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Anonymous at Sat, 9 Nov 2024 01:08:56 UTC No. 16467809
Is it just me or would a lot of things in math make more intuitive sense if the concept of a "set" was replaced with the concept of a "label" (or something similar)?
It makes me realize that the whole idea of a set needing to "contain" its elements is kind of just made up. There is no need for a "containment" type of relationship.
So instead of paradoxical ideas like sets containing themselves, you would simply have labels that apply to themselves. That's way easier to think about.
It's also more truthful to the concept of infinity. Because infinity is not supposed to be an actual number, there is no point at which you've reached infinity and therefore the idea of anything infinite being "contained" (or "in the set") is counter-intuitive.
But if it was just called a "label" then there is no confusion. Any number you can write down simply fits the rule of the label, there is no need to say that it was "in the set", or "in" anything at all.
Yes, I know this doesn't change anything about how math is actually done. I just think it would be more intuitive, especially when first learning the concepts.
Anonymous at Sat, 9 Nov 2024 01:31:15 UTC No. 16467831
>>16467809
I think that is fair, and especially is a useful way of thinking about continuous sets (which are a bit puzzling to really think about even in the simplest cases as intervals). With that said, I think this intuition is more or less the way that abstract sets are written about in measure theory and functional analysis texts. If you have a measurable function, the pre-image of a particular interval is simply whatever Borel set fits the rule. It's really just a label applied to whatever you rule need it to fit rather than some clearly defined "container" relationship.
With that said, I think it's sort of a duality thing. The notion of a subset is very useful an important for just about every analytic discipline. Sub-label is maybe less intuitive than simply allowing for some "generalized" notion of a set containing another set.
Anonymous at Sat, 9 Nov 2024 01:56:23 UTC No. 16467862
>>16467809
I think the intuitiveness is quickly going to fly out the window when you have to start defining set operations