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comment Is $\aleph_0 = \mathbb{N}$?
@GitGud: There's Scott's Trick, which is useful when working in ZF without choice but with foundation. In this representation $\aleph_0$ is the set of all countably infinite sets of rank $\omega$, so instead we have $\omega\in\aleph_0$.
3m
revised Is $\aleph_0 = \mathbb{N}$?
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5m
comment Is $\aleph_0 = \mathbb{N}$?
Obviously not, since a set cannot contain itself. In axiomatic set theory we have $2=\{0,1\}$ by definition, but $2\ne\{1,2\}$.
6m
comment Is $\aleph_0 = \mathbb{N}$?
@Renato, It may be if $a, b, c, d$ happen to equal $0$, $1$, $2$ and $3$.
1h
comment Positive integer solutions to $a^{a^a}=b^b$
Special case of can different power towers have the same value?.
2h
revised Show that there is a number on the form $11 \dots 000 \dots 0$ divisible by 2014
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2h
answered Show that there is a number on the form $11 \dots 000 \dots 0$ divisible by 2014
2h
comment $f^{-1}(S)$ of a recursively enumerable set
@querty: A function is total computable if it is computed by some Turing machine that halts and outputs $f(n)$ no matter which natural number $n$ it is given as input. This is indeed often shortened to simply "computable". A partial computable function can have a subset of $\mathbb N$ as its domain, and there must be a Turing machine that halts and outputs $f(n)$ when the input is in the domain, and otherwise doesn't halt.
2h
revised $f^{-1}(S)$ of a recursively enumerable set
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2h
answered $f^{-1}(S)$ of a recursively enumerable set
2h
comment $f^{-1}(S)$ of a recursively enumerable set
@AndresCaicedo: Church's thesis will no doubt be appealed to somewhere in the proof if the underlying definitions are based on Turing machines -- but the tricky part is to figure out what to apply it to.
3h
answered Are vectors $e_i$ linearly independent if and only if the matrix $A$ is nonsingular?
4h
comment Is it true that $|f(x)|\leq |f^2(x)|$?
Note that this has nothing in particular to do with functions. Whether $|y|\le|y^2|$ depends only on which number $y$ is; it is not influenced by $y$ being the value of a function at some $x$.
6h
comment Can 720! be written as the difference of two positive integer powers of 3?
Interesting how your solutions aim for a contradiction from the sizes of the numbers involved, whereas I'm looking at the ternary expansons from the least significant end ...
6h
comment Can 720! be written as the difference of two positive integer powers of 3?
@R..: And my answer basically argues that the last nonzero ternary digit of $720!$ is $1$ whereas the last nonzero ternary digit of $3^x-3^y$ is $2$, so they can never be equal.
6h
revised Is the free abelian group of rank 2 linear?
added 292 characters in body
6h
answered Is the free abelian group of rank 2 linear?
8h
awarded  Nice Answer
10h
revised Probability of randomly selecting one student from each of three cities
Don't shout, please.
10h
answered What is the meaning of Right Hand Limit at $\infty$?