From another thread, severad discussions were made - and questions asked - about "infinity" or "infinite".

A complete study of infinite numbers was made in late XIX century by George Cantor - who died crazy.

About finite numbers, nothing special to say: we all know what they are. So, we have so many feet as shoes because it is possible to make a correspondence ( a so called one-to-one function) between feet and shoes: to each foot we assign one shoes, and viceversa.

And, if a set has , say, 3 elements, it cannot have such a one-to-one correspondence with a set having -say -4 elements.


What happens about infinite sets? The first amazing thing is that it is possible to have two sets, one contained in another, and still such it is possible to find such a correspondence:

for example, {even numbers} is a subset of {positive integer numbers}, but there is a one-to-one correspondence:
define, for every even number n, its half n/2.

We say that even numbers are so many as positive integers numbers, or that they have the same "cardinality" = kind of infinity.

The interesting thing is the rational numbers (fractions m/n) are so many as integers numbers (such a correspondence can be built), and real numbers are not (it can be proven that such a correspondence cannot exist)

If it can't be represented in the physical world does it really exist? [eg]

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My post in emanuela's aformentioned thread:

"Infinity" was once *disproved to me by this explaination:

Take your thumb and your forefinger and spread them apart as far as possible. This represents a distance. Now make that distance one half as large. Continue to cut the distance between them in half.

Theoretically, your thumb and forefinger will never actually® touch each other.

Does my thumb and forefinger ever touch as I understand touch? Or is my perception not only not real but not representable by mathematics (as could be the only proof of touch *contextually)?

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Isn't the infinity of integers 2x that the infinity of only odd integers? If not, then why bother with proving infinity?

Theoretically, your thumb and forefinger will never actually® touch each other.

Infinity is not constrained by your physical limitations.

Theoretically you could move your thumb and forefinger closer and closer together, with each move halving the distance between them. Practically you don't have good enough control to manage more than about five or ten moves.

Practically you don't have good enough control to manage more than about five or ten moves.

Oh, you noticed that too, did you? Personally, I think he should just lay off the booze, I do ...

- Pfranz

Take your thumb and your forefinger and spread them apart as far as possible. This represents a distance. Now make that distance one half as large. Continue to cut the distance between them in half.
This is a version of the Zeno paradox!
About Achilles, and the arrow never reaching the target.
It was the root of millennial discussions.

The answer satisfying me is that there is un underlying problem about time: are you making each move in the same time, say a second?In that case, yes, they never touch.

But the model of real motion is such that you are making every step in a time that is half of the previous time.

So now they will touch, and this corresponds to the fact that
1 + 1/2 + 1/4 + 1/8 + ... is equal to 1
The meaning of the infinite sum above is the limit of the partial sums of the first n terms, as n is infinitely growing (same subject as in 0.999....=1 question)

Pfranz, you lying sack of sh.... well, on the booze part, anyway...

In the morrow...

Warning!
[There is a slight danger that your viewing machine will go wide
if you view this post.
Unfortunatly this is necessary because what we attempt
here is to write out the largest number
ever written in a single line since the world began.
So be cooperative.]

Can "Infinity" have a plural form? Of course it can.
We can't imagine one infinity so why can't we not imagine two?
So emanuela, the answer to your question is yes.
In an endless series of numbers all combinations
of all numbers will repeat within that series
as well as those that don't.

But what I attempt here is much more modest.
I hope to get in the Guinness Book of World Records by writing
a single number on a single line that will reach
from Birmingham Alabama to the site of the Wordapallosa
on the west coast of Ireland by June, so here I go...

The number that I have chosen is a Perfect Number,
that is, a number that equals exactly the sum total
of all of it's divisors, for example, the first Perfect
Number is 6 (six). 1, 2,and 3 are its divisors and if you add
them, you get 6. The second perfect number is 28, and so
on... But of course my perfect number will be bigger.
Lots bigger. But to make it even bigger I will exibit
my number as the multiplier of "T",
with "T" being the total number of atomic particles in the
universe. So without further ado here is my number
(I have left off comma (,) dividers for expediency
and to save electronic ink.)...

T x 7609323273987546351218765946376229987865673113702470328764332667578544321098765435628712109876890978654324567899900098987653355432346789086765541113426673345999878721212345456676544332232346788909009877654232467890987653432233333443434566677788889987655433234543221114778878898536543324234564123403578778343458807645322664432256554337684320982113456543232679789796785543346579900099876665223444232146788965438653763542668559097333545666677890008675342126645678762343778809900246576980912568980737521029384756102983765231879873456877108432866578398764839299983334261836290923417093245674666666699876534329294567643218765931286541954232768954362112121358964365012465483320987656546742390034214544220099007656643435678865342219823739353414580096421357909765323567653217878694035342612626373848475322678065369009765443411113358760900580005547765264375109890548854541346788909078877743652178895466300999908876655442434325988701234467899006658899094443902973648556718110987652200998876765443245564321356567885545767434927162143148765545270861213141526577665488760900987657887765543232098295872508671567345890078676565...
                                                                                                                                                                                                                                 -->  to Ireland -->  

milum, I'm no expert, but you seem to have made a common
misstep of the amateur mathematician: overloading your terms;
thus rendering your statement to be unreadable.

-ron

ermm, should that be an ellipse at the end, or three full stops?
…
...

it's taking me forever to figure this out...…