Intereting Posts

The equation $x^3 + y^3 = z^3$ has no integer solutions – A short proof
The product of all the conjugates of an ideal is a principal ideal generated by the norm.
Free Group Generated By Image
Clarification of use of Cauchy-Riemann equations
How can I show that $\sum \limits_{n=2}^\infty\frac{1}{n\ln n}$ is divergent without using the integral test?
On a non-standard approach to the classification of conics?
Does the integral $\int_0^\infty \sin(2x^4) \, dx$ converge absolutely/conditionally?
Calculating probability of 'at least one event occurring'
Choose 100 numbers from 1~200 (one less than 16) – prove one is divisible by another!
Is the partial sum of cosine bounded?
Linear Homogeneous Recurrence Relations and Inhomogenous Recurrence Relations
Show $ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.$
Points of discontinuity of a bijective function $f:\mathbb{R} \to [0,\infty)$
Numbers as sum of distinct squares
How to logically analyze the statement: “Nobody in the calculus class is smarter than everybody in the discrete math class.”

Prove the following statement on negation of quantifiers:

*Statement:* To negate a statement of the form

$$

Q_1x_1 Q_2x_2 \ldots Q_nx_n\; P(x_1,x_2,\ldots,x_n),

$$

where $Q_i$ is $\forall$ or $\exists$ for $1 \leq i \leq n$, we do the following:

(i) Change every $\forall$ to $\exists$ and every $\exists$ into $\forall$.

- Proof negation in Gentzen system
- What is a proposition? Conflicting definitions.
- Can unprovability unprovable? Is there an $\omega$-fold unprovability?
- Prove $\langle \mathbb{N}, x \mapsto x +1, 1 \rangle$ is a Peano system without circular reasoning
- if $f: (0,\infty) \to (0,\infty)$ is a strictly decreasing then $f \circ f$ is decreasing?
- Every sentence in propositional logic can be written in Conjunctive Normal Form

(ii) Replace $P$ by its negation.

**Background:** This problem appears in the book *How to Think Like a Mathematician* by Kevin Houston, where it comes up in a chapter with an introduction to induction. I remember spending considerable time on this problem and not being able to come up with anything satisfactory. It seemed like you had to go into some deep logic to try to come up with an answer (something the author did not intend). I communicated with the author and noted the exercise, but he never got back to me. I took this as a sign that the exercise was flawed (or at least was very inappropriate for where it was placed in the text). I am still curious as to how this problem may be solved, if at all possible. Or is it something more axiomatic in nature and not something you can really deduce per se?

- Is the negation of the Gödel sentence always unprovable too?
- Is there a first-order-logic for calculus?
- $e^{e^{e^{79}}}$ and ultrafinitism
- How to formalize $\text{span}(S)=\{c_1v_1+\cdots+c_kv_k\mid v_1,~\cdots,~v_k\in S,~c_1,~\cdots,~c_k\in F\}$ rigorously in first order language?
- How can I correct my wrong Intuition that $\forall \, x \, \in \,\emptyset : P(x) \quad $ is false?
- The Language of the Set Theory (with ZF) and their ability to express all mathematics
- Does elementary embedding exist between two elementary equivalent structures?
- Why does one have to check if axioms are true?
- This sentence is false
- Can a model of set theory think it is well-founded and in fact not be?

Here’s the argument spelt out in my Gödel book — $\varphi$ is the predicate for which we aim to show by induction that $\forall n\varphi(n)$

- Minimal polynomial of restriction to invariant subspace divides minimal polynomial
- Help me complete finding the Reduction formula of $J_n=\tan^{2n} x \sec^3 x dx$?
- The integral $\int_0^1 \frac{(x+1)^n-1}{x} dx$
- a sequence of polynomials converges to $0$
- Theorem that von Neumann proved in five minutes.
- A subset of a compact set is compact?
- Proving limit doesn't exist using the $\epsilon$-$\delta$ definition
- Evaluation of $\int\frac{dx}{x+ \sqrt{x^2-x+1}}$
- Why is the 3D case so rich?
- How can a high schooler get more involved in mathematics?
- Condition on degrees for existence of a tree
- Are Continuous Functions Always Differentiable?
- Integral of Sinc Function Squared Over The Real Line
- basic calculus/analysis question. why is $\frac {dy}{dx} dx = dy$?
- solution of Diofantine equation