3,100 reputation
1427
bio website dorais.org
location Hanover, NH
age
visits member for 4 years, 5 months
seen 7 hours ago

I like math and a few other things...


Mar
8
comment Are there statements that are undecidable but not provably undecidable
@JDH's objection is correct but there is more. The implicit assumption is actually that ZFC+Con(ZFC) is $\Sigma^0_1$-sound: that this theory does not prove any false $\Sigma^0_1$-statement. If ZFC+Con(ZFC) is not $\Sigma^0_1$-sound then there is a Turing machine $M$ such that ZFC + Con(ZFC) proves that "$M$ halts" but $M$ doesn't actually halt...
Dec
23
comment Problems with introducing ordered pairs axiomatically
Andrej, this doesn't answer the question as posed. The OP is asking about a global pairing function, not about the existence of products, which only gives local pairing functions.
Aug
26
comment Special subgroup of a group of order $n$
@RobertM: When in doubt, flag for moderator attention and explain what you want to do. There are mechanisms set up to transfer questions from one site to another.
Aug
17
comment Category of profinite groups
If you want this question to be on MathOverflow, ask for it to be migrated. Please don't crosspost!
Jul
28
comment clearing doubt over a definition
The description on Wikipedia seems pretty understandable to me. The original paper could also be helpful - dx.doi.org/10.1109%2FSFCS.2000.892006 The bracket notation is a quirk of computer science and should ideally be described in any textbook that uses it.
May
8
comment A natural example in category theory
Oh! This is excellent! I think this has finite products but not all finite limits. Right?
May
7
comment A natural example in category theory
This is not an explicit requirement but it is implied: if $X$ is itself inhabited then it is clearly isomorphic to a subobject of an inhabited object.
Apr
26
comment What is Baire's zero-dimensional metric space?
Additional context would help, but it's usually this one - en.wikipedia.org/wiki/Baire_space_(set_theory)
Apr
5
comment Uniform distribution with probability density function. Find the value of $k$.
How many types of mathematicians are there?
Jan
7
comment What are the differences between rings, groups, and fields?
That used to be the case but most authors today define a ring to have $1$. The unusual looking term rng is sometimes used for the concept without $1$.
Dec
27
comment Is empty set a proper subset of itself?
$B \setminus A \neq \varnothing$ does not imply that $A$ is a subset of $B$. (But, if it is, then it is indeed a proper subset.)
Aug
19
comment Borel linear order cannot have uncountable increasing chain
I had missed the link! The author cites Harrington and Shelah, who did prove the result I recalled earlier. I guess it would be best to check that reference. @William: No. There are no uncountable wellordered Borel chains at all so that weaker variant is vacuously true.
Aug
19
comment Borel linear order cannot have uncountable increasing chain
This is true if chain is replaced by wellordered chain. In other words, a Borel linear order cannot contain a copy of $\omega_1$ or its reverse. I would guess that's what is meant, otherwise "increasing or decreasing" is not very meaningful. Where is this from?
Aug
16
comment What is actually “relatively consistent”?
"If a system is not complete, then it is consistent." Seeing that complete usually means "proves $\sigma$ or $\lnot\sigma$ for every sentence $\sigma$" and that consistent usually means "does not prove every sentence $\sigma$," a system that is not complete must be consistent. There are variations but, in any case, I don't think your second sentence exactly says what you meant to say.
Aug
6
comment Axiom of choice - to use or not to use
Also, the axiom of choice is not necessarily non-constructive. For example, full choice is valid in constructive type theory.
Aug
3
comment Proving that the set of algebraic numbers is countable without AC
And yes, countable means $\leq \aleph_0$ and, in particular, infinite Dedekind finite sets are "uncountable" in ZF. So one shouldn't think that "uncountable" means "large" in ZF...
Aug
3
comment Proving that the set of algebraic numbers is countable without AC
Yes, your answer is perfectly correct, my comment was just an addendum. Since I had a chance to look it up, the reference is: Hodges, Läuchli's algebraic closure of $\mathbb{Q}$, Math. Proc. Cambridge Philos. Soc. 79 (1976), 289-297. ams.org/mathscinet-getitem?mr=422022
Aug
3
comment Proving that the set of algebraic numbers is countable without AC
This is correct if by "the algebraic numbers" you mean the algebraic closure of $\mathbb{Q}$ contained in $\mathbb{C}$. However, Hodges has shown that ZF does not prove that this is the only algebraic closure of $\mathbb{Q}$. In particular, since ZF proves that any two countable algebraic closures of $\mathbb{Q}$ are isomorphic, there is a very wild model of ZF where $\mathbb{Q}$ has an uncountable algebraic closure!!!
Aug
1
comment Solution space to a functional equation
Perfect. Thanks!
Jul
31
comment Is the Collatz conjecture in $\Sigma_1 / \Pi_1$?
The usual statement is $\Pi_2$, but since the Collatz conjecture is a sentence it is equivalent to either $0=1$ or $0=0$... (Assuming we're working in the standard model. If you're asking whether the conjecture is provably equivalent to a $\Pi_1$ or $\Sigma_1$ sentence over PA or ZFC, that's a different matter.)