That depends if you are doing sequenctial filling or filling in parallel.
Who said we have to go left to right?

Ahhh, but if we could get on board a spacheship and travel really really fast ...

Having done some programming, I agree. When a program hits an infinite loop we often say "and so it never gets to there". We used to also call that a "dynamic halt"!

I was taught about not knowing how "big" infinity is, so that ∞/∞ might be 7, or maybe -2, or, or ... , so it is undefined

How about 1^∞?

Here is one I thought of.

Is a pyramid with infinite height a prism?

Cantor created real numbers, and real numbers have these kinda "original sin" --my belief:D

So, do you think we should change how we teach Infinity?

"Infinity goes on and on and on ..." is the normal idea. But it makes people think of infinity as a "dynamic" thing, and they want to see what is happening at the "unfolding edge".

But isn't Infinity the idea of where you will end up after having gone "on and on"?

Thoughts please , as I may design a page about Infinity

"Infinity goes on and on and on ..." is the normal idea. But it makes people think of infinity as a "dynamic" thing, and they want to see what is happening at the "unfolding edge".

Well, this defination was invented by Cauchy,A.L. and improved by Weierstrass,A.a

You have a very good question, Cauchy's defination has a flaw: Given the limit value, the variable can be as close to it as you want--but where do this limit value come from? do you predict it? or do you simply see it in a graph?

Weierstrass, Meray,C, Dedkind,R and Cantor,G gave their solutions in between 1850 and 1900, they claim that according to Cantor's and Dedkind's defination, real numbers are continuous, hence limit manuplition can be applied to real world. Here real numbers are no longer simply rationals plus irrationals.

Their flaw was attacked by Russel, the famous British mathematician, logician and poet. He attacked that reals are dummies of rationals, and proved circular logic of barbar's(representing rationals) paradox. During half a century since then, mathematicians loving calculus and those loving logic had been quarraling about validity of reals. Logicians also took part, including  Charles Sanders Peirce...

In the last years of Cantor, he proved a point equals to the whole universe according to his defination(Dedkind's defination was homogenous). He can only refer to God to defend his proposition and argue with challengers. He finally died with mental disorder.

David Hilbert is the one who ended this quarral, he proved Cantor's defination homogenous to the ancient Greeks geographic propositon a line is consist of numerous neighboring non space points. They are both true or both false.

It's upon a mathematician to freely chose to believe or disbelieve these propositions. They are the begining and cannot be proven, though with logic flaws.

Obviously, recent prof in universities won't persue logic, and according to their favor, they'd rather believe.

Here real numbers are no longer simply rationals plus irrationals.

So what other type of number is in the set of reals?

Or an even better question.  The normal defintion of irrational (in my experience) is not rational.  So are you saying something can be both rational and irrational?

I mean, they would not bother

Did anyone hear Larry's comment on numbers last night?

"Ya know Aristotle said infinity is the lack of limitation. Which I suppose...is a definition of evil."

EEK! CREEEPYYYY!!! I knew calculus was evil!

krassi_holmz wrote:

Is the property of the real number set is aleph_1?
for better understanding, here's a link: Wikipedia

is the cardinality of the set of all countable ordinal numbers, called ω1 or Ω, which set is uncountable, the second smallest infinite cardinal number. Infinity isn't that big
That is a nice link, thanks, krassi_holmz!

Why does an infinite pyramid have a top (pointy top), but has no base that one can imagine?
The base is really big, and bigger than that, but don't forget , fully grown.
I like the fully grown idea.
It's interesting that we can imagine more than one infinite set, co-existing.
Good thing they are fully grown, or one infinite set would get in the way of the other one.
For example, the real numbers between 1 and 2, and the real 3-D coordinates between (1.5 +/- 0.5, 1.5 +/- 0.5, 1.5 +/- 0.5), which is a cube with sides equal to length 1 and all the points inside it.
Neat how you can have an infinite set that is a subset of another infinite set.
However, the different infinities are all the same size: just infinity.
Pretty neat, huh?

to me it seems mroe symbolic than mathematical, and ironically, if something is infinite, then it is beyond our understanding anyway, and so there is no point ever trying to consider what it is like because we will always fall short


P.S i don't like thinking about infinite, too many people have spent all their lives thinking aobut it, and though it is everything, it is nothing at the same time.