# for all and for every , what's wrong with this simple contradiction?

For these two statements,
(1). There exists a positive real number $\dfrac {\varepsilon } {2}$ smaller than every positive real number ${\varepsilon }$ .

(2). There is no positive real number smaller than every positive real number ${\varepsilon }$ .

Questions :

1. (1) contradicts with (2), which is wrong and which is right ?
2. What I got confused to get this contradiction?
• (1) doesn't make much sense. If you say "there exists...", then you implicitly assume that the number $\varepsilon/2$ is fixed, and hence so is $\varepsilon$, so you can't really say that this then holds for all $\varepsilon$. Nov 18, 2016 at 8:51
• In order for (1) to contradict (2), it would need to establish a positive number (not an expression) that is smaller than every positive real number. What number would that be? $\varepsilon/2$? That's not smaller than $\varepsilon/4$, though. Nov 18, 2016 at 9:06

The order of your logical symbols is important.

1. For all $\epsilon > 0$ there exists a smaller real number. $$\forall\epsilon>0\exists\delta>0 \text{ such that } \delta<\epsilon$$
2. There does not exists a number that is smaller for all $\epsilon>0$. $$\nexists \delta > 0 \forall \epsilon>0 \text{ such that } \delta<\epsilon$$

Note that these statements don't contradict eachother. Actually, they are the same statement. Call the second statement $A$. Then $A = (A^c)^c$.

\begin{align} A^c &= \exists \delta > 0 \forall \epsilon>0 \text{ such that } \delta<\epsilon. \\ (A^c)^c &= \forall\delta > 0 \exists\epsilon>0 \text{ such that } \delta>\epsilon \end{align} This is the same as statement 1 with $\epsilon$ and $\delta$ reversed.

• Then how to tell the order of logical symbols in a statement ? Nov 18, 2016 at 9:36
• @iMath The author should make that clear, by making the order of the sentence follow the order of the symbols: e.g. they should have written "For every $\epsilon>0$ there is a positive real smaller than $\epsilon$". In certain cases, a failure to do so is okay when there are other textual clues. For example, in the statement (1) above, "there exists a positive real number ${\epsilon\over 2}$" references $\epsilon$, so must come after the quantifier introducing $\epsilon$. This is, however, pretty bad writing, and I find the language of statement (1) fairly unclear. Nov 20, 2016 at 0:05

The subtlety here fooled me once to.

The key to understanding the contradiction is noticing what is fixed first.

For "There exists a positive real number $\dfrac {\varepsilon } {2}$ smaller than every positive real number ${\varepsilon }$ " we first set $\epsilon$, any epsilon, and then say $\dfrac {\varepsilon } {2}$ is smaller, which is true.

For "There is no positive real number smaller than every positive real number ${\varepsilon }$" we are saying that, there is no number that we can fix that will be smaller than any other. They are saying that whatever, for how many small, $x$ is, there will be a smaller $\epsilon$.