Infinity in mathematics

[Edgard Neuman]

Hi,

I don't know if I post in the right section..

 

Infinity is a big part of mathematics.
I've known for long time all the assertion in mathematics based on bijection : for instance the idea that the number of pair integer is equal to the number of integer because you can create the bijection n -> 2*n.

This type of idea is mainly accepted as true, and used in different forms that conducts to strange ideas which are counter intuitive.. (the idea that infinity contains several different part of itself is counter intuitive).

I've never believe in such ideas : and here's why I think we should reject the idea of bijection :

To reason in infinite sets, we often start of a set of mathematical objects (like integer) with a property and apply a mathematical induction to apply the property to a deductible bigger set of object.
For instance, you think of the bijection (1-> 2) and then,( 2-> 4) and then (3->6) etc.. and so by "induction" you deduce that n->2n for the whole set of integers.

But I think that when we do this, we actually use different sets of object. For instance (3->6) use at least the set of integer [0;6] which also contains 1, 3, 5.
If we take a set [0;n], for this set of integer, the bijection is in fact not true, because you can't find the bijection for the numbe 5 (5->10) in this set. So we really can't build a set of integer for which the bijection is true. If we try to add "10" to the set, we need at some point to define 7, 8, and 9. (for instance if we use peano arithmetic to define integers), and so the bijection, for this new set is still false (because now 6, 7, 8, 9 and 10 have no bijective partner) . So the reasoning is false, and so it should be for the infinite set.

Should we reject assertions based on this kind of bijections ? (and look much carefully at all those "mathematical induction" made from finite to infinite set of objects)

[John]

It seems like you're treating infinity as simply a very large integer, whereas infinity is in fact a concept defined as being larger than any number. While we do run into problems using f(n) = 2n using finite subsets of the integers like [0 .. 6] in the sense of your post, we will never run into similar problems using the entire set of non-negative integers, for instance, since there is no highest integer beyond which we'll run out of outputs.

 

I'm not sure what your meaning is with regards to mathematical induction. The principle of mathematical induction (PMI) simply states that given some natural number n and property P, if P(n) holds, and if P(n) implies P(n + 1), then P holds for all natural numbers greater than or equal to n. This is taken as an axiom in Peano arithmetic, but can also be proven from the well-ordering principle (and indeed, the WOP can also be proven from the PMI, i.e. the WOP and PMI are logically equivalent). The validity of these statements is somewhat built into our definition and concept of the natural numbers, though I'm sure there are some logicians out there who question their validity.

[Dekan]

It seems like you're treating infinity as simply a very large integer

That's a very good observation. A lot of confusion is caused by abstract nouns like "infinity".

 

The noun makes us think, that there's an actual thing called "infinity". Which could eventually, in theory, be reached. Even if it's a very, very, long way away.

So when we say: "natural numbers extend to infinity", we think of "infinity" as a kind of far-away goal - an end-point, an ultimate destination.

 

But suppose we replace the noun "infinity", by the adverb "endlessly".

And say: "Natural numbers go on endlessly".

 

Doesn't that get rid of the phantom "infinity". And allow us to think more clearly.

[Edgard Neuman]

http://en.wikipedia.org/wiki/Cardinal_number#Motivation

 

My point is that I think bijections are not valid in any infinite set when they are not for a finite part of it.
I just use integer as an example.
What I try to explain is that I don't believe that card(N)=Card(N²) or =Card(pair number) or anything like that. I don't believe neither that card([0;1])=card([0;2]) in R for instance.

It's usually said that Card(N) = Card(N²) because we can find a bijection between the two.
I think the bijection can't be use because in the reasoning is false : there's a confusion between the steps (where a finite subset in used) and the final proposition (when it's applied to the infinite set).

In fact, I think Cantor was wrong.

I think the kind of reasoning using bijection (like in this : http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel) should simply be considered false, so we should only deduce properties of "infinite" set using "finite" sets.
In the Hilbert Paradox, for instance, it is said that you could move each guest from the room "n" to the room "n*2" or "n+1".
I think this is incorrect : you can't do that because the room n*2 is occupied. If in the beginning every room of the infinite hotel is occupied, you can't find any "n*2" room that is free for the nth guest.
My idea is that proper reasoning, using induction, should use a finit set at each step. So only if a proposition is true for a finit set, and the next one, it is for the infinite set.

Edited July 25, 2014 by Edgard Neuman

[John]

I think this is more a philosophical discussion than a mathematical one, really. While infinity doesn't actually exist (at least for all practical purposes) in our universe, we are able to reason about it in a logically sound way in mathematics. There is an entire school in the philosophy of mathematics called finitism, which rejects the idea of infinite objects. But it sounds like you do accept infinity, but simply disagree with some of the consequences of its existence.

 

As for Cantor, you're certainly not the only person to have ever doubted the validity of his work, but it's pretty well accepted by the mathematical community at this point.

[studiot]

Others have already observed that you bijection statement is unsupported.

 

For some examples see my post#8 here

 

 

1+1=2 and hypothesis - Speculations - Science Forums

http://www.scienceforums.net/topic/84274-112-and-hypothesis/

 

 

It seems like you're treating infinity as simply a very large integer, whereas infinity is in fact a concept defined as being larger than any number. While we do run into problems using f(n) = 2n using finite subsets of the integers like [0 .. 6] in the sense of your post, we will never run into similar problems using the entire set of non-negative integers, for instance, since there is no highest integer beyond which we'll run out of outputs.

 

Yeah +1

The edit button doesn't seem available at the moment but you might like to look at Professor Ian Stewart's semi mathematical book

 

From Here to Infinity.

 

Go well

[Edgard Neuman]

My ideas are not expressed with the good symbols.

Let's say that the proposition

 

(Exist bijection X->Y) => Card(X)=Card(Y)

 

Seems wrong to me when X include Y or Y include X.

 

[ and we could have this other proposition (X include Y)=> Card(X)>Card(Y) ]

 

That's the case for [latex]\mathbb{N}[/latex] and even numbers (I mistakenly used the word "pair" because in french even number are called "paire")

 

I found the solution :

In fact we must use a bigger set of axioms to avoid doubts :

http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory

Edited July 26, 2014 by Edgard Neuman

[Mr. Astrophysicist]

Infinity is really unknown and generally unclear. It appears in several papers of mathematics and physics, and expands to such a quite large range, from size and respective fields of physics. We do know in theory that infinity is not affected by numbers, and only affects numbers.

[ajb]

Infinity is really unknown and generally unclear.

Can you elaborate on this?

 

The concept is usually formalised in terms of the cardinality of sets and this is clear, if like a lot of mathematics quite abstract.