40,069 reputation
374145
bio website math.lsa.umich.edu/~ablass
location US
age
visits member for 4 years, 4 months
seen 5 hours ago

1d
comment Definability of arithmetic functions and relations
@HansStricker The existence of those automorphisms is essentially the fundamental theorem of arithmetic, that every natural number admits a unique factorization into primes. So, for multiplicative purposes, all the information about a natural number $n$ is in the exponents $e_2,e_3,e_5,\dots$ of the primes $2,3,5,\dots$ in the factorization of $n$. Multiplying numbers amounts to adding corresponding exponents. From this viewpoint, it's obvious that permuting the primes amounts to just listing the exponents in a different order and doesn't affect the structure.
1d
comment In the category of sets epimorphisms are surjective - Constructive Proof?
@JochenWengenroth This proof would be the same as the one in your comment if classical logic were available. As Emil pointed out in the comment after yours, your formulation assumes that every $y$ is either in $f(X)$ or not, so that the two cases in your definition of $g$ cover every $y$. Andrej's answer avoids that case distinction, so it is valid even in constructive logic. The price for that is that, unlike your $g$ whose values are 0 or 1, his $f$ takes values not (constructively) known to be $\varnothing$ or $\{b\}$.
1d
answered Definability of arithmetic functions and relations
1d
comment Are there fragments of set theory which are axiomatized with only bounded (restricted) quantifiers used in axioms?
@FrançoisG.Dorais I must confess that I don't see much similarity between this answer and Ackermann set theory. In particular, important features of Ackermann's class of all sets are that it's not definable unsing only $\in$ and it's a member of various higher classes. Martin's $U$ might well be the only non-set in some model, so it might be definable and might not be a member of anything else.
Aug
17
comment Squares in a square grid
What is a "square $a\times a$ permutation"?
Aug
17
comment How short can we state the Axiom of Choice?
If $\iff$ counts as a logical symbol, then you can say "nonempty pairwise disjoint sets" faster by saying that two of the sets have a common member if and only if they are equal.
Aug
17
comment How short can we state the Axiom of Choice?
I conjecture that the selection operators that you want to exclude are things like Hilbert's epsilon operator (also known as Bourbaki's tau operator) that are built into the language and thus try to make AC part of the underlying logic. In other words, I conjecture that you would not object to a formulation asserting, in the usual language of set theory, the existence of some sort of selection function.
Aug
17
comment Directed subposet of a poset containing the minimal elements
Are you allowing the possibility that, for certain $x$ and $y$, the set $\{z:x<z>y\}$ has no minimal elements? You assumed only that the whole $P$ has plenty of minimal elements, not that its subsets do.
Aug
15
comment A question about ordinal numbers and sub-theories of ZF
"Close (2)" means that 2 people have voted to close the question. It takes 5 votes (or a moderator) to actually close the question.
Aug
15
comment How many turns will it take to draw all cards from a deck with shuffling and replacement?
I think this is known as the coupon-collector problem.
Aug
14
comment action of structure group on pullback of torsors
In addition to Jason's suggestion to write out definitions carefully (with which I agree), I have two more suggestions. (1) You presumably meant that $Y\times_XW\to W$ (not $Y\times_XW\to X$) is a $G$-torsor. (2) Remember that the action of $G$ on $Y$ commutes with the projection to $X$.
Aug
13
comment A question about ordinal numbers and sub-theories of ZF
"Ordinal numbers are defined by such formulae" could mean (in analogy to what you wrote for cardinals) "there exists a formula ... containing x as its one and only free variable, which expressed the statement 'x is an ordinal number'." Or it could mean "for each ordinal $\alpha$, there exists a formula, containing x as its one and only free variable, expressing $x=\alpha$". The former is true; the latter is false.
Aug
13
comment Density with infinite cardinals
@user57037 I edited some more information into the answer.
Aug
13
revised Density with infinite cardinals
added 969 characters in body
Aug
12
answered Density with infinite cardinals
Aug
12
comment A question about ordinal numbers and sub-theories of ZF
If I remember correctly, Lévy's theorem is not about defining an individual number (cardinal or ordinal) but rather about defining what it means to be a cardinal.
Aug
12
comment A question about ordinal numbers and sub-theories of ZF
What does it mean for the existence of an ordinal number $\alpha$ to be provable in ZF? If we have a definition of $\alpha$, say $\alpha$ is the unique ordinal satisfying $\phi(x)$, then provability of $\exists x\,\phi(x)$ makes sense. But what of undefinable $\alpha$? (Ignore them?) Worse, what if the same $\alpha$ has several definitions and ZF proves existence for one definition but not another? As a platonist, I understand "$\phi(x)$ defines $\alpha$" in terms of truth of $\phi$ in the real world; what can a non-platonist do?
Aug
9
answered angle inequality in n-dimensional vector space
Aug
8
comment Is it possible to write a research statement which is too short?
Longer proofs can also justify a short research statement. Andrew Wiles's research statement could perfectly well be "I proved Fermat's last theorem and I plan to continue working in number theory." But if you're not Wiles, you'd better say more.
Aug
7
revised Question about higher inductive types and computational rules
changed "Computation" to "Introduction" at end of 2nd paragraph