About the properties of infinity

[shivakumar06]

If we consider a number 9999999………. infinite times then no other number that can be represented bigger than that without plus or multiply operation. So we can be sure infinity is an odd number am I right

 

[jedishrfu]

your notion of infinity needs to be expanded:

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

You're thinking of a countable infinity and the notion of even or odd makes no sense. Why not add zeros to 1 to produce a million, billion, ... googleplex... all even but no different than 999999...

 

[micromass]

If we consider a number 9999999………. infinite times then no other number that can be represented bigger than that without plus or multiply operation. So we can be sure infinity is an odd number am I right

Except that 99999999... is not actually a number.

 

[shivakumar06]

may i just know why 9999... not a number? can you please throw some light on it?

 

[micromass]

may i just know why 9999... not a number? can you please throw some light on it?

Well, the first thing you will need to do is to actually define what you mean with 9999...

 

[schtruklyn]

Hi. You are trying to assign a value to a notion of "infinity". There is no value for "infinity". "Infinity" is a process, a limiting process, a keyword for:

What happens if a quantity grows ever larger and larger?

There is a point at infinity in complex analysis, but to analyse it, You need to use limits, so "Point at infinity" is a mis-nomen, I'd say, just a shorthand notation for something more subtle.

This is all according to Weirstrass and analysis.

And yes, there are "countable" infinities and "continuous" infinities. If You decide to explore Cantor's set theory about types of infinities, be warned: set theory is inconclusive in Godel sense. Continuity theorem is both provable and not provable. This drove Cantor mad and put him into asylum.

I deliberately use layman jargon here for obvious reasons.

Good luck!

 

[micromass]

be warned: set theory is inconclusive in Godel sense.

Yes, and so is the theory of natural numbers. I don't see anybody warning people for using natural numbers.

Continuity theorem is both provable and not provable. This drove Cantor mad and put him into asylum.

First of all, it's the continuum hypothesis. Second of all, it is not "both provable and not provable", that would be a contradiction. Rather, the continuum hypothesis is consistent with set theory and the negation is also consistent. So it is not provable, using the current axioms.

Cantor himself never realized this about the continuum hypothesis. That the continuum hypothesis was consistent was known much later by Godel. That it was unprovable, even later, by Cohen. What drove Cantor mad was rather that his theory was not accepted by his peers.

 

[schtruklyn]

Yes, and so is the theory of natural numbers. I don't see anybody warning people for using natural numbers.

Yes, awkward situation

The point is: explaining some issue by introducing another, even more complicated issue, is not a good way to explain anything. Therefore the warning about diving into the Cantor's set theory For instance, I can still today find manuscripts on-line trying to prove or disprove Cantor's continuum hypothesis, even on arxiv.

Cantor could not prove nor disprove his Continuum hypothesis. One day he wrote a letter to his publisher and editor that he finally managed to prove it. The very next day he urgently sent another letter apologizing for such a childish excitement, for the very last night he actually proved it wrong. Then the next day... And so on. For Cantor himself, it was a hypothesis alright.

Besides, the original question was not quite technical, so my answer was not quite technical either.

Finally, only Cantor knew what was the issue within his mental reasoning that drove him mad -- even maybe Cantor himself could not articulate it. His late father; his childhood; religion; math as religion; loneliness; how can math be inconclusive; if conclusive -- then what's wrong with him; how could God allow this; not eating well... The point here being: we will probably never know.

Regards.

 

[micromass]

The point is: explaining some issue by introducing another, even more complicated issue, is not a good way to explain anything.

I don't see why not. In physics, we explain classical mechanics by introducing relativity and quantum mechanics, which certainly are much more complicated.

For instance, I can still today find manuscripts on-line trying to prove or disprove Cantor's continuum hypothesis, even on arxiv.

Those manuscripts are not of professional mathematicians then. For mathematicians, the issue is solved. Unless of course you work in other axiom systems, but that is a different issue entirely. Just because some crackpots out there try to disprove an entire theory, doesn't mean that the issue isn't resolved for serious mathematicians.
Also, being published on the ArXiV doesn't mean that the paper is of a good quality, let alone that it is true.

 

[schtruklyn]

True that. I was simply preventing possible future confused questions. We both agree. Regards :)

Edit: by the way, do You happen to know the answer to my post "Reference or explanation on zeta Mellin transform in critical strip"? It would be really helpful to me if someone could give me a book reference for it. It was posted today in Number Theory section. Thanks in advance.