Infinite infinities

[MrMongoose]

Maths doesn't let you subtract infinities as infinity is not a number and subtraction is an operation that acts on numbers.

[Fred56]

Math does 'let' you do this. You should take a closer look at the posts before yours.

There are an infinite series of real values between 1 and 2. There are also an infinite series (of real values) between 2 and 3, (with all values real numbers). So what's the difference between them (is it zero or something else)? Although they are bounded differently, the intervals are the same size...

[D H]

No, it does not let you subtract infinities. Subtraction is not defined for the cardinal numbers. You can't even subtract aleph_null from aleph_null. For example, the disjunction of the integers and the even integers is the odd integers. All three sets have the same cardinality, aleph_null. Problems arise no matter how one tries to define subtraction of cardinals. So mathematicians explicitly leave subtraction of one cardinal from another undefined.

 

Subtraction of one cardinal from another is not needed or used to enable someone to say that two sets have the same cardinality or that one set has a smaller cardinality than another.

[MrMongoose]

Precisely.

 

Also I don't see what the difference between two infinite series has to do with summing a made up number called infinity.

[Fred56]

it does not let you subtract infinities

How do you guys get a series of (potentially) infinite terms to converge? Or are mathematicians who do this deluding themselves, maybe? Alternatively, are you claiming that an indefinite integral is an illusion of some kind?

[D H]

Limits have absolutely nothing to do with subtracting infinities. What in the world makes you think that there is some connection between the two? The concepts of limits is very well-defined mathematically. The concept of subtracting cardinal numbers is intentionally not defined. One last time, subtraction is not defined for the cardinal numbers.

[Luminal]

No evidence for the infinite has ever been brought forward to the extent of my knowledge.

 

Everything is finite in reality, and infinity is just a tool that humans use to approximate trends (such as a series).

 

Sure, you could say that traveling on a circle is infinite, but the circumference is finite and no traveler could ever travel an infinite period on that circle.

 

In short, infinity does not exist, but it is useful to extend our calculations to trends way beyond what would be feasible to manually measure.

[Fred56]

Then calculus is surely in big trouble, if there is "no evidence" infinity actually exists.

This must mean that an indefinite integral is impossible (except I do it all the time, so do lots of other people, what do you think they should be told about this stunning revelation?)

The concepts of limits is very well-defined mathematically

Really? Does it include a definition of, um, an infinite series, by chance?

What brought you to the conclusion that I am saying subtraction and integration are 'connected'?

And why is there no response to my query about infinite series that converge (which must be impossible, according to your objections)?

And you appear to be saying that [math] (2-1) -(3-2) \neq 0 [/math]

[Luminal]

Then calculus is surely in big trouble, if there is "no evidence" infinity actually exists.

This must mean that an indefinite integral is impossible (except I do it all the time, so do lots of other people, what do you think they should be told about this stunning revelation?)

 

Really? Does it include a definition of, um, an infinite series, by chance?

What brought you to the conclusion that I am saying subtraction and integration are 'connected'?

And why is there no response to my query about infinite series that converge (which must be impossible, according to your objections)?

And you appear to be saying that [math] (2-1) -(3-2) \neq 0 [/math]

 

Math is a collection of symbols used to represent something (hopefully something in reality) to humans or human devices. It doesn't actually exist.

 

Because infinity can be represented in mathematics is not somehow evidence that it exists.

 

Integrals... let's use those as an example since you mentioned them.

 

Those are approximations to the area under a curve. If you kept making rectangles smaller, even with unlimited technology at your disposal, they would reach a physical barrier (such as the Planck length) in which nothing could be broken down further and everything was discrete. This would be where you had a inconceivably large number of small rectangles, not an infinite amount.

 

Thus, as I said, no evidence for infinity has even been brought forth; it is only a concept to help approximate solutions to near perfection. If an experiment has ever been performed that relies on an infinite number of anything in reality, or an infinite span or measurement of any kind, I'd be greatly interested to hear about it.

 

"Approaching" infinity being the key word in all of these calculus concepts.

[D H]

Really? Does it include a definition of, um, an infinite series, by chance?

 

If you claim to know how to perform integration, certainly you know about Weierstrass limit theory? The infinite series [math]\sum_{r=0}^\infty a_r[/math] converges to some value [math]S[/math] if the sequence of partial sums converges to [math]S[/math].

 

More formally, define the n^th partial sum as [math]S_n = \sum_{r=0}^\infty a_r[/math]. The series converges to [math]S[/math] if for every [math]\epsilon>0[/math] there exists some integer [math]N[/math] such that [math]|S-S_n|<\epsilon[/math] for all [math]n>N[/math]. Note that the word "infinite" is not used anywhere in the formal definition of convergence of an infinite series (or of an infinite sum, which is just a special kind of infinite series). There is no addition of infinities, let alone subtraction.

 

 

What brought you to the conclusion that I am saying subtraction and integration are 'connected'?

I said

Limits have absolutely nothing to do with subtracting infinities.

in response to this exchange,

No, [ math '] does not let you subtract infinities.

How do you guys get a series of (potentially) infinite terms to converge? Or are mathematicians who do this deluding themselves, maybe? Alternatively, are you claiming that an indefinite integral is an illusion of some kind?

 

It is appearing that you are being intentionally obtuse.

 

Back to your last post,

And you appear to be saying that [math] (2-1) -(3-2) \neq 0 [/math]

 

I take it you don't know what the cardinal numbers are. The cardinal numbers include not only 0, 1, 2, but also things like [math]\aleph_0[/math]. Both wikipedia and mathworld have fairly good entries on the cardinal numbers. One last time, subtraction is not defined for the cardinal numbers.

[MrMongoose]

How do you guys get a series of (potentially) infinite terms to converge? Or are mathematicians who do this deluding themselves, maybe? Alternatively, are you claiming that an indefinite integral is an illusion of some kind?

 

I'm sorry, but how does "the sum of a finite number of infinite numbers" mean the same thing as "the sum of an infinite number of finite numbers"?

 

There is no problem with calculus, because while the sum of a finite number of infinite numbers doesn't exist, the sum of an infinite number of finite numbers does exist, so can you please stop arguing against a point that I didn't make?

[Fred56]

I believe, nonetheless, that there is a mathematical definition of infinity. This is the definition used in calculus, and defined by the concept of limits (of an infinite series).Series can converge or diverge all by themselves, right? The infinite series of fractions that add up to the real value 1, for example, how do you explain that this is (actually) a set (of fractions) with infinite cardinality, but a finite limit, neither of which needs to consider the 'physical' limits of measurement (the Planck constant or the wavelength of light), because we can conceive of something smaller than the smallest physical 'thing'?

It's a bounded set. So infinity (as you already know) can be bounded, as discussed previously.

I think you are splitting philosophical hairs.

Of course, actually infinite values are undefined, or unreachable...

Your statements so far lead me to think that there must be an objection to the following claim:

"The set of natural numbers is defined as positive integers (which, according to the definition used includes or excludes zero), and are used for counting. Since we can count backwards (the concept of subtraction), then either we are subtracting positive natural numbers from some set, or we are adding negative natural numbers."

Is there a difference? Is there a genuine philosophical or mathematical objection to claiming that natural numbers can be negative (as well as positive) integers?

[uncool]

Usually, the definition of infinity actually doesn't define infinity itself - it defines limits to infinity, instead. However, in some non-standard structures, there's a definition of infinity - it's just an extra point on the extended real line. Or it's a point in the projective real line. Or perhaps it's the compactification of a topological space...

=Uncool-

[MrMongoose]

What I'm getting at is that you can't say anything is equal to some number called infinity. i.e 1+2+3+4+...=infinity. It would however be correct to say that 1+2+3+4... tends to infinity, but thats not saying that there is some number infinity which the sum gets closer to everytime, it's just strange terminology to say the series gets bigger.

 

On the other hand, the adjective infinite and adverb infinitely are quite unambiguous and useful for describing limits.

[Fred56]

But infinity isn't one thing, its an infinity of things, there are an infinite series of alephs, an infinity of limits, yes? Unfortunately (or whatever), we use the same word for all of them.

Or maybe someone can explain transfinite sets...

[MrMongoose]

Doesn't each series have one limit?

 

Unless you're talking about an infinite number of series, im afraid you've lost me.

[Mr Skeptic]

I think that there's two kinds of infinity: the one used in limits to mean "as the number gets arbitrarily bigger and bigger", and the ones used in sets that specify just how hard it is to "count" sets.

[Fred56]

I like to think of something that is actually infinite, as a set of numbers to which no number can be added, because there is no number we can add which is outside an (actually) infinite set. Such a set already contains any number we might think of.

Then there's the infinite series of numbers which approaches, but doesn't become infinite, which we say has a limit at (a mathematical) infinity.

 

Limits have absolutely nothing to do with subtracting infinities

Details, details:

 

There is, one more time, an infinite number of real values (points coinciding with the real number line) between any adjacent 'whole' numbers, on the same line. The interval between any two adjacent real values in the set between such whole values (1 interval), might not be equal to the first or last such interval, but if all the intervals (which are ordered, or have ordinality, along with their infinite cardinality), are added together, a whole value is 'restored'. This addition, or summation, can be done between any two such 'whole values', which cannot otherwise be constructed (starting with real values), but must necessarily be defined (like the single real interval is) first, as particular, but arbitrary points. Therefore this is the idea of two limits (of infinite series) being added or subtracted (or multiplied or divided).

 

Doesn't each series have one limit?

Indeed, each particular series (of which I assume there is an infinite variety) can have only one limit, more exactly, taking a particular limit can have only one result. This is our concept of unity. Perhaps Elvis has left the building after all (to see his hairstylist)...

[D H]

Fred, part of your confusion lies in the fact that the term "infinite" has multiple meanings. A quantity is infinite (or unbounded) if no finite number can act as an upper bound on the absolute value of the quantity in question. Think of the "point at infinity". A set is infinite if a proper subset of the set can be placed in one-to-one correspondence with the set itself. This meaning is quite distinct from the first. The set of real numbers in the closed interval [ 0,1 ] is a bounded infinite set.

 

There is, one more time, an infinite number of real values (points coinciding with the real number line) between any adjacent 'whole' numbers, on the same line. The interval between any two adjacent real values in the set between such whole values (1 interval), might not be equal to the first or last such interval, but if all the intervals (which are ordered, or have ordinality, along with their infinite cardinality), are added together, a whole value is 'restored'.

 

Try that again. What you said doesn't make much sense.

 

I think what you are trying to say here is addition of cardinal numbers can be defined. This is true. Addition of cardinal numbers is a well-defined operation. For example, [math]10+\aleph_1 = \aleph_0+\aleph_1 = \aleph_1 + \aleph_1 = \aleph_1[/math]. Note well: There is no unique cardinal number [math]x[/math] such that [math]x+\aleph_1 = \aleph_1[/math]. This lack of uniqueness means subtraction cannot be defined for the cardinal numbers.

[bob000555]

To say that subtraction isn’t defined is not accurate, it is how ever proper to say that subtraction does not exist, nor division for that mater: to subtract is to add a negative number, to divide is to multiply by a subtraction (it is true that you could reverse this and say that addition is subtraction by a negative number). Well now that im done with the longest runon sentence in the history of my writing.

[Fred56]

OK, well it looks like it's all about what someone who has done set or number theory (which isn't seen as a necessary goal when learning about "infinite" series or limits in calculus), has to say about the infinite sets and the transfinite numbers. I have no real concrete idea what Cantor means by "transfinite", except that it looks like another word for infinity.

Really you don't need to go beyond the ideas of the integers extending (countably) to +- "infinity", or some limit, and the idea of real numbers (an always larger set), being uncountable. Beyond that it's pretty much custard all the way.

What I was getting at, in my own inimitable fashion, was that the idea of a series, or list (which is what a set of real values along the number line is), must have finite, real intervals between them, and adding them together (or just saying that one of them is a whole number, since any interval also contains an infinite set of points with an infinite set of intervals), must make some 'bigger' value. Integrate some interval between any two points (by adding all the intervals together), and call it '1'. then double it to get '2', or subtract it to get '0'. Is this making any more sense?

[MathNerd]

hehe i got headaches from this topic

[1bobwhite]

The concept of infinity in mathematics has of necessity to be constrained to a set in order for finite numbers to be applied in the formulas, and have any meaning in the resultant answers.

 

Infinity is not null set, or undefined as is Zero, but a constraint on the endless progression, for the formula math to have a definable result.

 

Numbers such as PI, or ones that are endless such as 12.666666.... to "infinity" are truncated for ease of use. After a determined amount, the remainder is of no value to the degree of accuracy required.

 

For that matter all math that uses the concept of infinity is an approximation of the actual. Its is a matter of determining the level of accuracy required for the solution.

 

Any manipulation of infinity by addition, subtraction, multiplication, or division, will give the same result unless it is constrained to a set, and thus limited to a resultant that may now be manipulated.

 

In the physical world the infinities are distance and position from the observer. All matter and energy has been determined to be "countable" quanta with a space reference for both position and time. The infinity of distance is what is beyond the observable.

 

The possible positions within the observable range, are supposedly infinite however with physical units the actual range of possible positions may not be infinite but constrained by their dimensions and their reactive behavior. Most of these actual numbers however are beyond our practical mathematics, and are just treated as infinity. For instance the number and frequency of collisions of molecules of a gas against a vessel wall that gives us a "pressure".

 

Bob