Reputation
759
Top tag
Next privilege 1,000 Rep.
Create new tags
Badges
4 11
Newest
 Curious
Impact
~29k people reached

  • 0 posts edited
  • 0 helpful flags
  • 91 votes cast
Dec
9
comment First Order Logic Example of Halting Problem
Take a look at the following file math.psu.edu/simpson/courses/math457/trakh.ps
Nov
26
comment Is there a decidable theory in propositional logic whose consequences are not decidable?
Hint: Let us consider for every $X \subseteq \mathbb{N}$ the theory $T_X$ which is the one generated by the set $\{p_n:n \in X\}$ of variables. Can you consider some X such that $\{p_n:n \in X\}$ is not decidable but $T_X$ is recursively enumerable?
Nov
22
comment Who first explicitly noted that second-order logic is unaxiomatizable?
Indeed, Henkin says "This follows from results of Godel concerning systems containing a theory of natural numbers, because a finite categorical set of axioms for the positive integers can be formulated within a second order calculus to which a functional constant has been added." So Henkin does does not give a reference, but he provides a proof (without the details, but it is quite clear this is a proof).
Sep
25
comment Difference between elementary logic and formal logic
@Peter: You are absolutely right that metalogic is in general in a much wider sense. I should have said "elementary logics refers to a part of metalogic" (the easiest part).
Sep
25
comment Difference between elementary logic and formal logic
I suspect that your answer to question 1 (that is, elementary logic refers to metalogic) is what Kelley had in mind. Concerning question 2 I suggest you look at en.wikipedia.org/wiki/Consequence_operator
Aug
21
comment Is the set of all definable subsets of the natural numbers recursively enumerable?
@Thomas: I do not follow your last edition, it has the same troubles. I can understand what you mean by a natural number which is well formed (although I dislike your terminology and proposal), but I do not understand your definition for subsets of natural numbers. Let me do some concrete questions: how do you codify the subset of odd numbers? how do you codify the subset of even numbers? how do you codify the subset of prime numbers? etc.
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 $\forall x \forall y \exists X (Xxy)$.