Are these statements about real numbers erroneous?

[bomba923]

*Note: I think questions 2-4 are erroneous?

1) There does NOT exist any $$x \in \mathbb{R}$$ such that [itex] x \in \left( {1,1} \right) [/tex] ?

2) The product of all reals in $$( 0 , 1 )$$ is zero, right?

3) The product of all reals in $$( 1 , 2 )$$ is infinity, right?

4) The product of all reals in $$(0 , \infty )$$ I suppose is one, because each real greater than one has a reciprocal less than one (and vica versa). Their products equal to one. Therefore the product of all reals in $$(0 , \infty )$$ is one?

 

[Zurtex]

Number 1 seems o.k, particularly if you know how the real numbers are defined.

The problem with number 2 is that how exactly would you multiply all real numbers greater than 0 and less than 1 together?

It's quite easy to multiply a sequence of real numbers together that are greater than 0 and less than 1, but the answer could be any real number equal or greater to 0 and less than 1. However, you that's a lot different from multiplying "all real numbers". Same goes for number 3 and 4.

Also something you don’t seem to understand about adding or multiplying an infinite number of elements, order is quite important. You can add the same set of numbers together in a different order and the result may be entirely different.

 

[bomba923]

Also something you don’t seem to understand about adding or multiplying an infinite number of elements, order is quite important. You can add the same set of numbers together in a different order and the result may be entirely different.

Exactly why I think Question #4 is wrong
Depending on how I represent this crazy product (in this case, which order I multiply), indeed I would get another value. For example, I could multiply each recipocal-number product by 2, and overall still represent a product of unique reals.
(these weren't my questions, but I could not at the time offer much of a counterargument//->which is why I'm here at PF!)

 

[hypermorphism]

Exactly why I think Question #4 is wrong
Depending on how I represent this crazy product (in this case, which order I multiply)...

The set of all reals in a continuous interval cannot be ordered into a sequence (see Cantor's diagonal method), and as such cannot be indexed in order to be put into a product in the first place. Is this a question in a text ?

 

[Hurkyl]

The set of all reals in a continuous interval cannot be ordered

You mean they cannot be arranged in a sequence. (indexed by the naturals)

Of course they can be ordered... for example, < is a perfectly good ordering on the reals in an interval.

 

[matt grime]

and they can even be well ordered too, modulo the axiom of choice, that is it is possible to define an order relation where it makes sense to talk of there being a 'next' real number after another one. doesn't agree with the usual ordering, of course.

 

[hypermorphism]

You mean they cannot be arranged in a sequence. (indexed by the naturals)

Of course they can be ordered... for example, < is a perfectly good ordering on the reals in an interval.

Silly me. :rofl: I meant arranged in a sequence.