infinity to the power of infinity

[gaara]

Hello,

 

i heard from my friend that infinity to the power of infinity is greater than infinity. (e.g opposed to infinity plus 2 would = inifinity).

 

he said there is some mathematical model to prove this.

and analogies.

 

i know that the maths is over my head but could some people give me information about this concept

[DV8 2XL]

Transfinite numbers, also known as infinite numbers, are numbers that are not finite. These numbers were first considered by Georg Cantor.

As with finite numbers, there are two ways of thinking of transfinite numbers, as ordinal and cardinal numbers. Unlike the finite ordinals and cardinals, the transfinite ordinals and cardinals define different classes of numbers.

The lowest transfinite ordinal number is ω.

The first transfinite cardinal number is aleph-null, , the cardinality of the infinite set of the integers. The next higher cardinal number is aleph-one, .

The continuum hypothesis states that there are no intermediate cardinal numbers between aleph-null and the cardinality of the continuum (the set of real numbers): that is to say, aleph-one is the cardinality of the set of real numbers.

In both the cardinal and ordinal number systems, the transfinite numbers can keep on going forever, with progressively more bizarre kinds of numbers.

Beyond all these, Georg Cantor's conception of the Absolute Infinite surely represents the absolute largest possible concept of "large number".

 

See: http://en.wikipedia.org/wiki/Transfinite_number

 

[gaara]

thanks brother.

[ElijahJones]

I had no idea it had gone so far, follow some of those links and you find like 30 different types of cardinals. I like the argument though that you cannot form the set of all sets because it would itself be a set you did not include. Its a nifty little logical circle from one perspective, but I never really saw how it was necessarily a problem because every set is considered a subset of itself. So the set of all sets by definition should already contain itself.

[matt grime]

there is no bijection from any set A to P(A) (in any model of ZF). Thus if there were a set of all sets, U, then card(U)=card(P(U)) since P(U) must be a set larger than U and hence U. But then there would be a bijection between U and P(U), contradiction, thus there is no set of all sets in a model of ZF (Cantor's paradox). Plus if there were a set of all sets then we can use restricted comprehension to create the set of all sets that do not contain themselves and we are led to Russell's paradox.

 

 

If you wish to know how we get round this apparent problem without appeal to the theory of types (which roughly orders mathematical objects as elements, sets of elements, classes of sets of elements and so on), then the analogy to bear in mind is that a vector is not a vector space. I can give you a link that explains that idea if you like.

[ElijahJones]

there is no bijection from any set A to P(A) (in any model of ZF).

 

You are accepting of course the empty set {}.

 

P({})={}

 

Thus if there were a set of all sets, U, then card(U)=card(P(U)) since P(U) must be a set larger than U and hence U. But then there would be a bijection between U and P(U), contradiction, thus there is no set of all sets in a model of ZF (Cantor's paradox).

 

But that makes perfectly good sense, its seems obvious that for any set with cardinality greater than zero card(u)<card(P(U))

 

 

Plus if there were a set of all sets then we can use restricted comprehension to create the set of all sets that do not contain themselves and we are led to Russell's paradox.

 

this I am not familiar with 'restricted comprehension'?

 

 

If you wish to know how we get round this apparent problem without appeal to the theory of types (which roughly orders mathematical objects as elements, sets of elements, classes of sets of elements and so on), then the analogy to bear in mind is that a vector is not a vector space. I can give you a link that explains that idea if you like.

 

If you don't mind I would take a look at that link.

 

Thanks

[matt grime]

THe empty set is not an exceptin. the cardinality of the emty set is zero, the cardinality of its power set is 1.

 

restricted comprehension means that we can only specify sets in terms of other sets. Thus we can talk of the set of real numbers that have real square roots because we are restricting to "the set of real numbers"

 

approximately, you shuold think of "a set theory" (for there are many) as a way of assigning objects with labels, some have the labe "set" some do not. we call this a model. sets in this model must obey the axioms of the theory, there is no way to create a model of ZF that has the collections of all things labelled with "set" also labelled with "set".

 

 

more explicitly the way the restricted comprehension way to define a set in a model of ZF is:

 

S={x in X:P(x)}

 

that is it is the set of x's in some other set X satisfying some property P. It may appear circular on first reading but it isn't becuase it is only a consistency relation.

 

the link can be fuond at

 

http://www.dpmms.cam.ac.uk/~wtg10

 

follow it to his discussions of basic mathematics a la polya.

[ElijahJones]

I was just on my way back to correct that P({})={{}} so the cardinality is one.

 

I'm not sure whether to be insulted about your polya comment or not that you would care.

 

What would be the value of having a different model of set theory?

[matt grime]

erm, a la means in the style of. what is insulting abotu describing some articles written abuot elementary mathematics (first year undergraduate mathematics) as being in the style of polya?