Linked Questions

2 votes
0 answers

Are all statements about math inherently formal? Can one do math without formal logic? [duplicate]

Are all people who do mathematics applying (whether they know it or not) formal logic? Does every statement someone may make about math, at its core, a formal statement in mathematical logic? (I'm ...
justin's user avatar
  • 4,651
43 votes
14 answers

Still struggling to understand vacuous truths

I know, I know, there are tons of questions on this -- I've read them all, it feels like. I don't understand why $(F \implies F) \equiv T$ and $(F \implies T) \equiv T$. One of the best examples I ...
user525966's user avatar
  • 5,451
11 votes
7 answers

Why is "A only if B" equivalent to "(not A) or B"? [duplicate]

I've encountered this recently and I just can't wrap my head around it. My book states that $$A \rightarrow B \equiv \neg A \lor B$$ It's my understanding that $A \rightarrow B$ means that if $A$ ...
bones_mccoy's user avatar
5 votes
2 answers

If $\lim_{x\to 0}f(x)=L$ then $\lim_{x\to 0}f(cx)=L$ for any nonzero constant $c$.

I was just wondering if this proof is correct. I'm trying to prove that if $\lim_{x\to 0}f(x)=L$ then $\lim_{x\to 0}f(cx)=L$ for any nonzero constant $c$. Proof: If $\lim_{x \to 0}f(cx)=L$ ...
chrismc's user avatar
  • 621
0 votes
4 answers

Are there important situations where we study false statements as if they were true?

I know of two situations resulting from asserting that a false mathematical statement is true (by this we assume that the statement has been made to be a mathematical axiom and that it must be true ...
user64742's user avatar
  • 2,177
0 votes
1 answer

"If P, then Q; If P, then R; Therefore: If Q, then R." Fallacy and Transitivity

Two fallacious arguments: If P, then Q If P, then R Therefore: If Q, then R And If P, then Q If R, then Q Therefore: If P, then R However, if these particular propositions were interpreted as ...
David Marlowe's user avatar
4 votes
2 answers

What is a good example to show high school students why a proof for induction is a reasonable kind of proof?

I teach average-level high school students who have not had much beyond Algebra 1. I want to show them why induction makes sense. I want the sort of problem where it is intuitive that a statement is ...
MathTeacher's user avatar
  • 1,519
1 vote
2 answers

Intuitonism and metamathematics.

There are various reasons why one would want to reject the law of the excluded middle when doing "normal" mathematics, which I won't get to here, but accepting those, does the same reasoning hold when ...
Nathan BeDell's user avatar
0 votes
1 answer

Can't find the demonstration of a theorem about recursion [closed]

Here's the theorem : Let $E$ be a set, $g$ a function from $E$ into $E$, and $a$ an element of $E$. There exists a unique function $f$ defined from $\mathbb{N}$ into $E$ such that $f(0)=a$ and $f(n+...
toto's user avatar
  • 165
1 vote
2 answers

Understanding quantifiers through gaming metaphors

In the appendix about mathematical logic of his book "Analysis 1", Terence Tao explains quantifiers via gaming metaphors. For example he writes In the first game, the opponent gets to pick what x ...
user372129's user avatar
1 vote
1 answer

negating propositional formula with quantifiers

In order to solve an exercise in computer sciences (proving a language $L$ to not be context-free) I need to negate the Pumping-Lemma. I was provided with the definition in the following form: If $L$ ...
ljrk's user avatar
  • 113
1 vote
1 answer

Quadratic Logic Question on Implications

There are two statements P and Q, where P is $x^2-3x+2 =0$ and Q is $x=1$ or $x=2$ or $x=3$. The first statement, where $x$ $\epsilon$ $\Bbb R$, says $x^2-3x+2 =0$ $\Rightarrow$ $x=1$ or $x=2$ or $...
user258521's user avatar
0 votes
0 answers

Do we have to resort to intuition at the very root of our thinking?

Formal first order logic is the foundation of ZFC set theory. This gives me the impression that a theoretical system has to be based on a formal logical system, with its own axioms and deduction rules....
Ziqi Fan's user avatar
  • 1,786