2,223 reputation
1324
bio website math.berkeley.edu/~sramesh
location Berkeley, CA
age 28
visits member for 4 years, 2 months
seen 2 days ago
I am a graduate student in the Logic program at Berkeley, broadly interested in categorical logic and foundations of mathematics, as well as in applications of category theory to the semantics of programming languages.

May
24
comment Avoiding reflexive paradox in set theory
Richard, you are coming now quite close to a principle like "Sets are given by well-founded comprehension"; this is the spirit behind ZF, and once you have that, the rest of its axioms are not far behind [you'll find that the existence of infinite sets, powersets, etc., are not automatically assured, and you may want to assure them].
Dec
1
comment Application of polynomials with non-negative coefficients
Surely, the thing to do is to use q = p(1) + 1 rather than q = p(1), so as not to have to treat the case p(q) = q^n on an ad hoc basis.
Nov
18
comment There are two points on the Earth's surface that … ?
What does "separated by the same geodesic distance" mean for two points?
Nov
17
comment Tarski's Theorem and Gödel's Second Incompleteness Theorem
Well, one caveat: One needs that Tarski's result is constructive enough that we have not just "For every T, for every model, there is a G such that...", but in fact "For every T, there is a G such that for every model...". But, basically, were Tarski's result phrased as strongly as its proof actually warranted, it would give us everything.
Nov
17
comment Tarski's Theorem and Gödel's Second Incompleteness Theorem
I've modified the word of the second paragraph to alleviate this concern. The second incompleteness theorem follows from the simultaneous "external" and "internal" truth of Tarski's theorem (by which I mean, the fact that Tarski's theorem is both true and provable (or, just as well, true inside every model)).
Nov
2
comment Entailment and implication
It's also worth noting that the trivial topos, like all toposes, internally considers itself to be non-trivial. It just happens to also consider itself to be trivial...
Nov
2
comment Entailment and implication
In the internal logic of the trivial topos, "there is no rational square root of two" is true. It's just that "there is a rational square root of two" is also true. The internal logic of the trivial topos happens to validate all statements; that's what makes it so trivial. But it also means it doesn't harm anything, or make any difference to the model-theoretically valid statements, to consider it a legitimate model.
Oct
21
comment Properly “transfinitely” Euclidean domains
I don't think the omnific integers can admit an ordinal-valued (as opposed to omnific integer-valued) Euclidean function; an ordinal-valued Euclidean function should still lead to being a unique factorization domain, in the usual way, but the omnific integers lack unique factorization (e.g., 2, 3, 5, 7, ..., all remain prime, but $\omega$ is divisible by all infinitely many of them).
Oct
3
comment “Mathematics talk” for five year olds
Naive question: what is the sense in which these final tessellations live in FOUR-space?
Sep
23
comment Is it possible to have t triangles in some graph on n vertices?
It's not clear to me that {1, 2, ..., n} is contained in $T_n$. How do I get 2 or 3 triangles with just 3 vertices?
Sep
19
comment Reductio ad absurdum or the contrapositive?
("If a proof if" should be "If a proof of", of course. I am too lazy to delete the comment and retype the whole thing just to fix it)
Sep
19
comment Reductio ad absurdum or the contrapositive?
If a proof if $p$ implies $q$ by contrapositive (establishing $\neg q$ implies $\neg p$) is useful because one learns intermediately that ($\neg r_1$ implies $q$), ($\neg r_2$ implies $q$), etc., then a proof of $p$ implies $q$ by contradiction (establishing $\neg (p \wedge \neg q)$) is useful because one learns intermediately ($p \wedge \neg r_1$ implies $q$), ($p \wedge \neg r_2$ implies $q$), etc.
Sep
17
comment Groupoid interpretation of type theory
Does "groupoid" really need a diaeresis over the "i"?
Sep
2
comment Intuitionistic consistency of surjection from naturals to reals
(Also, returning to the dependent choice-based argument against a surjection from N to R, I'm not sure this argument would work when R = MacNeille reals, since it depends on the dichotomy "Either $a_n > (3u + 2v)/5$ or $a_n < (2u + 3v)/5$", which seems an example of precisely the sort of thing which isn't guaranteed for general MacNeille reals)
Sep
1
comment Intuitionistic consistency of surjection from naturals to reals
Well, that's the embedding I'm thinking of too, but it's not obvious to me that $1 \in e(q) \Rightarrow q$. After all, the (only classically injective) lattice homomorphism from $\Omega$ to $\Omega_{\neg \neg}$ is also given by $e(p) = \sup \{1 | p \}$ (as is any suplattice-with-top morphism on the domain $\Omega$).
Sep
1
comment Intuitionistic consistency of surjection from naturals to reals
I like that argument! But it's not obvious to me that the unique complete lattice homomorphism from truth values into MacNeille $[0,1]$ is an embedding. For example, the regular truth values form a nontrivial complete lattice, into which truth values map via double-negation. But this is only an embedding if truth values were Boolean to begin with. So simply being a nontrivial complete lattice is not enough.
Aug
31
comment Intuitionistic consistency of surjection from naturals to reals
Right... I guess what I'm wondering, then, is what goes wrong for injecting MacNeille reals into N in IITM realizability that doesn't go wrong for Dedekind reals. Because if the MacNeille [0, 1] does inject into N, then we will have a surjection from N to MacNeille [0, 1] by sending each natural to the supremum of its preimage. (Or did I misunderstand which notion of reals injects into N? Was that not the Dedekind reals?)
Aug
31
comment Intuitionistic consistency of surjection from naturals to reals
Also, motivated by the dependence of my original argument on the existence of suprema... is there anything interesting to say about intuitionistic surjections from N onto the MacNeille reals?
Aug
31
comment Intuitionistic consistency of surjection from naturals to reals
Also, just to make sure: the provided link only demonstrates an injection $\mathbb{N}^{\mathbb{N}} \to \mathbb{N}$. I'm assuming essentially the same ideas work when the domain is switched to $\mathbb{R}$?
Aug
31
comment Intuitionistic consistency of surjection from naturals to reals
Thanks! For what it's worth, the part of my self-convincing that doesn't go through must be the idea that subsingletons of reals surject onto reals (I was imagining this could work by sending a subset of the reals to its least upper bound (I suppose by "reals" here, I really mean something like $[0, 1]$)); had that been so, an injection from $\mathbb{R}$ to $\mathbb{N}$ could be reversed into a surjection from $\mathbb{N}$ to $\mathbb{R}$.