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"We encourage children to read for enjoyment, yet we never encourage them to 'math' for enjoyment. We teach kids that math is done fast, done only one way and if you don't get the answer right, there's something wrong with you. You would never teach reading this way." - Rachel McAnallen

"Mathematicians create by acts of insight and intuition. Logic then sanctions the conquests of intuition." - Morris Kline


22h
comment Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
How does one actually assert, in ZFC, that there exists an inner model in which (say) a measurable cardinal exists? We can adjoin a predicate symbol to do this, but can we do it without the predicate symbol? Also, is the mere existence of an inner model with a measurable (say) the right kind of axiom in your opinion, or would it be better to explicitly choose an inner model (say HOD) and assert that that particular inner model possesses a measurable?
22h
accepted Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
1d
comment Do large cardinal properties tend to be semiabsolute?
@AndresCaicedo, thanks. (Your knowledge never ceases to amaze me.)
1d
comment Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
@AndresCaicedo, interesting.
1d
comment Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
Yes, assuming the axiom of choice, they're precisely the infinite cardinals. (I'm not sure how I feel about this. A non-empty part of me wishes that the aleph numbers started at the cardinal $0$, so that $\aleph_n=n$ for all finite ordinals $n$.)
1d
revised Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
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1d
comment Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
@Martín-BlasPérezPinilla note that $\beth_1$ can vary greatly on the scale of aleph numbers, while $\aleph_1$ can vary greatly on the scale of ordinal numbers. They're different scales.
1d
revised Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
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1d
asked Are there axioms for $\mathrm{ZFC}$ that imply that $\aleph_1$ is very large?
1d
revised Do large cardinal properties tend to be semiabsolute?
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1d
asked Do large cardinal properties tend to be semiabsolute?
1d
comment Understanding infinity
Goldrei's "Classic Set Theory For Guided Independent Study" is pretty good.
2d
comment From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?
@Rahul, thanks, that is a useful comment.
Dec
15
awarded  Nice Question
Dec
15
comment Algebraic Structures Books
Try searching the website for book recommendations for learning abstract algebra.
Dec
15
comment Is there a concise way to notate 'There are exactly 482 x, such that Px…' in logical notation?
If you have access to the expressive resources of set theory, you can write "The cardinality of the set of all muppets is $2$," or more formally "$|\mathrm{Muppets}|=2$.
Dec
15
accepted (Non?)-uniqueness of sums of squares
Dec
15
comment (Non?)-uniqueness of sums of squares
@BalarkaSen, that is an answer.
Dec
15
asked (Non?)-uniqueness of sums of squares
Dec
15
comment Are there any articles (or otherwise) that discuss the idea that the set-theoretic universe should be as “free” as possible?
@AsafKaragila, haha yep. this is one powerful principle!!