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Dec
20
comment On a topological proof of the infinitude of prime numbers.
I am afraid to make this last remark true you have to define the characteristic function of a set as: 0 for elements in the set, and 1 for elements not in the set. In other words, this is the dual of the common way of defining characteristic function.
Nov
5
awarded  Enthusiast
Oct
30
comment What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?
@David: There is a misprint in your definiton of "Sierpinski number of the second kind". It says "for all k", and it should be "for all n".
Oct
30
comment What is the probability that some number of the form $10223\cdot 2^n+1$ is prime?
Which chapters of "Towards a Philosophy of Real Mathematics" you mean are interesting?
Oct
29
awarded  Tumbleweed
Oct
25
comment First-order logic: nested quantifiers for same variables
@user18180: I will just give you a hint. Use that in classical logic you can prove the sequent "$\Rightarrow \exists x \neg Px, \forall y Py$".
Oct
25
comment First-order logic: nested quantifiers for same variables
@user18180: Your answer is right, it is enough to consider the student with the smallest score. This student is a witness of the validity of this existential. By the way, the formula I wanted to write is $\exists x (P(x) \to \forall y P(y))$ (somehow I forgot the $x$ in the existential).
Oct
25
comment First-order logic: nested quantifiers for same variables
I always use this formula $\exists ( P(x) \to \forall y P(y))$ when teaching logic. It does a great job with students if you use the following interpretation of the predicate $P$: "student that will pass the exams". In other words, in this context the sentence $\exists ( P(x) \to \forall y P(y))$ says that: "there is some student such that if he passes the exams, then everybody pass the exams".
Oct
23
comment A question about the deduction theorem
Take $\psi$ as $Px$, and $\varphi$ as $\forall x Px$.
Oct
23
comment A question about the deduction theorem
@Matt: This is not a counterexample because $\exists y (x = y)$ is always true (it does not matter which is the element $x$). Take a formula with a free variable, but which is not always true.
Oct
23
comment A question about the deduction theorem
Hint: Forget about $T$ (i.e., take $T = \emptyset$), and take $\psi$ as a formula with free variables (i.e., it is not a setence) such that $\varphi$ is the universal closure of $\psi$.
Oct
22
revised Software for some universal algebra issues
added 1 characters in body
Oct
22
asked Software for some universal algebra issues
Oct
13
answered Study material for fuzzy logic
Oct
9
awarded  Critic
Oct
9
comment Confused about Wikipedia definition of NP
"Efficient" always means "verifiable in polynomial time in the length of the input". And in this case the input is the formula, not the ZFC-proof.
Oct
9
comment Confused about Wikipedia definition of NP
In the Wikipedia there is no formal definition since the term used is "efficiently verifiable". Do you guess how to translate this expression to a formal setting?
Aug
26
comment Proper classes and models of set theory
$V_k$ is a set (for all $k$), not a proper class. Thus, I do not see the connection with your question.
Aug
26
comment Proper classes and models of set theory
A class is just a formula with one free variable. Some formulas define sets, and the ones that do not define sets are called proper classes. Hence, think about your question when you consider the formula "$x=x$"; does it define a set or always a proper class?
Aug
9
comment Undecidability in ZFC of statements concerning logical validity
Very nice (and simple) way to answer the question. Finally, I have chosen Carl's answer because it produces an example of such formula.