This is an excellent question with many possible answers.

For an example of intuitive results based on the Zermelo-Frankel set theory, consider the Axiom of Abstaction.

Axiom of Abstraction.  Given any property, there exists a set whose members are just those entities having that property.

Example 1. 

Nearness Property: nearness of adjacent polygons.   The term nearness here is understood in terms of the Hausdorff metric that says that a pair nonempty sets are near, provided the greatest lower bound of the distances between pairs of elements in the sets is zero.

Theorem 1.   Adjacent polygons are near sets in Dirichlet tessellation of a plane surface.

Proof:  Let A, B be adjacent polygons in a tessellated surface.   Adjacency here means that A and B have a common edge.   In that case, we can find x in A, y in B such that d(x,y) = 0.   This happens for each x,y in an edge common to A and B.  Hence, by the nearness property, A and B are near sets.   Consequently, we can "grow" a set of polgons that are near a given polygon.   [ ]

Example 2.

Remoteness Property: remoteness of non-adjacent polygons.   The term remoteness here means that the greatest lower bound of the distances between pairs of elements in a pair of nonempty sets is greater than zero.

Theorem 2.  Non-adjacent polygons are remote sets in a Dirichlet tessellation of a plane surface.

Proof: Symmetric with the proof of Theorem 1, except that now we observe that d(x,y) > 0 for all x, y in edges of a pair of non-adjacent tessellation polygons.   From the Remotness Property, we can "grow" a set of polygons that are remote from a given polygon. [ ]

Since these examples have strong visual content, perhaps everyone will agree that these results are intuitive.   But in terms of the required understanding of metric spaces, the Hausdorff metric, and Dirichlet tessellation, it is also the case that these sample results are not so intuitive.  For more about ZF theory, see

http://tachyondecay.net/workspace/downloads/benbabcock_honours_thesis.pdf

The most un-intuitive for me is Serpinski's proposition that the 3-dimensional ball can be cut in several pieces and this can be rearranged in such a way as to make two balls with the same size.

For me this result means that transfinite induction is BS and would only ever use it for conter-examples.

Dear Milen Ivanov,

Many thanks for your very nice example.   Perhaps you will find the following video about Sierpinski fractals of interest:

https://www.youtube.com/watch?v=SsmP9BEY5Lo

and

http://fractalfoundation.org/resources/fractivities/fractal-cutout/

and

http://www.georgehart.com/rp/rp.html

The mathematics for a Sierpinski sieve is given at

http://mathworld.wolfram.com/SierpinskiSieve.html

Dear Milen Ivanov,

                                     What you mention is the Banach Tarski Paradox that requires AC. The Sierpinski-Mazurkiewicz paradox does not require AC (and yes, it is similar) and says that there exist subsets of the plane that can be partitioned into two copies of itself.

Su, Francis E., et al. "Sierpinski-Mazurkiewicz Paradox." Math Fun Facts.

The BT paradox uses non-measurable parts, while in the SM paradox the subset has measure zero. It can be argued that both paradoxes are due to the axiom of infinity. From a physicalist point of view BT paradox is not a paradox as the parts generated  are not actualizable.

But in this question we are talking of ZF + not AC

In models of ZF+ not AC, the reals form infinite sets with no countably infinite subset. This in my opinion is among the more counter-intuitive results.

Dear Prof. Mani,

Thank you for the clarification! I must have overlooked ~ sign.

I have come to accept CF as occupational hazard in Analysis. :)

Best Regards,

Milen