Assume x is the smallest real number greater than 0. Then x/2 is also a real number and is greater than 0 but less than x. Therefore, x can't be the smallest real number greater than 0.
Well only to the extent that you don't want to throw away any of the other axioms. Sometimes you do and there are some fun systems of math, but few have any practicality and those that do are often so advanced that even someone with an undergraduate focus in math can't appreciate those systems.
It is much the same with computer science. I personally enjoyed playing around with formal concepts of computation and adding some extras to see what happens. For example, what happens to a Turing machine if part of the machine can time travel or has access to an oracle. Does this make concepts like time travel inherently contradictory to our notion of computation?
But the practicality of these exercises does not exceed their entertainment value.
The problem is that if we include such a number in our formal system of math, we quickly find contradictions and the whole system falls apart. So such a number is incompatible with any formal system of math (though I guess you could start building one which does include such a number and see what properties it has).
Herein lies the problem, the people you are talking with do not use a form system. There system of math has something similar to the same flaw of their system of grouping of things, which would include the whole grouping that contains every grouping that doesn't contain itself. People rarely deal in formal systems and thus they can handle completely illogical statements fine as long they are protected from seeing the consequence of it.