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2d
comment What is an Empty set?
@BrianJ That's actually one of the most famous paradoxes in the development of set theory.
2d
comment What is an Empty set?
@ruakh What CarlMummert said. Except that I would add that "<X> of real numbers" is terminologically more akin to "grocery <x>" than "<x> of groceries", despite the phrasing.
Apr
30
comment What is an Empty set?
The confusion of "how can a bag of shopping not hold any shopping" only (or at least primarily) exists because "hold any shopping" implies that the items usage, while "bag of shopping" could mean either usage. Hence why I clarified in my first response to you that a bag for shopping (the action usage) is always a bag for shopping, regardless of its contents. (Even if "for shopping" is taken to mean for holding shopping items!)
Apr
30
comment What is an Empty set?
@ruakh I say "slippery" not because it's "incorrect" English, but because the implied analogy to formal mathematical concepts makes no sense due to the multiplicity of meanings involved. If "shopping" means both the action of shopping and the items acquired while shopping, then it's not actually parallel to "groceries" or "holding", since "groceries" only means items and "holding" (in the D&D sense) only means the action of holding.
Apr
29
comment What is an Empty set?
@ruakh I guess so, but "bag of shopping" strikes my ears as something like a D&D-ism, and the noun usage of "shopping" seems a bit slippery. A bag for shopping is still a bag for shopping even when no shopping has yet been done with it.
Apr
29
comment What is an Empty set?
@Tom Depends what axiom schema you're using.
Apr
29
comment What is an Empty set?
@MadeleineP.Vincent Asking isn't sufficient; carefully considering the advice and feedback given on that forum is necessary as well.
Apr
29
revised What is an Empty set?
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Apr
28
revised What is an Empty set?
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Apr
28
comment What is an Empty set?
@SufyanNaeem I think my response holds even if you replace "help" with "help much." In any case, in my answer, I did my best to provide comprehensive answers to each of your questions; did that help at all?
Apr
28
revised What is an Empty set?
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Apr
28
comment What is an Empty set?
@SufyanNaeem No one's saying they "don't believe you"; they're just saying that if you're having trouble understanding that definition (and yes, not understanding why the empty set is a valid set means you're having trouble with the definition), then you should consider some other definitions, or reconsider the definition without trying so hard to make it fit your preconceived notions based on non-mathematical language.
Apr
28
answered What is an Empty set?
Apr
28
comment What is an Empty set?
@SufyanNaeem If the analogy of "a bag is still a bag even when it's not holding anything" doesn't help, I'm not sure what will help. It's unclear what is actually causing the confusion here, since your question simply says that the definition appears to contradict particular examples, but the shopping bag example shows why the definition isn't contradictory.
Apr
28
comment What is an Empty set?
@SufyanNaeem Your other two "problems" are also just misunderstandings of definitions.
Apr
22
revised Is the number $-1$ prime?
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Apr
20
comment Should you ever stop rolling, THE SEQUEL
@Newb Thanks! Unfortunately I never got around to really finishing the answer to my satisfaction, and the question never garnered much interest anyway. Oh well.
Apr
15
comment Is the Banach-Tarski paradox realistic? Why is Volume not an invariant?
I take the paradox to be one of many indications that our concept of the "coninuum" (i.e. the real number line) is a very artificial construct that doesn't correspond to physical, spatial reality. (I personally only "believe in" Turing-computable numbers.)
Mar
8
revised Should you ever stop rolling, THE SEQUEL
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Mar
8
comment How large is the set of all Turing machines?
It might work as a helpful metaphor while trying to intuitively answer the question--e.g., one could ask, "do you believe that the set of all possible inputs for a compiler is countable? Why or why not?" But even there, it's clear that the solution to the initial question has to do with these inputs, i.e., the strings in the language, not with the compiler.