Re: Simple Set Theory question

Gerry Myerson wrote:
In article <1161805363.250372.102560@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
"MoeBlee" <jazzmobe@xxxxxxxxxxx> wrote:

Ron Jones wrote:

Can anyone give me proofs of the cardinality of all FINITE subsets of (a)
natural numbers N, (b) real numbers R, (c) set S with cardinality > cardR,
or alert me to where I can find proofs. Thanks.

It is given as an exercise in Enderton's text to show:

If x is infinite -> x is equinumerous with the set of finite subsets of
x.

And conversely.

Right. And easy even without choice: For a finite set x, the set of
finite subsets of x is the power set of x, and no set is equinumerous
with its power set. Thanks for pointing it out; I always like to make
the theorems into double arrows when possible.

My own question is how to prove:

N is equinumerous with the set of finite subsets of N,

without using even the axiom of countable choice or the prime
factorization theorem, but okay to use the theorem that NXN is
equinuerous with N).

How about matching the finite set {a, b, c, ..., z}
with the natural number 2^a + 2^b + 2^c + ... + 2^z ?

Okay, I see how I can formalize that. But how to prove it is an
injection? I pretty much see intutitively that it is an injection, but
it's not obvious to me how to prove it.

Thanks,

MoeBlee

.

Relevant Pages

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