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Jan
15
answered Is there an uncountable Borel almost disjoint family?
Dec
29
comment Unconventional types of induction
Thanks for fixing it.
Dec
17
revised Are undecidable consequences of Con recursively enumerable?
\\, -> \, (TeX)
Dec
11
comment Under what conditions can we put a complete norm on a linear subspace of a separable Banach space?
If you drop the separability assumption, a similar fact is true: An infinite-dimensional vector space over the reals of dimension $\kappa$ admits a complete norm if and only if $\kappa = \kappa^{\aleph_0}$. (Arthur Kruse, Badly incomplete normed linear spaces. Mathematische Zeitschrift, 1964.)
Dec
6
comment Distribution of definable integers
These functions will typically turn out to be non-computable, I think. You may find relevant information by searching for "Chaitin's constant".
Nov
20
comment Basic results with three or more hypotheses
@PeterLeFanuLumsdaine: Second: You really think that if we let $\alpha_n:=n$, and $\alpha_{\omega+\beta}:= \aleph_{\beta}$, then the scale of alphas would be "more natural" than the scale of alephs? For "more natural" it is not enough for the definition to look nicer, it also has to be useful. Given that most theorems of set theory deal with infinite cardinals only, I would expect that most definitions and theorems would become more complicated. A characterisation (or definition) of GCH, for example, would now read "$2^{\aleph_\alpha}=\aleph_{\alpha+1}$ for all $\alpha\ge \omega$".
Nov
20
comment Basic results with three or more hypotheses
@PeterLeFanuLumsdaine: First: Even if you would use a different indexing for the alephs, this would not invalidate my point. There still are many other examples where you have to distinguish between 0 and nonzero limit ordinals in inductive definitions/proofs, such as the inductive definition of ordinal arithmetic.
Nov
16
comment Is any function taking compact sets to compact sets, and connected sets to connected sets, necessarily continuous?
Not in general. For example, let $X= \omega+1$ (a convergent sequence), and let $f$ be discontinuous with finite (hence compact) image.
Oct
5
comment Essays and thoughts on mathematics
Hardy's apology is available here: math.ualberta.ca/~mss/misc/A%20Mathematician%27s%20Apology.pdf
Sep
30
comment Homogeneity of a variant of Prikry forcing
A more general version would allow the ultrafilters to be indexed by elements of $[\kappa]^{<\omega}$.
Sep
28
comment Who needs RCS iterations?
When I wrote about "Kunen" conditions, I did not mean to restrict only to conditions $p$ such that each $p(\beta)$ is in $dom(Q_\beta)$ (as Kunen does). This would lead to "Kunen's pathology" (exercise E4; one could almost call it the "Kunen inconsistency":-) that a CS iteration of proper (even ccc!) forcings might collapse $\omega_1$. - I only use "full names $Q_\beta$", or equivalently, ALL names $q(\beta)$ are allowed, at least below a certain rank. - I think that then the discrepancy disappears.
Sep
26
comment Who needs RCS iterations?
Jech requires $1_\beta\Vdash p(\beta)\in Q_\beta $ (Def 16.29, page 280). Kunen requires only $p\mathord\restriction\beta \Vdash p(\beta)\in Q_\beta$. (Def 5.8, p. 273) - But Jech's conditions are dense in Kunen's: Let $p$ be a "Kunen" condition, and define a ("Kunen", for the moment) condition $p'$ with the same support as follows: $p'(\beta):= p(\beta) $ if that happens to be in $Q_\beta$, and $1_\beta$ otherwise. (Use existential completeness, of course.) By induction on $\beta$, show $p'\mathord\restriction \beta \le p\mathord\restriction \beta$. - Now $p'$ satisfies Jech's condition.
Sep
26
awarded  Enlightened
Sep
26
awarded  Nice Answer
Sep
25
revised Who needs RCS iterations?
tag: lo.logic
Sep
25
answered Who needs RCS iterations?
Sep
19
answered Examples for “nice” Boolean algebras that are not complete or not atomic
Sep
6
revised Fuzzy logic of Godel
tags added
Sep
6
answered Fuzzy logic of Godel
Aug
31
comment Notation: $Sigma$ and $Pi$ of intersections
Some people may write $\bigvee A\cap M$ or $\bigvee_{x\in A\cap M} x$ instead of $\sum(A\cap M)$.