Different sizes of infinity

SLUSK

There are apparently different sizes of infinity. Say you got an infinite amount of odd numbers because you can always add another odd number and the same with the amount of even numbers. This would mean that the amount of all numbers are a bigger size of infinity.

I remember something like this during some logic lecture when they went through set theory.

Isn't it funny that infinity comes in various sizes?

masherella88

does the infinity that you added one to not become negated as an infinity by the addition of another unit to it? surely the only infinity that exists is the bigger one as the smaller one can't be infinity as there is an infinity larger than it? i'm just wondering

Joe1919

masherella88 wrote: »

does the infinity that you added one to not become negated as an infinity by the addition of another unit to it? surely the only infinity that exists is the bigger one as the smaller one can't be infinity as there is an infinity larger than it? i'm just wondering

But, does infinity ACTUALLY exist. Perhaps infinity is only a concept.
Mathmatical concepts, such as perfect squares, circles etc. may be only figments of the imagination of the mathmatican, as these are not found in nature.
Even the grains of sand in the Saraha could be counted in theory, so we can't presume that infinity actually exists.

Its an old argument. Aristotle argued that infinity only existed potentially but not in actuality. St.Bonaventure had more or less similar arguments as above. e.g.
http://books.google.ie/books?id=UB0Tao4oikEC&pg=PA238&lpg=PA238&dq=st.+bonaventure+infinity&source=bl&ots=u7wRnbc3I0&sig=iJhUWib8f_mndjJke7jh_dt9nT0&hl=en&ei=D4CUStbnMuSfjAett8HsDQ&sa=X&oi=book_result&ct=result&resnum=8#v=onepage&q=st.%20bonaventure%20infinity&f=false
The Christian ideology is, of course that only God is infinate and this may have been the motivation, but nevertheless it does appear that the notion of infinity is absurd.

masherella88

but if the universe is constantly expanding that must mean that there is something that is infinite, that keeps getting larger, no? if somebody started counting and that's all they did for their life, when they died somebody else continued to count from where the last person had stopped and when that person died somebody else continued to count and so on and so forth there would never come a point when somebody would say okay that's it i can't keep adding another one to the last number i just counted, would there? so in some way, even if only in concept, infinity exists, no? if it even exists as a concept, the idea exists so it too in some way exists even if its existence is only in the form of an idea, right?

sorry, that reads kinda confusingly...i find it all fascinating but i'm not great at wording my questions!

paulhealy1991

thats some crazy **** man!! if both sets of numbers are never ending then neither can be bigger than the other.

Joe1919

masherella88 wrote: »

.........so in some way, even if only in concept, infinity exists, no? if it even exists as a concept, the idea exists so it too in some way exists even if its existence is only in the form of an idea, right?

What you are saying has many similaraties to the famous 'ontological proof' for God. However, there is a difference between 'infinity' as a concept and in reality.

Similarly, when it comes to religion, most people would agree that God (or Buddha or Santa or the tooth fairy) exist as a concept.

nyarlothothep

Doesn't infinity occur with the observation of objects entering blackholes or their gravitational pull? or trying to reach the speed light? Would both sets of numbers be ultimately infinite and that they would just expand at different rates?

ChocolateSauce

Yep. There are an infinite number of numbers between 0 and 1, and 1 and 2, and so on. In maths, there are orders of infinity. Mind boggling really!

Fremen

SLUSK wrote: »

There are apparently different sizes of infinity. Say you got an infinite amount of odd numbers because you can always add another odd number and the same with the amount of even numbers. This would mean that the amount of all numbers are a bigger size of infinity.

I remember something like this during some logic lecture when they went through set theory.

Isn't it funny that infinity comes in various sizes?

What you're saying is true, but the example you give is not quite correct.
We say two sets have the same cardinality (read: two sets are the same size) if there exists a rule which "pairs up" elements from one set with elements from the other set, such that every element in both sets has a unique partner. Here's a picture of the concept:



In this way, we can say that the set of even numbers and the set of whole numbers have the same cardianlity: the rule is,
"take an even number and divide by two".

The cardinality (size) of the set of whole numbers is called "aleph-0"

However, it is possible to show that there is no rule which pairs up everything from the real numbers with everything from the whole numbers uniquely. There are just "too many" real numbers. We call the cardinality of the real numbers "aleph-1".

Now, if your brain didn't explode from the idea that there are different sizes of infinity, try this:
Suppose we ask if there's an infinity less than aleph-1, but greater than aleph-0. Call it aleph-1/2. You can prove that if you assume existence of aleph-1/2 you will never get a contradiction in your logic.
However, you can also prove that if you assume aleph-1/2 does not exist, you'll never get a contradiction in your logic.

In other words, the statement "Aleph-1/2 exists" is both true and false at the same time.

Took me a little while to get over that

Joe1919

ChocolateSauce wrote: »

Yep. There are an infinite number of numbers between 0 and 1, and 1 and 2, and so on. In maths, there are orders of infinity. Mind boggling really!

But can we say that numbers actually exist?

Numbers only describe a quantity of 'something' that exists but the existence is in the 'something' and not in the numbers.
Two empty buckets have the same quantity as a million empty buckets.(which is nothing).
If I said I have 7 in my pocket, you would ask 7 what! as 7 on its own is meaningless in describing something. A quantity (number) is always a quantity of 'something' to be real. The reality is in the 'something'. The number merily describes the quantity.

Secondly, the continuum between 1 and 2 contains an infinite amount of 'potential' numbers, but we are talking potentially here and potential is different to actual in that what is potential does not actually exist.

An analogy can be made here to a woodcarver with a large block of wood. Potentially, it is within his power and his imagination to create any shape within an infinite range of possibilities. He has an almost infinite choice and range of shapes. His knife can fall anywhere and cut anywhere. However, in actuality, he only creates one shape or carving with that block of wood.

Anyhow, here are some links that may help.

http://uk.geocities.com/frege@btinternet.com/cantor/Phil-Infinity.htm

http://platosheaven.blogspot.com/2005/12/do-numbers-exist.html

Fremen

Joe1919 wrote: »

But can we say that numbers actually exist?

Could you clarify what you mean by existence?
This is the trouble I have with philosophy. I find the terms to be so ill-defined that the questions are impossible to answer.

Joe1919

Fremen wrote: »

Could you clarify what you mean by existence?
This is the trouble I have with philosophy. I find the terms to be so ill-defined that the questions are impossible to answer.

I dont think its possible to give a full clarification of what to exist means.
Existence is often taken to mean 'to appear' or 'to stand out' or 'to be'.

Of course, because of this, one could argue that existence is subjective in that if something is to appear, it must appear to someone and in their mind and hence it could be argued that everything that can appear in the mind has 'being' and exists. e.g. fairies, goblins etc. because 'being' has a certain subjectivity and can appear in different ways. (Extreme Idealist position?)

However, my argument about numbers presuppose the existence of the world and that certain items in the world exist (e.g. Apples) and that certain other items don't exist e.g. Fairies.

If I say to you that I have 40 apples in a box, I can presume that this statement is possible because, for me apples have existence and the number 40 is my way of conceptualising the quantity and hence 40 Apples may exists.

However, if I say to you that 40 Fairies exist, I can presume that 'nothing exists' in the box because I am prejudices in believing that fairies do not exist and hence the box is empty.

Numbers do not have to refer to real entities. 5 fairies + 3 fairies =8 fairies
Numbers are human concepts and are created by the intellect to describe real and unreal things (that may or may not exist).

For the purpose of my post then, Apples exist and Fairies do not exist.
(My apologies to people who believe in fairies).

Fremen

Seems like you're saying numbers only exist insofar as they describe things in the physical world. There are lots of branches of mathematics which deal with entities which have no physical interpretation at all, but which (I would say) certainly exist.

Joe1919

Fremen wrote: »

Seems like you're saying numbers only exist insofar as they describe things in the physical world. There are lots of branches of mathematics which deal with entities which have no physical interpretation at all, but which (I would say) certainly exist.

I suppose it depends on what it means to exist. Wittgenstein thought that most philosophical problems and disputes were language problems. To some extent, he is right. My use of the word 'exist' may be different than your use of the same word.( The meaning of the word is in its use ?). Hence, we may not be playing the same language game so to speak.
I suppose, I am fond of nature, I enjoy the outdoors and hence, for something to exist, it must exist in nature as a particular instance .( If you can kick it it's real.)
Someone else may take a more 'internal' or idealist approach to life and be more inclined to thoughts and concepts and see this as real existence.
My particular problem with this, is that if you take this latter approach, then everything is real, i.e. Dragons, deamons, goblins etc. and the word 'real' means nothing anymore as there is nothing unreal.
Just because we can conceptualise something does not make it real.