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Aug
20
comment Is the set of all definable subsets of the natural numbers recursively enumerable?
As Carl Mummert says in his answer, your question is ambiguous (I disagree with his interpretation of the question, but what he says is completely fine). Problems about decidability (and recursive enumerability) must be presented in the form of a subset of natural numbers (another option is to consider a subset of the set $\{0,1\}^*$ of finite binary strings). Thus, your question must be rewritten saying which is the subset of natural numbers you are wondering whether is recursively enumerable. Notice for instance that in Carl's answer he has considered the subset $\{ n_{i,j}:i,j\in\omega\}$.
Aug
20
comment Is the set of all definable subsets of the natural numbers recursively enumerable?
Let me remember that the "Turing machine" model is a finitistic one; it only accepts finite inputs and it only produces finite outputs. How do you want to codify in a finitistic way a subset of natural numbers? The more natural way is to use a finitistic formula to describe the set, but then it is obvious which is the answer to your question (YES). If you do not want to use the natural way then you must add to the question what is the finitistic codification (for subsets of natural numbers) you want to use.
Aug
16
comment Totally ordered set with greater cardinality than the continuum
Although no Axiom of Choice [AC] is needed if you use the theory of transfinite ordinals, let me give a "simple" example (simple in the sense that you do not need to know ordinals) using Zorn's Lemma (which is known to be equivalent to [AC]). First consider the partial $( \mathcal{P}(\mathbb{R}), \subseteq )$, and then use Zorn's Lemma to prove that every partial order can be extended to a linear order (look at en.wikipedia.org/wiki/Linear_extension and en.wikipedia.org/wiki/Szpilrajn_extension_theorem )
Aug
9
answered Introductory text for lattice theory
Jul
24
comment Aftermath of the incompletness theorem proof
In this context (Peano arithmetic, natural numbers, ...) it is common to use the word "true" to mean "holds in the standard model of natural numbers". In other words, claiming Gödel sentence is true is just claiming that it holds in the standard model.
Jul
16
comment Where are good resources to study combinatorics?
@ Tom Au: a finite number of sets? I suppose you mean finite sets..
Jul
9
comment Is there an easy way to see associativity or non-associativity from an operation's table?
Paper sent. Right now I am too busy to write down the details of this method. If you go through the paper I suggest you add the details in an answer to your own post.
Jul
9
answered Is there an easy way to see associativity or non-associativity from an operation's table?
Jul
6
comment Where can I find (download) a book about basics of discrete mathematics?
Try courses.csail.mit.edu/6.042/spring12/mcs.pdf
Jun
8
awarded  Caucus
May
22
comment Variety generated by finite fields
@Magidin: I have always believed that Birkhoff proved that a class of algebras is a variety iff it is closed under $H$, $S$ and $P$. And, Tarski proved that the variety generated by a class $K$ of algebras is $HSP(K)$. Anybody can confirm this?
May
12
comment Second-order logic - monadic version and Henkin semantics
@Levon: It seems you didn't get my point. Even if you want to use your "parentheses convention", your formula is not well written: it should be written as $\forall x \forall y \exists X X(x,y)$.
May
12
comment Second-order logic - monadic version and Henkin semantics
Maybe I misunderstood the question, but I thought he was asking for an explicit formula which is valid in monadic second order logic and is not valid in full Henkin semantics. Any idea how to write down one of these formulas?
May
12
comment Second-order logic - monadic version and Henkin semantics
Be careful with your example, what you have written down is not a formula; essentially because $(x,y)$ is not an atomic formula. A formula capturing your idea is $\forall x \forall y \exists X (Xxy)$.
May
11
accepted Mutual Uniqueness of Operations in PA models
May
11
comment Mutual Uniqueness of Operations in PA models
Great. This paper is exactly what I was looking for.
May
11
revised Mutual Uniqueness of Operations in PA models
added 5 characters in body
May
11
asked Mutual Uniqueness of Operations in PA models
May
3
comment Is there a branch of mathematics that requires the existence of sets that contain themselves?
Since your question considers paradoxes, let me point out the book "The Liar: An Essay on Truth and Circularity" where the authors use non-well founded set theory (i.e. Aczel's one in Mark Dominus' answer) to discuss this paradox (which is obviously a non set-theoretic paradox)
Apr
25
comment Is there a decision procedure for intuitionistic propositional logic?
Let me add that "intutionistic propositional logic with one variable" is decidable in linear time, while "intutionistic propositional logic with two variables" is Pspace-complete. The first result is the classical one by Rieger-Nishimura, and the second one is a recent result by Mikhail Rybakov.