# Questions about the logic of infinity

1. Mar 30, 2012

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 ∞/∞ $\neq$ 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

2. Mar 30, 2012

### 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.

3. Mar 30, 2012

### 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 $\frac{\infty}{\infty}$ are undefined.

Last edited by a moderator: May 5, 2017
4. Mar 30, 2012

### 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.

5. Mar 30, 2012

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

6. Mar 30, 2012

### 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.

7. Mar 30, 2012

### 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.

8. Mar 30, 2012

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$\sqrt[]{}$n , n$\subset$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.

9. Mar 30, 2012

### jgens

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

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

Last edited by a moderator: May 5, 2017
10. May 4, 2012

### logics

A couple of simple questions:

1) ∞ is not a real number, is there a definition of what it is?, ∞ $\notin$ ℝ, ∞ $\in$ ?, (I suppose $\hat{ℝ}$$\bar{ℝ}$ 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 ?

Last edited by a moderator: May 6, 2017
11. May 4, 2012

### micromass

Depends on what you mean with ∞. If you mean the extended real numbers, then $\infty\in \overline{\mathbb{R}}$. And yes, $\overline{\mathbb{R}}$ 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

$$\lim_{x\rightarrow 1}{x^2+5}$$

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

$$\lim_{x\rightarrow +\infty}{2x+5}$$

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

$$\lim_{x \rightarrow +\infty}\frac{x}{x+1}~\text{and}~\lim_{x \rightarrow +\infty}{\frac{2x}{x+1}}$$

Substituting $x=\infty$ in both expressions both yield $\infty/\infty$. But the first limit would indicate that this equals 1, and the second would indicate that this equals 2. So since any definition of $\infty/\infty$ 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 $\infty$ and in the second it is $2\infty$. So we are not working with the same infinity here!! OK, but we defined earlier that $2\infty=\infty$, so we are working with the same thing here.
However, this observation is a valid one. Perhaps we could define a system such that $2\infty$ would not equal $\infty$ 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 $\mathbb{N}$ is already infinite and would have to be prohibitied.

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12. May 4, 2012

### logics

Thanks, 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$\in$ 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 ∞ $\in$ $\bar{ℝ}$), 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.

Last edited: May 4, 2012
13. May 4, 2012

### micromass

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

14. May 4, 2012

### 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 $\in$ ℝ, but 0 shares 'some' properties of the reals, then I make a subset in ℝ.

Last edited: May 4, 2012
15. May 4, 2012

### Staff: Mentor

logics, I think you missed micromass's point. In your previous post in this thread, you gave a link to a different thread (https://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?

16. May 4, 2012

### 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 "$\infty+ 1= \infty$". How is this any different. Having said "both series diverge into $\infty$" so why are you now talking about them as if they were different "infinitys"?

17. May 5, 2012

### logics

'definitions' is my LOB, by I am no authority;

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 ......

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 ? :uhh:

Last edited: May 5, 2012
18. May 5, 2012

### logics

I'm maintaining that:

- calling an integer a number is not an arbitrary act: (baz. is meaning-less, like 0, and if we call it a cod one might deduce it swims). We call it a number because a numeral has denoted a quantity for ages
- the properties of numerals, numbers and of operation(s) are not arbitrary, the object of definition, they are dictated by reality
- 0 cannot be considered a number because
a) does not denote a quantity, denotes a non-quantity, is meaning-less sign
b) its properties and properties of the operations involving 0 are different from numbers, or even arbitrary: left to the whim of the individual (0/0=54)
- if we are expressing principles, concepts in term of sets, we cannot include or adjoin alien elements.
can someone tell me if I'm wrong?

My question to micromass (who's an authorithy about sets) was: if we say that an entity is an element of a set, can we accept the fact that it is even slightly different ?
In the set fruit, we have different subsets for apples, pears and 'orange-lemmon etc' are together in a different subset of citrus, could we put them all together? Can we adjoin, just by definition, a dog to the set 'fruit', is this a 'lawful' logical or technical operation?

Last edited: May 5, 2012
19. May 5, 2012

### Staff: Mentor

Why all the random colors?
But micromass did NOT replace the word "integer" with a different word that has meaning. He used "bazalbieba", which I believe is a made-up word that is meaningless.
This is a very limited notion of what it means to be a number. Using this reasoning would you accept -3 as being a number? If so, would you also accept i (the imaginary unit) as being a number? How about omplex numbers? I certainly can't say that I now have 3 - 2i apples, but so what? If you can somehow accept complex numbers as being numbers, what about quaternions or octonions?

Mathematicians consider all of these to be numbers.
Yes. You seem to be thinking that numbers and operations are inseparable. They are not. Mathematicians have developed a number of structures, each of which involves a set of things (e.g., integers, real numbers, matrices, ...) and one or more operations (e.g. addition, multiplication, scalar multiplication).

Here are a few of these structures.
group - a set of things together with one operation.
ring - a set of things together with two binary operations, usually denoted by + and X.
integral domain - a ring in which multiplication satisfies a number of additional properties.
field - an integral domain in which every element except one (sometimes denoted z) is a unit.

It's not clear to me that you understand what being a member of a set entails. By definition, elements of a set are different from one another. From the wikipedia definition - "set - a collection of well defined and distinct objects." If the objects weren't distinct, we would have multiple copies of the same thing in the set. The objects in the set are distinct, so we can tell them apart. So a set is just a collection, which means you can put pretty much whatever you want to in a set.

For example, this is a perfectly valid set:
A = {apple, pear, lemon, orange, dog, cat}

Set A has a few obvious subsets:
{apple, pear, lemon, orange} - fruits
{apple, pear} - pomes
{lemon, orange} - citruses
{dog, cat} - mammals

20. May 5, 2012

### SteveL27

That is not a useful definition of number.

First, a numeral is not a number.

Secondly, there is no general definition of "number" in math. The following can all be called numbers: the integers, the reals, the complex numbers, the p-adics, the integers mod 5, the transfinite ordinals, the transfinite cardinals, the Conway surreals, the Robinson nonstandard reals. That's not even an exhaustive list.

All of those can be called "numbers" within particular contexts. There is no general definition of numbers. Nor do numbers need to indicate quantities. If you imagine a number line, zero is just a point on the number line. It's like relabeling your address zero, the house to your left at -1, the house to your right as +1. It's just an arbitrary point on the line from which you begin defining left and right.

If numbers are a quantity, then do you regard the imaginary unit $i$ as a number? What quantity does it represent?

And what on earth can "ages" have to do with it? How long must a mathematical concept be accepted before you personally accept it?

Last edited: May 5, 2012
21. May 5, 2012

### nucl34rgg

It isn't enough to speak of "infinity." One must specify which infinity he/she is talking about. In other words, there are many different kinds of infinities.

(Here I will assume ZFC axioms.) For example, the cardinality of the real numbers is infinite, but it is also uncountable. The cardinality of the power set of the real numbers is also uncountable and infinite, but is greater than the cardinality of the reals. Also note that neither of these cardinal numbers are elements of the sets they describe. Another neat thing is that one can't consistently define such a thing as "the set of all cardinal numbers." Such a definition for a set leads to a paradox like the one described by Russel's Paradox.

Note that when one starts trying to describe "transfinite" numbers," it starts to become very important which axiomatic system one chooses to adopt. Certain axiomatic systems can lead to different conclusions.

If you are like me and are fascinated by these types of things, the area of math that deals with these types of questions is set theory.

Also, as SteveL27 has stated above, the concept of "number" itself has been difficult to define. It seems to be something primitive akin to that of "set" from which one must simply start.

Last edited: May 5, 2012
22. May 6, 2012

### logics

I am grateful to you all for your contributions, I'll reply in the thread: qualitative vs. quantitative . . My concern is not about the definition of number, but the logics of ∞. Could you tell me why the answer to that question should be no, if ∞ is not necessary?

just two notes:
- Steve, 'numeral' is a hyper[o]nym: a term that includes what you call now numbers (1,2...) , other numbers (I,V,X...) and their language 'word-form' (one, two,....)
- Mark44, a set is indeed made up by different elements, but is identified by Cantor's 'sentential formula', [Edit ,i.e.: the rule for set membership : {x | x is a natural number/fruit/numeral...} ], the set 'numeral' has subsets identified by hyponyms: one may be 'glyph-numerals' and another can be 'letter/word-numerals' the subset 'glyphs', in its turn, has subsets ''Indian/Arabic-number/glyphs', 'Latin glyphs' etc . We surely cannot make a jumble.

Edit: If one wants to adjoin 0 to the set { x | x is a natural number} he must prove it's a number. In the FAQ 0 is not defined,
micromass
hasn't replied directly to post#18
but in this post has just confirmed my thesis, if 'non-quantity' means 'not measurable'. Now we must only find out if we can adjoin 0 to N when we think 'it makes no sense to make 0 a number':

- Mark44 you may make a jumble set if you list all its elements : {0, 1, 2 ,..} but I maintain you must change the rule: {x | x is a baz.albieba} [a baz. is either 0 or n !(more or less so)] }

I hope HallsofIvy will tell me where I am wrong!

Last edited: May 6, 2012
23. May 6, 2012

### SteveL27

New term to me, but I'm looking at this:

http://en.wikipedia.org/wiki/Hyponymy

and I do not believe that a number is a numeral. It's a very basic fact of mathematical culture that a numeral is not a number; a numeral is a representation of a number.

In hyponymy, we are talking about types. A rectangle is a shape. This is not the relationship between a numeral and a number. I wonder if you are not stretching the concept a bit. If there is some field of knowledge that considers a numeral a number, that's fine; but mathematics is not such a field of knowledge. A numeral is not a number. Of that I am certain.

The fact that you consider V and 5 to be different numbers is a clue that you are stretching the concept too far. V and 5 are different numerals representing the same number.

24. May 6, 2012

### Staff: Mentor

I have no idea what you mean by "logics of ∞".
We surely can. Whether it suits a purpose is a different discussion. Again, you misunderstand what a set is, which is nothing more than a collection of things. A set can contain whatever you choose to put in it.

25. May 6, 2012

### Staff: Mentor

I'm with Steve on this: the terms number and numeral are two different things. Numerals are the symbols that we use to represent a particular number.