Questions about the logic of infinity

n_kelthuzad

I asked a question related to infinity a few weeks ago, but the answer I got really lead me to a confusion. Is there any way, that infinity can be compared in another plane or whatever. So here is something paradox if you treat infinity as it is in the set of real numbers.
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if there exists a number ∞, than the number must equal to it self.
so "∞ = ∞" ( this and all the following are one-sided equations)
if ∞ has a 'logical maximum value' of a, than ∞=a
however, a is bigger than any other numbers in this plane. Then (a+1) must have the same properties as a.
so ∞ also equal to (a+1)
however if a=a+1, 0=1. such a number does not exist.
or for ∞/∞ [itex]\neq[/itex] 1:
a+1/a > 1 and a/a+1 < 1.
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so the number ∞/∞ (or all the other primary calculations of ∞, and 0/0) must be 'any real number' and also 'not a fixed number'. this leads me to the thinking of a higher dimension of the numeral system. Is there any way other than limits that can describe different ∞'s?
You cannot put ordinary logic into ∞, since the concept or property of infinity is:
A NUMBER THAT CAN NEVER BE PHYSICALLY REACHED.
however when I look some series:
e.g. the harmonic series 1+1/2+1/3...
and the "harmonic + 1" series that I just made up in mind
2+3/2+4/3+...
both series diverge into ∞.
however, the 2nd series is bigger than the first one, in n terms, the difference is (1n).
Does this provide a way to compare infinitys as the difference between series in a finite number of terms?

Thanks
Victor Lu, 16
BHS, CHCH, NZ

Number Nine

The short answer is no. The long answer is no with a lot of elaboration. What you mean to say about those series' is that the partial sums of the series grow without bound as the number of terms increases. All this means is that both series will eventually surpass any positive real number x. The second one is "greater" than the first only in the sense that, for any n, the sum of the first n terms of the first series is greater than the sum of the first n terms of the second. It says nothing about "infinity".

There are, however, different "sizes" of infinity that arise in set theory and have nothing to do with what you're describing here.

micromass

Indeed it is a paradox. The solution is of course that infinity is NOT a real number. So you can't work with it like you work with real numbers.

Furthermore, things like [itex]\frac{\infty}{\infty}[/itex] are undefined.

See the FAQ: http://www.physicsforums.com/showthread.php?t=507003

chiro

There is not really a paradox because infinity is not a normal number or quantity in the way that we are used to and expect quantities to be.

I you are treating infinity like a standard number, then it means that you have not understood infinity and what it represents.

n_kelthuzad

Yes, I understand what ∞ means, that was just a proof.
and as I said, infinity is logically impossible to approach, so does that make 1^∞=e true, since the calculation is not logical, one cannot say the answer is logical.
(1+1/m)^m=e

chiro

It's not just in terms of a measure like a number, it's a concept.

Infinity doesn't just relate to quantities, it relates to a higher understanding that helps us understand convergence, structure, and things that are a product of these two things.

The fact that we consider say a vector space with infinite dimensions but having properties that 'make sense' (like norm convergence or Cauchy-completeness) is one way of understanding infinity. These kinds of situations help us think about how we can really and truly make sense of something that is really hard to define in the first place.

jgens

I am really not sure what you mean by this statement. In ordinary language, logical tends to mean a very different thing than it does in mathematical language. There are plenty of axiomatic set theories (ZFC for example) which are strong enough to have very well developed theories involving infinity (like the theory of ordinal numbers in ZFC). That infinity may not be physically realizable -- whatever you define that to mean -- is irrelevant to whether or not theories involving infinity in mathematics are logical.

n_kelthuzad

sorry my fault cause I am still in high school so a lot of advanced math things are still a blur to me. And I do not know how to speak proper mathematical language.
Anyway:
a[itex]\sqrt[]{}[/itex]n , n[itex]\subset[/itex]ℝ;n>0
as a gets bigger , a√n converges to 1;
so can this be true:
∞√n = 1?
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when I look at another equation: n/∞=0
I noticed that no matter how many times 0 multiplys itself, it will always be 0 - except ∞;
so as for the above equation:
1*1*1*1*1*1*1*1*1*1... will always equal to 1 when the terms are finite;
however when infinite terms, can it be any number(n)? just like n/∞=0.

jgens

Your notation is really poor, but it looks like you are noting that [itex]\lim_{n\to\infty}n^{1/n} = 1[/itex] and asking that if we fix [itex]t \in \mathbb{R}[/itex] such that [itex]0 < t[/itex], then is it true that [itex]\lim_{n \to \infty}t^{1/n} = 1[/itex]. The answer to that question would be yes.

This is the problem with reasoning with [itex]+\infty[/itex] as a real number. It does not work. You should read the FAQ micromass posted earlier: http://www.physicsforums.com/showthread.php?t=507003

logics

A couple of simple questions:

1) ∞ is not a real number, is there a definition of what it is?, ∞ [itex]\notin[/itex] ℝ, ∞ [itex]\in[/itex] ?, (I suppose [itex]\hat{ℝ}[/itex][itex]\bar{ℝ}[/itex] are systems, not sets?)
b) probably your answer to 1) will explain the fact that operations are possible
2) The link shows that all operations are possible except division, why so?
3) In physics ∞ non datur, why we need this concept in maths? What happens if we do without it?
Thanks,
Edit: what if we chose L = (10¹⁰)¹⁰¹⁰ ?

micromass

Depends on what you mean with ∞. If you mean the extended real numbers, then [itex]\infty\in \overline{\mathbb{R}}[/itex]. And yes, [itex]\overline{\mathbb{R}}[/itex] is a set. In mathematics, everything is usually a set (or has a set theoretic description).

Possible is a bad word. Everything is possible, you just need to define it. If we wanted to do division by zero, then this is possible as well, we could just define it as 0/0=54 for example. However, there are good reason not to define division by 0 or division with infinities. The reason usually is that not everything will behave nicely if we define such things. Why is that??
Well, extended real numbers are being defined to be able to be compatible with limits. For example, if we want to calculate

[tex]\lim_{x\rightarrow 1}{x^2+5}[/tex]

then this is easy, we just need to substitute x=1 and see what we get. We want to do the same thing with

[tex]\lim_{x\rightarrow +\infty}{2x+5}[/tex]

And indeed, if we define [itex]2\infty=\infty[/itex] and [itex]\infty+5=\infty[/itex] then we indeed do get the right answer. However, if we dare to define [itex]\infty/\infty[/itex] then this will not be compatible with limits anymore. For example

[tex]\lim_{x \rightarrow +\infty}\frac{x}{x+1}~\text{and}~\lim_{x \rightarrow +\infty}{\frac{2x}{x+1}}[/tex]

Substituting [itex]x=\infty[/itex] in both expressions both yield [itex]\infty/\infty[/itex]. But the first limit would indicate that this equals 1, and the second would indicate that this equals 2. So since any definition of [itex]\infty/\infty[/itex] is incompatible with the limit situation, we prefer not to define that expression.

However, you might add, there is a difference. In the first limit, the numerator is [itex]\infty[/itex] and in the second it is [itex]2\infty[/itex]. So we are not working with the same infinity here!! OK, but we defined earlier that [itex]2\infty=\infty[/itex], so we are working with the same thing here.
However, this observation is a valid one. Perhaps we could define a system such that [itex]2\infty[/itex] would not equal [itex]\infty[/itex] but that would give us the right answers every time?? Such a system is possible. When working with the so called hyperreal or surreal real numbers, then this can be done. But that's an entirely different story

The ∞ of the extended real numbers is not needed in mathematics at all. I think that most result in mathematics can be accurately described without using infinity. However, using ∞ is preferable as it makes our life much easier most of the time. If we did not use this, then we would have to define various special cases every single time something becomes "unbounded". So can we do without it? Yes, but it would be annoying.

Note that there are multiple kinds of infinity possible in mathematics. Some infinities would be very hard to do without. For example, if we were to work without any notion of infinity, then most of math would fall down. Indeed, a "simple" structure such as [itex]\mathbb{N}[/itex] is already infinite and would have to be prohibitied.

logics

ℝThanks, micromass, for your (as usual) clear replies.

Before I can make further questions, please clarify this 'question of principle':

you also refer to 0 in another post and say definition of 0 as a 'number' is so arbitrary you might call it baz[albieba, no difference!. That is perfecly all right logically and semantically: 'baz', 'number', 'cod' can be considered just a 'sign/scribble' and not a Linguistic_sign.

But if you forget it and then say 0[itex]\in[/itex] N , I may say, by the same logical disconnect, 0 is a fish, (baz is meaning-less, ergo, no set).If we assign an element to a set, shouldn't it share all properties of that set?
If we say 0 is a number, (or ∞ [itex]\in[/itex] [itex]\bar{ℝ}[/itex]), shouldn't it always behave like a number, (or to be able to swim like any other cod)? At least, you ought to make a subset with limited properties, but brobably that is not enough.
Where do I go wrong?
Thanks a lot.

micromass

What do you mean with "share all the properties of the set" and "behave like a number" in the first place??

logics

If I say "Socrates is a man", authomatically I assume he shares ALL attributes of a man, including being mortal. I deduce: if "Socrates is a man" → "he is mortal"
I cannot say ,
" He is a man but is immortal". right? If that is right.
Then you cannot say : definition '0 is a number' is arbitrary and then say 0 [itex]\in[/itex] ℝ, but 0 shares 'some' properties of the reals, then I make a subset in ℝ.

Mark44

logics, I think you missed micromass's point. In your previous post in this thread, you gave a link to a different thread (http://www.physicsforums.com/showthread.php?p=3362301). Here is what I believe you are referring to - post 107 in that thread. If you are referring to a different post in that very long thread, please let me know.
micromass is not saying that how 0 is defined is arbitrary, but is saying that the name "number" is arbitrary. It doesn't matter whether you call them "numbers" or "numeros" or whatever, they still have the same properties.

I don't see that he said that 0 shares "some properties of the reals". Can you provide a link to where micromass said that?

HallsofIvy

what do you mean by "logical maximum value" of a number?

What "plane" are you talking about? I thought you are talking about the real numbers.

I have no idea what "physically" can mean here.

You just said before that "[itex]\infty+ 1= \infty[/itex]". How is this any different. Having said "both series diverge into [itex]\infty[/itex]" so why are you now talking about them as if they were different "infinitys"?

logics

'definitions' is my LOB, by I am no authority;
to your question I'll let a Science advisor reply.

N , n , 'numbers' (baz., cod, numero...), a quantity, has the 'best possible' definition in the world, the 'ostensive' definition. Math is indeed an arbitrary, symbolic, axiomatic system, but arbitrary only to a certain point: its (vocabulary, lexicon,"words") 'numbers' are abstract concepts but they are linked to reality, to the abacus, otherwise it is a useless mind game. Be O any object, a circle, a marble....

O → 1 (circle), OO → 2 (circles) ,..... 3,.....,4 .....OOOOO → 5 (circles)..........9 ;

and so are their ('behaviour'): attributes, properties, operations...etc.,: dictated by reality :

OO|OO|O (5) = OO (2)+ OO (2) + O (1) = O|OO|OO =; OOO|OO = 5 = OO|OOO → 3+2 = 2+3 ......

I'd rather return to the logics of ∞2) Arguably, one might, but only because 54 is like king-of-France's hair, which is the colour one chooses !); it's also arguable that really 'everything' is possible, can we define that 3/3 = 54 ?

1)The link says: "arithmetic operations can be partially extended to ∞".
The problem I highlighted is right in that adverb: 'Can be" (why a bad word?) and partially.
Partially, I suppose, means it doesn't 'fully' belong there (post #14), exactly like 0, which shares only 'partially' the properties of 'regular' numbers, as you just said (0/0..).

(*note: I tried to prove in the other thread that 0 is 'literally' meaning-less, but useful if if it is considered just a sign and I'll be glad to continue that discussion there, if you wish.)

Thank you, micromass, for your attention , if you agree, I'll ask some more questions about (3) the logics of ∞, such as : with 10¹⁰⁰ you can count all elementary particles in the Universe billions of times over, could we trade ∞ with 10¹⁰⁰⁰ ?