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1d
revised How can mathematics work in wildly different set theories?
Fixing a piece of careless language.
1d
comment How can mathematics work in wildly different set theories?
@AsafKaragila: I get predictable when NF comes up :P
1d
answered How can mathematics work in wildly different set theories?
1d
answered I don't understand the axiom schema of separation.
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comment I don't understand the axiom schema of separation.
Suppes allows that non-sets exist--that is, objects which have no members and are distinct from the empty set. If you omit the first part of that conjunction, then it's not possible to infer that, say, $\{x\in A:x\neq x\}$ and $\{x\in A: \forall y(y\in x)\}$ are the same object. Which makes things weird.
1d
comment I don't understand the axiom schema of separation.
What book are you working from? This looks like Suppes, in which case the first clause of the conjunction is saying that separation doesn't give you an urelement.
Apr
23
comment Why are there no “continuous maps” in algebra.
Can you elaborate on the kind of behavior you're talking about?
Apr
23
answered Composing functors with natural transformations
Apr
20
comment Set theory question on the definition of surjection
Showing once again that an author that would use no delimiters between quantifiers and their scope would also steal sheep....
Apr
20
revised question about a theorem in Maclane-Moerdijk's “Sheaves in Geometry and Logic”
added 322 characters in body
Apr
19
comment question about a theorem in Maclane-Moerdijk's “Sheaves in Geometry and Logic”
Let me know if I need to expand on any of this. I realize it may be a little dense.
Apr
19
answered question about a theorem in Maclane-Moerdijk's “Sheaves in Geometry and Logic”
Apr
17
comment “There is no set containing everything”?
@NoahSchweber: $U$ properly containing $\mathcal{P}(U)$ is a different statement than $|\mathcal{P}(U)| < |U|$. The former is definitely true with urelements even without choice, but the latter doesn't obviously follow without Choice.
Apr
17
comment “There is no set containing everything”?
@Noah: I overstated the case; at least as of the last revision of Elementary Set Theory with a Universal Set, Holmes was not aware of any proof that did not use Choice, but comments that it is unknown whether you can get that result without Choice. I believe last time I spoke to him this hadn't changed, but I wouldn't put much stock in my memory.... :P
Apr
17
comment “There is no set containing everything”?
@NoahSchweber: Specifically, that happens in $\mathsf{NFU+Choice}$. $\mathsf{NF}$ just proves $\mathcal{P}(U)=U$, which is slightly less odd looking.
Apr
17
answered “There is no set containing everything”?
Apr
17
comment Are all algebras groups?
What you describe above is a commutative magma; commutativity ($a*b=b*a$) is not a necessary property of groups, and the properties of being associative, having a unit, and having inverses would still need to be verified. It's easy to confirm that a Boolean algebra is not a group under either alternation or conjunction.
Mar
7
comment Show that (p ∧ q) → (p ∨ q) is a tautology?
@TRiG - I'm pretty sure it's just a tearful face emoticon.
Jan
31
answered Defining natural transformations based on generalized elements?
Jan
31
comment Defining natural transformations based on generalized elements?
When you express a doubt about the existence of "this morphism" in $\mathbf{C}$, which morphism in particular do you mean?