Prove infinity+1=infinity (1 Viewer)

[withoutaface]

I was wondering if there is any way to prove that oo+1=oo other than 'by convention'. I've included my attempt which is probably wrong, so yeah, any takers?

 

[Xayma]

Infinity isn't a number, it is a concept, your trying to combine numbers and concepts which dont like being combined.

 

[withoutaface]

Infinity isn't a number, it is a concept, your trying to combine numbers and concepts which dont like being combined.

I know that, but that's why I'm so interested in finding out.

 

[Xayma]

Well consider the non-negative integers 0,1,2,3,4,5.....
Now consider the positive integers 1,2,3,4,5....

Now there are ∞ non-negative integers.
There are ∞ positive integers.

However the non-negative integers contain all positive integers + one extra term.
Therefore ∞ +1=∞

 

[wogboy]

Strictly speaking ∞ is only allowed as a limit, at the end of an arrow e.g. lim{x -> ∞} 1/x, not in an equation nor an expression e.g. ∞ + 1

 

[Archman]

well you can form a bijection between the set of positive integers and the set of non-negative integers, i guess thats enough.

 

[abdo]

well you can form a bijection between the set of positive integers and the set of non-negative integers, i guess thats enough.

what the? what is this bi... jection? not to worry... i got my trusty merriam-webster... eureka!!!

bijection: a mathematical function that is a one-to-one and onto mapping

 

[Affinity]

Strictly speaking ∞ is only allowed as a limit, at the end of an arrow e.g. lim{x -> ∞} 1/x, not in an equation nor an expression e.g. ∞ + 1

it's more than a limit sometimes.. such as the sizes/cardinality of some sets.

oh.. by the way, with the bmp file, deducing a true statement from what you want to prove does not prove it..

as wogboy mentioned, you can't use inifinty as an ordinary number, it doesn't satisfy basic properties under + and X that numbers would.

 

[Iota]

Consider the set of all cardinal numbers:
{1,2,3,4,5...}
This set has a one-to-one relationship with the set
{2,4,6,8,10...}

Each element in the set matches. Hence, we can say that the two sets are equal.

Now, consider the sets
{1,2,3,4,5...}
{shoe,1,2,3,4...}
Again, we can say that each ellement in the set matches, and hence, the sets are equal.

So it follows that the two sets:
{1,2,3,4,5...}
{2,3,4,5,6...}

Are also equal.

inf + inf = 2*inf = inf
inf + c = inf
inf * c = inf
inf * inf = ? (one would say infinity, but what degree of infinity? Perhaps it is merely aleph-null, in which case infinity is the correct answer, again. I actually forget this one)
inf / inf = ?
inf - inf = ?
inf / c = inf

It's called transfinite arithmetic.

Do a Google search on George Cantor. He introduced mathematical rigour into the concept of infinity, and it is in his works which you shall find your answer.

 

[gman03]

Consider the set of all cardinal numbers:
{1,2,3,4,5...}
This set has a one-to-one relationship with the set
{2,4,6,8,10...}

Each element in the set matches. Hence, we can say that the two sets are equal.

No they are not equal thou both are infinitely countable sets. They are NOT equal for sure. Two sets are bijection does not necessary mean they are equal. Read your sources again. In fact post your sources.

2*inf = inf

wtf? 2*inf = inf? so 2*inf - inf = inf - inf? so inf = 0

I so damn dislike you, you are ruining the art of mathematics.... what you have typed is called <b>false implies true</b>

Wogboy is right, you can't put numbers and concepts into equation.

"Strictly speaking ∞ is only allowed as a limit, at the end of an arrow e.g. lim{x -> ∞} 1/x, not in an equation nor an expression e.g. ∞ + 1"

 

[Iota]

No they are not equal thou both are infinitely countable sets. They are NOT equal for sure. Two sets are bijection does not necessary mean they are equal. Read your sources again. In fact post your sources.

"Equal" was a bad term to use, but since the target audience is not university students, equal is what I stick by. The correct term is "one-to-one correspondence".

wtf? 2*inf = inf? so 2*inf - inf = inf - inf? so inf = 0

It's called TRANSfinite arithmetic for a reason. It doesn't obey normal mathematics.

2*inf = inf, but 2*inf - inf is not a definable operation.

Here's a source for you. Note that the strange X-like character to base 0 is aleph null, which is also known as cardinal infinity.

http://cs.wwc.edu/~aabyan/CII/BOOK/book/node42.html

http://mathforum.org/library/drmath/view/51472.html

However, I will admit my error in saying that divison is an allowable operation.

I so damn dislike you, you are ruining the art of mathematics.... what you have typed is called <b>false implies true</b>

I am sorry for that, as I have nary even a trifle of desire to "ruin the art of mathematics". However, I aslo believe I am not doing so, since I am only showing you a mathematical concept which you are not familiar with. Is this wrong of me?

Wogboy is right, you can't put numbers and concepts into equation.

Oh, you can to some degree. And George Cantor proved this. Just remember that "normal" arithmetic laws are not the same as transfinite arithmetic laws.

Have a nice day.

 

[Archman]

"Strictly speaking ∞ is only allowed as a limit, at the end of an arrow e.g. lim{x -> ∞} 1/x, not in an equation nor an expression e.g. ∞ + 1"



ye infinity its not really anything, i mean, u need to define something more specific if u want to do calculations and stuff. i mean u can say there is infinitely many integers and reals, but there are a LOT more reals. (or more technically, you cannot find a bijection between reals and integers, only reals and the power set of integers)

 

[gman03]

I am only showing you a mathematical concept which you are not familiar with. Is this wrong of me?

Nope there is nothing wrong with you demonstrating concepts. However I believe there is some problems associated with your understanding of such concept.

For example:

Consider the set of all cardinal numbers:
{1,2,3,4,5...}
This set has a one-to-one relationship with the set
{2,4,6,8,10...}

and from http://cs.wwc.edu/~aabyan/CII/BOOK/book/node42.html (42 )

the set of all subsets of the integers <b>cannot</b> be put in one-to-one correspondence with the integers, that this set is really a different size of infinite set, truly larger

clearly you misunderstand some part of it. And I was a little bit piss about that. THat just my attitude, please do not take offense from my capital letters and bold letter. I apologise.

Have a nice day.

You too

 

[Iota]

Hmm. I see. Sorry, I was explaining as it was taught to me.

T'is all good.

 

[xiao1985]

i would think infinity + any finite number = infinity
infinity minus (or divide) infnity is undefined...

sigh~~~~~~~~ i love pure mathematics... but then above are mere speculations...