That’s the beauty of anonymous commenting ;) It enables you to be as politically incorrect as you like to be.

FYI, finite degree polynomials represent a very limited set of curves. Even a curve as simple as a sine or cosine or exponential requires an infinite degree (Taylor’s expansion) polynomial.

okie can you construct infinity larger than set of all possible curves in a plane or space ?

Absolutely bullshit question to have been asked by someone at IIIT. Its somewhat like asking whether 99999999999999 is the largest integer. You can always construct a bigger set. For a trivial example let ‘B’ be a set which has a lone ‘piece of shit’ as its member and call the set of all curves as ‘A’. Then the cardinality of ‘A U B’ is obviously greater than cardinality of A.

And If I am not wrong the example given earlier explains the existence of such type of sets — “if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is ‘abnormal’ ”

@Author — Realy an eye opener. Nice!

I think…

We have got a paradox here. This means something must have been wrong/inconsistent somewhere before . I am considereing the paradox as a contradiction. And the fact that abnormal set is a set at the first place as the hypothesis. An abnormal set has itself in it, how can we accept that to be a set so straightaway. Afterall, a set is a collection of well defined objects. A question which should be asked is “If a set has itself in it, then is it a set?” or “Is the set as an element of itself a well defined object?”. So, an abnormal set might not be a set!

Now, consider the paradox to be contradiction. Then the hypothesis that Abnormal set is a set wrong. That is to say abnormal set is not a set!

So, this paradax was actually the proof of “Abnormal set not being a set”.

A Corollary would be, For a set to be a set it must not contain itself.

Maybe such kind of “not set” sets could be called pseudo-sets in future.

Here is a formal proof for the theorem, “A set cannot contain itself.”

Hypothesis: let a set contain itself and still be called a set. (i.e. Abnormal sets are sets)

Then, All the paradox matererial comes here.

Since we get a contradiction (the paradox), the hypothesis is false.

Therefore, a set can’t contain itself.

The members of power set of A and those in A are incomparable — rather, they represent different things altogether. For instance, consider the following example to illustrate the difference between a set and a power set. Let A represent the computers in a network. Say, each of them has a secret key to access something. Let P` be a collection subsets of A. I design a system as follows - Let x be an element in P`; When all players in x use their secret keys together can open the secret. In a way, I can view P` as an Access Structure defining access to some secret. The elements in A and P` are incomparable in this current form.

Probably, your question might be motivated by looking at natural numbers and real numbers. Even here, there is not direct co-relation. For example, can we find an element in the power set of natural numbers missing from natural numbers? We can however attempt to seek an answer for the above question when we represent — the power set natural numbers as a real numbers. So, representation is the key here.

In general, I think it will not be fair to compare the elements of a set and its power set.

Another one would be, consider the space you are considering to be n-dimensional — Then every curve is a polynomial over (x1, x2, x3,…….. xn). Let the degree of the polynomial be m. So, every polynomial can be represented as a 2^m + 1 tuple with each number of the tuple indicating the coefficient in the binomial expansion.

So, in a way any curve can be written as a real number with the understanding that — a real number can be split into 2^y+1 parts and each part gives the coefficient in the binomial expansion. For making the real number unique, you may use a separator between. But, once you use a separator — each one happens to be a collection of 2^y+1 reals.

Hence, it is easy to see that the power-set of Real Number is of the order of these curves. Therefore, power set of power set of Reals is larger infinity than the set of curves in space.

You are due a treat from me :-)

The set of Integers/natural numbers is the smallest infinity and the set of reals is of the order of their power sets.