Problem with Infinity

[pallidin]

I have often heard throughout these forums that in order for a concept to be considered true, there must be some evidence of it.
Great. Makes sense to me.
So, what evidence is there for Infinity?
Hmmm... I truly cannot think of a single process or event in the real world which is not constrained in some way by finiteness. Can you?
Curiously enough, I believe that an experiment to prove infinity is inherently impossible.
Therefore, I conjecture that infinity is an assumed concept without any basis or potential of verification.

[NateTG]

I'm not sure why you put this in the QM section.

That said, the concept of infinity as you describe it is not sufficiently well defined for a strong response, but there are some indications that certain types of infinity have physical instantiations. If you want a better response, it may be constructive to clarify what you mean by infinity.

We certainly think of physical space as continous -- so there should, in theory, be an infinite number of orientations an object can have. Unfortunately I have no idea whether modern physics makes any predictions about this.

As a more esoteric example, if you want to have a quantum theory that uses hidden variables, then you need these hidden to be in an unmeasurable domain, which, strongly indicates an infinite number of possible states.

[Integral]

The only thing that really matters about "infinity" is its mathematical definition. All else is but mutterings of philosophers.

Mathematically infinity is defined as an extension of the Real Number system the basic definition is:

\infty > x \forall x \in \Re

Along with this definition are specifications of how to deal with the symbol in the basic mathematical operations.

[NateTG]

Originally posted by Integral

Mathematically infinity is defined as an extension of the Real Number system the basic definition is:

\infty > x \forall x \in \Re

Along with this definition are specifications of how to deal with the symbol in the basic mathematical operations.

The fact is that there are many different infinities in mathematics, and there are different definitions of infinity depending on what kind of mathematics are being done. Typically what quetions like this refer to is more closely related to infinite cardinals.

The \infty that you describe above cannot be added to the reals without causing the reals to cease being a field so any specification for subtraction or division involving it will be problematic.

[Integral]

That is how it is done in all of my Real Analysis Texts. With as I stated above definitions for the basic operations.

As far as physical usefulness as far as I know only \aleph_0 is used. Since it is a mathematical construction it underlies the existence and uniqueness theorems for much of the mathematics on which physics is based.

[NateTG]

Originally posted by Integral
That is how it is done in all of my Real Analysis Texts. With as I stated above definitions for the basic operations.

I expect that it's used only in the context of limits and that the basic operations only work in the context of limits as well. (I should go home and burn the Real Analysis texts that I own if they contain something this bad.)

As an example of how broken this \infty is:

\infty+1 > x \forall x \in \Re
so by your definition
\infty+1 = \infty

If \infty-\infty \neq \infty
then \infty-(\infty+1)\neq (\infty-\infty)-1.

So \infty-\infty = \infty
if we can distribute
(1-1)*\infty = \infty
0*\infty=1*\infty
divide both sides by \infty
0=1

Technically, you should be OK if you avoid \infty-\infty and \frac{\infty}{\infty} but investigating why is related the cardinal arithmetic.

[Integral]

Perhaps you should go home and read your Real Analysis text. I find this definition, (I did not list the rules of operation, they do not correspond to your ad hoc ones!) In Real Analysis by Royden and Principles of Mathematical Analysis by Rudin

Here is the definition from Royden (http://home.comcast.net/~rossgr1/extendedreals.pdf)

[master_coda]

NateTG is quite correct in showing that you cannot add infinity to the real numbers and still have a field.

And Integral is correct in showing that you can add infinity to the real numbers with well defined operations.

I fail to see the problem.

[Hurkyl]

Right, the extended real numbers are used because of their nice topological properties, not because of their (not so) nice algebraic properties.

[NateTG]

Originally posted by Integral
Perhaps you should go home and read your Real Analysis text. I find this definition, (I did not list the rules of operation, they do not correspond to your ad hoc ones!) In Real Analysis by Royden and Principles of Mathematical Analysis by Rudin

Here is the definition from Royden (http://home.comcast.net/~rossgr1/extendedreals.pdf)

Royden does not define \infty. He defines an extension to > to cover \infty, and he does not claim that "Mathematically infinity is defined as an extension of the Real Number system." I would probably be ready to burn (well, actually, sell) any text that made those claims.

[Integral]

Pardon the semantics. Perhaps you need to go back and read my posts again.

[NateTG]

Originally posted by Integral
The only thing that really matters about "infinity" is its mathematical definition. All else is but mutterings of philosophers.

Mathematically infinity is defined as an extension of the Real Number system the basic definition is:

\infty > x \forall x \in \Re
...


Maybe I'm misreading that, but the text above:
1. Implies that the mathematical definition you're about to give is canonnical.
2. Claims that this (allegedly cannonical) definition is of \infty as an extension to \Re
3. Claims that \infty > x \forall x \in \Re is a mathematical definition of \infty

It's also not particularly constructive.

[Integral]

I will stand by post as it is, not sure where you get the cannoical bit. I believe that that form was out of Apostals book, don't have it on hand, nor can I find the scan to give the complete wording.

My point is that Real Analysis is Mathematics, in Real analysis, the symbol \infty is defined. It is an extension to the Real Numbers and it comes with defined operations. I choose not to list the operations as this was meant to be the basic 25 word or less intro. Not a text book. Granted I did not deal with negitive infinty. My definition is essentially the same as that given in Royden and Rudin.

I am very puzzeled as to what your problem with this is?

[NateTG]

I guess it's that if you're going to take a mathematical approach, then you should be more carefull about the phrasing.

The problem is that it seems like you're representing a definition of infinity as the definition of infinity. Which is a substantive distinction, considering that you're also using phrases like "All else is but mutterings of philosophers."

I don't expect you to change the post, I just wanted to point out what bothered me about it.