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The Concept of Infinity

SteveElektro said:
The only way to comprehend what mathematicians mean by Infinity is to contemplate the extent of human stupidity.
- Voltaire
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This is going to be my motto from now on =D

And this:

vegan said:
"Only two things are infinite, the universe and human stupidity, and I'm not sure about the former."
eintein
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ive always thought of infinity to just "be" without question...i belive that the reason people find infinity hard to grasp is the fact that people question it...the fact that something is infinite means it is impossible to fully understand

~GH~
 
no I dont believe in infinity...because
1) Aristotle's unmoved mover disproved it
2) Hilbert's infinity hotel paradox disproved it
3) Thomas Aquinas's infinite regress (essentially the same as the unmoved mover) proves it. Cause-effect

Honestly after seeing these arguments its a none issue. There is clearly a concept of infinity, but it is not real.

InFInITY is countable....why?

Hilberts hotel, mathematical proof...
There is a hotel with a finite number of rooms, so when that number is reached, no more guests can have a room.
but what about a hotel with an INfinite number of rooms?well whenever u have a new guest, move them up one room. Ex, dude in 1 goes to 2, 2 to 3, etc.


Okay but what if an infinite number of people come for a room?
well, double the room number, therefore odds become evens and no one occupies an odd numbered room. Giving an infinite number of odds for the infinite group

There is a shitload more to this argument that I dont feel like writtings, look it up. But in math lingo...
-cardinality of the subset of odd rooms is equal to total number of rooms
-infinite sets are sets of same subset cardinality
 
^ That hotel thing makes absolutely no sense. Were you sober when you posted it?

I mean, I can disprove anything by creating a large set of assumptions.
 
*** ya, I wasn't sober so I may not have explained it completely right...here it is again, in full
 
In a hotel with a finite number of rooms, it is clear that once it is full, no more guests can be accommodated. Now, imagine a hotel with an infinite number of rooms. One might assume that the same problem will arise when a new guest comes along and all the rooms are occupied. However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent. There is a way to solve the problem: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1.

It is also possible to make room for a countably infinite number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, and so on, and all the odd-numbered rooms will be free for the new guests.

Now imagine a countably infinite number of coaches arrive, each with a countably infinite number of passengers. Still, the hotel can accommodate them: first empty the odd numbered rooms as above, then put the first coach's load in rooms 3n for n = 1, 2, 3, ..., the second coach's load in rooms 5n for n = 1, 2, ... and so on; for coach number i we use the rooms pn where p is the (i + 1)-th prime number. You can also solve the problem by looking at the license plate numbers on the coaches and the seat numbers for the passengers (if the seats are not numbered, number them). Regard the hotel as coach #0. Interleave the digits of the coach numbers and the seat numbers to get the room numbers for the guests. The guest in room number 1729 moves to room 01070209 (i.e, room 1,070,209. Leading zero added to clarify we take the first digit of the coach number first.) The passenger on seat 8234 of coach 56719 goes to room 5068721394 of the hotel.

Some find this state of affairs profoundly counterintuitive. The properties of infinite "collections of things" are quite different from those of ordinary "collections of things". In an ordinary hotel (with more than one room), the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the "number" of odd-numbered rooms is as "large" as the total "number" of rooms. In mathematical terms, this would be expressed as follows: the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. In fact, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets, this cardinality is called ℵ0 (aleph-null).

An even stranger story regarding this hotel shows that mathematical induction only works from an induction basis. No cigars may be brought into the hotel. Yet each of the guests (all rooms had guests at the time) got a cigar while in the hotel. How is this? The guest in Room 1 got a cigar from the guest in Room 2. The guest in Room 2 had previously received two cigars from the guest in Room 3. The guest in Room 3 had previously received three cigars from the guest in Room 4, etc. Each guest kept one cigar and passed the remainder to the guest in the next-lower-numbered room.
 
also on top of the Hilbert hotel argument, u have aristotle (and aquinas, among others) pushing the unmoved mover shizzle


((((((((((((
Aristotle's argument for the existence of the unmoved mover:

There exists movement in the world.

Things that move were put in motion by something other than themselves.

If everything that moves was caused to move by something else, there would be an infinite chain of causes. This is impossible.

Thus, there must have been something that caused the first movement.

From 3, this first cause cannot itself have been moved.

From 4, there must be an unmoved mover.
 
****ALSO, sorry about posting it unsober, i hope i didnt confuse u too much cuz i think i contradicted myself and shizzle in it...if this isnt clear, ill try to explain it better
 
*********also, what do you mean you can disprove anything with a large set of "assumptions?" whats the assumption? its a thought experiment
 
However, in an infinite hotel, the situations "every room is occupied" and "no more guests can be accommodated" do not turn out to be equivalent. There is a way to solve the problem
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what problem?

neither of these situations exist in a hotel with an infinite number of rooms

i'm really tired and couldn't make any sense out of your explanation, so i read the link you gave.

it doesn't try to disprove infinity at all.
it seems that you misunderstood what it is about
the last sentence is : This property is sometimes taken as a definition of infinity
 
i think the problem many people - myself included - with infinity is that the first thing they try to do to conceptualise it internally is to quantify it. we think of a huge number and add one to it. or we think of that huge number and we multiply it by itself. this is futile.

there's a quaint story which tries to help people understand how 'long' forever is. i'm paraphrasing but it goes something like this:

"imagine a fly on a planet a billion light years (that's 5,878,499,799,944,615,912,808 miles) from earth. once every billion years (that's 1,000,000,000 years or, say, 12,500,000 lifetimes), the fly flies a billion light years to earth and pick up one atom. it then flies a billion light years back to its own planet. it repeats the journey once every billion years until every one of the countless billions of atoms in the whole of planet earth are transported to the fly's home planet.

can you imagine how long that is? is it even conceivable? well, however long it is, it doesn't even begin to scratch the surface of "forever"."

futile, right?

maybe part of the problem is that we're not even wired to grasp the concept in certain ways.

when i read of dimensions beyind the ones i experience and understand, i struggle. what is the meaning of a 5th, 6th, 7th dimension to me?

so, in trying to understand concepts like infinity, i think you need to go beyond the thing itself and try to understand it from a different perspective - try to learn why it's so hard to understand.

i think that asking yourself what it means to draw a four-sided triangle or reading "flatland" is a much more productive way of understand infinity than trying to picture a million billion trillion gazillion times a million billion trillion gazillion. plus one :)

alasdair
 
maybe i misunderstood it...i always took it to disprove the concept of infinite regress because there could never be a starting point in time. Then, you can jump to the unmoved mover conversation because of that.

idk, i'm not saying i'm 100% right, and will change my opinion, but i think it disproves the concept
 
yougene said:
Infinity is a hard concept for humans to grasp. Infinity has no beginning and no end. It never began and never ended. Mathematicians say the number 0 embodies the nature of infinity. You multiply infinity by 5 and you still have infinity. You add 5 to infinity, you end up with infinity and 5 more, which is still infinity.
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I think infinite and nothing are two parts of the same thing. How many sides does a circle have? It has infinite sides and it has no sides. If you trace the digits of pi, it goes on forever. At the same time, it never ends. Therefore it is never complete, and ceases to be at all. You can walk an infinite distance around a circle, but you will never go anywhere. Nothing is infinite, and infinite is nothing.

yougene said:
This is why I think humans cannot comprehend infinity. The way our brains work is by making constant comparisons and connections within our realm of consciousness. To view infinity you must look at the collective whole of the universe that you are trying to observe. Infinity is however reported to be observed while under the influence of psychedelics although it always lacks coherent description. I think this happens because under the influence of psychedelics you view things as a whole. Not seperate parts of the universe, you simply look at the universe. It is like there is nothing to compare to so u simply look at infinity.
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We have to separate the universe into parts to understand it, I agree completely. In it's pure nature, the universe is infinite. Anything you find in it will have an infinite. We are vessels of finite energy, the universe's infinite nature is overwhelming for us. But we can comprehend it. Meditation allows you to feel your energy with the whole infinite, eternal energy of the universe. Feeling is knowing is comprehending. And when you have that feeling you will know, that is absolutely everything while at the same time it is absolutely nothing.
 
sunset117 said:
There is clearly a concept of infinity, but it is not real.
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What do you mean by "real?" Isn't infinity real inasmuch as it is a real concept? Take the number 4. Is it real? I say yes because 4 is always the same no matter who thinks of it. In other words, 2+2 always equals 4 because 4 is real--it's not dependent on someone's subjective mind.

Even things that seem to be obviously not real, like fiction, might be real if you believe in the framework of possible worlds. But this is a whole other subject that has nothing to do with infinity. [BTW, does anyone know of any good threads on "possible worlds"?]

What I'm trying to say is, you assume that the only real things are physical things that can causally interact with matter. But why can't numbers be real. Yes, they are concepts, but that doesn't automatically make them not real.


BurnOneDown said:
Feeling is knowing is comprehending.
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Hold on there. Feeling is not equal to knowing. Some might say that it is because intuition (or feeling) can guide people just a good as knowledge, but knowledge is knowing why you know not just knowing that you know.
 
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listerbean said:
What do you mean by "real?" Isn't infinity real inasmuch as it is a real concept?
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He's saying there can be 4 apples, 5 apples, any number of apples. But there can't be an infinite number of apples. The number of apples/atoms/anything is always tangible even if we don't have the tools to grasp it and label it infinite.

PGTips said:
Loads of people don't get that "0.9999r = 1 (r means recurring)"
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That's not the problem. The problem is that when you want to apply it to an infinite universe(everything).

If .99999r = 1, then we have the singularity. However, as a friend said, it follows that 1 - .99999r = 0. So outside the singularity there should be nothing, as defined, but there is a 1 at the end of that .0000...1, and that 1 is a something that is outside of the everything, and a something in a nothing. I find it much easier to deny the existence of infinity than write of that 1 as a property of infinity.
 
the definition of infinity: the length of time it would take to comprehend the "infinite"

but seriously, some definitions are quite simple. moving in a straight line on the surface of a sphere has neither a beginning nor end.

where does a doughnut begin and end?

edit: funny, moment after making this post, the missus changed the channel on the tv to a doco about coffee talking about the boom in doughnuts in canada.
 
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I think that even if the universe isn't spatially infinite, or even if it doesn't contain an infinite amount of matter, the universe is still "infinite" in the sense that different perceptions of it are arbitrarily extendable (through meta-cognition and new perspectives). Because cognition must also be part of the universe, the universe by extension is infinite.

ebola
 
^yes but in a finite amount of space and matter there still is a finite sum of possible perceptions.

i think it's bloody annoying when scientists use the word infinite when they actually mean "really fricken big"

to me that's like making light of the term. almost disrespectfully actually. should the infinite contain consciousness and it noticed this occuring, i'd be surprised if it wasn't thinking "hey, screw you, buddy!".
 
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