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Sep
18
answered Does every nonempty definable finite set have a definable member?
Aug
19
answered Borel linear order cannot have uncountable increasing chain
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
14
awarded  Quorum
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
awarded  Revival
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
awarded  Scholar
Aug
1
comment Solution space to a functional equation
Perfect. Thanks!
Aug
1
accepted Solution space to a functional equation
Jul
31
revised Solution space to a functional equation
minor correction
Jul
31
awarded  Student
Jul
31
asked Solution space to a functional equation
Jul
31
answered Is the Collatz conjecture in $\Sigma_1 / \Pi_1$?
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.)
Jul
29
comment All real functions are continuous
A translation of Brouwer's original paper can be found in van Heijenoort's From Frege to Gödel: a source book in mathematical logic, 1879-1931 books.google.com/books/about/…