0
votes
1answer
44 views

Proving property for all predicates in first order logic

Let's consider language with predicate $P$ and following derivation $${{{[P[a/x]]^1} \over {P[a/x] \rightarrow P[a/x]}}\rightarrow I^1 \over {\forall_x (P(x) \rightarrow P(x))}}\forall I$$ Doesn't ...
-2
votes
2answers
70 views

What's wrong with this induction based proof?

Claim: $\forall x \in \mathbb{R^+} ,$ $ x^n=1 $ $where$ $ n\in \mathbb{N}$ Proof by induction on n: Basis step: $\forall x \in \mathbb{R^+} ,$ $ x^0=1 $ Induction Step: Let this holds for all ...
1
vote
4answers
554 views

Proof of Drinker paradox [duplicate]

I searched all over the internet but didn't find a formal proof for this paradox, so here is my attempt: $\exists x[P(x)\implies \forall yP(y)]$ Let $x=x_0$. Thus $P(x_0)$ is given. Let $y$ be ...
17
votes
9answers
4k views

Is Lewis Carroll's reasoning correct?

A bag contains 2 counters, as to which nothing is known except that each is either black or white. Ascertain their colours without taking them out of the bag. Carroll's solution: One is black, and ...
0
votes
2answers
107 views

Why is this proof wrong?

I am taking a course on logical equations and I found this exercise while reading about proofs and how to prove a given sentence and what kind of mistakes usually occur when you are trying to prove ...
0
votes
5answers
124 views

point out the mistakes $1=2$? [closed]

Q:prove $1=2$ ? method1:- let us consider $x=1$ then $x=x^2$ $x-1 =(x^2)-1$ $x-1=(x-1)(x+1)$ $1=x+1$ finally $1=2$ i have little confusion here, is the mistake is considering $x=1$ or else ...
5
votes
3answers
411 views

Gödel's Paradox — Every set of formulas is consistent

I am sure I have made a gross misunderstanding of Gödel's completeness theorems, as to me, it seems to follow that all sets of formulas are consistent. Let $\Gamma$ be a set of formulas. If ...
1
vote
2answers
156 views

Paradox - What is wrong with this proof that proves a false assertion?

Theorem: Let $a_{n}=a_{n-1}+1, a_1=1$. For any $n$, in order to compute $a_n$, it is necessary to compute $a_i$ for each $i=1,\dots,n-1$, which takes $\Theta(n)$ time. Proof: This is vacuously true ...
15
votes
3answers
974 views

Proof that $\mathbb N $ is finite

Obviously this is a false proof. It relies on Berry's paradox. Assume that $\mathbb{N}$ is infinite. Since there are only finitely many words in the English language, there are only finitely many ...
11
votes
7answers
1k views

“Proof” that ZFC is inconsistent using Turing machines

I came across the following "proof" for the inconsistency of ZFC and can't find the flaw in it (if there is one...): Construct a Turing machine A which sequentially runs on all proofs in ZFC and ...