubt.
General purpose dictionaries good for defining words in every day usage. They're notoriously bad at defining technical words. (After all, their purpose is the former, not the latter)

How would you define "finite", exactly?
Depends on the context. When dealing with sets with an ordering and contain integers, by far the most typical definition is:
x is finite if and ony if it lies between two integers
or something equivalent; for example, I would be entirely unsurprised to see a textbook define a finite extended real number simply by "it's neither +\infty nor -\infty".

When dealing with sets, the typical definition is
S is finite if and only if there is a 1-1 correspondence between S and a bounded interval [0, n) of natural numbers, for some natural number n
(Or something obviously equivalent) (Note that [0,0) has a 1-1 correspondence to the empty set)
And for cardinal numbers,
A cardinal number x is finite if and only if it is the cardinality of a finite set
Or, sometimes, I've simply seen it defined by the equivalent statement that a cardinal number is finite if and only if it's a natural number.

Zero is certainly a peculiar finite number, if you consider it to be such. No sign, no multiplicative inverse,
Every number has its own pecularities.

and semantically meaning the absence of quantity.
Nononono. First off, it can only possibly have any relation to the idea of quantity in the particular case we are using a number to quantify something. Quantification is not inherent to the mathematical notion of number.

Secondly, a quantity of zero is not the "absence of quantity". After all, if the quantity is zero, then there is certainly a quantity involved. :tongue:

"The number of coins in my pocket" is a quantity, and that quantity can be zero.
"Blue" is not a quantity. It would be nonsensical to say "Blue" is zero.

Don't confuse yourself by the fact natural language has evolved to special-case zero.

I would imagine that one could also go off into fantastical reveries (much like has been done with infinity)
:rolleyes:

What are infinitesimals if not a kind of zero?
Nonzero. :tongue: (Actually, zero is an infinitessimal. All other infinitessimals would be nonzero)


Anyway, done with this rant. Jeez, you guys take things so personally sometimes. It's not about trying to say you're suckers for seeing things one way.
You claim to be a CS professor... what if I came into your class and tried to tell you that 1 is not O(x), or that the halting problem was computable? And then when you corrected me, I simply accused you of mindlessly 'parroting conventional wisdom'?

[csprof2000]

"You claim to be a CS professor... what if I came into your class and tried to tell you that 1 is not O(x), or that the halting problem was computable? And then when you corrected me, I simply accused you of mindlessly 'parroting conventional wisdom'?"

I think that unless you are an unreasonable human being, you will admit that what we are discussing here is not as open-and-shut as what you are saying. If I were arguing that 7x + 2y = 3 was a parabola and not a line then I would be in trouble. That's essentially the level of discourse you're suggesting in CS.

Definitions of vague notions are always open to interpretation. If you came into one of my classes and suggested any number of "vague" definitions in computing may be flawed (or, if not flawed, perhaps able to be improved upon) then I'd be willing to have a legitimate discussion.

It seems dogmatic, to me, to pretend that the current definition is necessarily the right one. Other definitions are sometimes possible (even for some things in mathematics!) Is that really so hard to stomach?

[Colin1]

...Infinity is a special class of number...
How so?
Surely to call it a number, even a special class of number, is to imply that infinity is numbered?
Isn't infinity more of a concept, something that we use to help us get by in everyday technical or mathematical situations?

A most elementary case in point: A circuit designer with a 100megohm resistor in series with a 1ohm resistor can safely disregard the smaller resistor; the 100megohm resistor would appear infinite in value and thus precluding the 1ohm resistor from circuit calculations will not affect the outcome. This does not imply that the larger resistor is infinite in resistance (good luck with the current value if it is), simply that from the point of view of the 1ohm resistor, it can be regarded as such.

Infinity explained as I see it, anecdotally:
A bus carrying an infinite number of passengers pulls up at a hotel with an infinite number of rooms. The driver asks the hotelier if he has room for all of his passengers, to which the hotelier replies "of course, easily, we have an infinite number of rooms".

Later, another bus carrying an infinite number of passengers pulls up at the same hotel and asks the same question, to which the hotelier replies "of course, easily, we have an infinite number of rooms".

So how'd you do that, if the infinite number of rooms are already taken by the first infinite influx of bus passengers?

I'm a simple engineer, not a high-browed mathematician, but I don't believe you can deal with infinity like a number (ie numbered); infinity is a concept.

[Hurkyl]

It seems dogmatic, to me, to pretend that the current definition is necessarily the right one.
Just what do you mean by "right"?

If I am talking about a mathematical notion characterized by a particular definition, and you are talking about some other mathematical notion characterized by an inequivalent definition, then we are talking about different things, plain and simple. There is no dogma involved.


If that's not what you meant -- instead, you were just questioning whether my estimation of the typical usage of the word "finite" is accurate -- I'm simply going to have to invoke authority here. I can also offer a few citations:

* Keisler's calculus textbook (http://www.math.wisc.edu/~keisler/calc.html) defines "finite" so as to include zero and all infinitessimals

* Wikipedia's disambiguation page (http://en.wikipedia.org/w/index.php?title=Finite&oldid=268507083) which is consistent with my description of how the word is used

* I can cite a few texts that deal with sets in an incidental fashion, which clearly consider the empty set a finite set.


Other definitions are sometimes possible
Of course. For example, each of the definitions I gave earlier -- "finite set", "finite cardinal number", "finite extended real number", "finite point of the projective plaine" -- those are all different definitions of different things.

[csprof2000]

What I meant to say, and I apologize if this was actually unclear, is that sometimes the currently accepted definition is not the final word in the definition. Perhaps a better, more useful definition is possible. I think I'm done with this thread.

[CRGreathouse]

To be fair, there are also uses (convergence of infinite products, for example) where 0 is not considered a finite number. But the usual definition of finite number, to me, is "between two integers" which of course includes 0.

[arildno]

What I meant to say, and I apologize if this was actually unclear, is that sometimes the currently accepted definition is not the final word in the definition.
Final word??
THIS is what you actually said:
How can you call 0 finite?

Have you ever given it any thought, or are you just parroting conventional wisdom?
This is very clear. Here, you are advocating the ideas that:
1. It is FALSE to call 0 a finite number
and
2. That the evil "establishment" somehow have brainwashed people into thinking that 0 might be a finite number.

There is nothing ambiguous in what you wrote, and your last post is just a shameful cop-out.
Perhaps a better, more useful definition is possible.
As in your previous empty blather?
I think I'm done with this thread.
how considerate of you.

[matticus]

This is the most bipolar thread I've ever seen, intensely funny and depressing. Everyone knows more about math than the mathematicians. Especially CS people (no offense to those who actually take the time to understand the concepts they are abusing, if you exist). The last time I read a post by CSProf he was trying to convince me that the reals are countable!

[Unknot]

Hasn't people yet learned that mathematics is not science and we somehow have "arbitrary definitions"?

Do they know the definition of what definition is?

I think we defined things way they are because... because... damn I cannot remember. But I am sure there's a reason! And I am sure they will keep discovering reasons why it should be wrong! [/sarcasm]

[Winzer]

I suggest that we adopt the following rule:
\frac{\Infinity}{\Infintity}=1

[HallsofIvy]

You mean as in

3237+ 2343= 1

213/234= 1

e3243= 1?

Certainly would simplify arithmetic!