(Is it a set?) Set of all months having more than 28 days.

Question

Set is a well defined collection of distinct objects.

Is the following is a set?

Set of all months having more than 28 days.

I'm confused here. Because on one hand I think that it is well defined because from person to person its meaning is not changed. On the other hand I think that it is not well defined because if we consider a leap year then February is included else not. Note that the year is not specified, I'm you cannot surely say that February is included or not.

Set of eleven best cricketers of the world.

This is not a set because the criteria for best cricketer changes from person to person.

So the set of all months having more than 28 days. Is it really a set?

Answer 1

In order to answer your question, you need to preliminary answer this one: What is the length in days of each month?

It seems to me that the function length on the set of months is not defined (it could serve as an example of something that taking two different values on a point in the domain is not a function).

Let $\cal M$ be the set of months. In order to fix the problem you have, imho, two possibilities.

Either you restrict the question to the subset of $\cal M$ on which length is defined, or you replace $\cal M$ with a bigger set $\cal M^\prime$ endowed with a surjective map $\pi:\cal M^\prime\rightarrow\cal M$ on which the function length can be extended.

Answer 2

Given that you say that the "Set of eleven best cricketers of the world [...] is not a set because the criteria for best cricketer changes from person to person," then wouldn't the "Set of all months having more than 28 days" also be not a set because the number of days of the month changes from year to year?

Answer 3

Sets are collections of mathematical objects. Months are not mathematical objects, they are an arbitrary period of time relevant to the physical universe.

If you model the universe as something with a timeline, and define some partition of the timeline to intervals which mimics the idea of months and days, then the answer is positive. But once you get down to brass tacks you will see that the question, as statement is not well-defined, since there is no obvious way to model this which is truly compatible with our experience (did months exist before humans did? what happens when the sun dies out? etc.).

But if you say something like "Consider $\Bbb N$ as the timeline, where $n$ represents the $n$-th Planck time from the big bang. One day is an interval of length $k$, and so on and so forth. Once you do that, you can easily define the set of all months having more than 28 days. It's just a set of sets of natural numbers.

You can use $\Bbb R$ instead if you want, or $\Bbb Q$. Or any other way which is compatible with how you think about time. If you've used a set, then the collection of "months" with such and such property will also be a set.