The Argument

Why is the second part where the quantifiers are interchanged false? Is there a concrete example for this?

  • $\begingroup$ The statement $(\exists m \in N)(\forall n \in N) m > n$ means that there is an upper bound on $N$, within $N$. If this statement were true, then it's true for $n = m$; but $m > m$ is certainly false. $\endgroup$ – user61527 Mar 14 '14 at 1:02
  • $\begingroup$ Thank you! I will reiterate my understanding: The highlighted sentence says "for every natural number 'n' (individual cases), there exists atleast one natural number 'm' such that m>n" while the second part states that "there exists atleast one 'm', for every (all) natural numbers 'n' such that m>n" which is definitely false. I think I was getting confused between 'every' and 'all' in the sense that 'every' & 'all' could mean individual cases or a collective group. Please let me know if you think my understanding is not correct. $\endgroup$ – va4az Mar 14 '14 at 2:20
  • $\begingroup$ For a lot of practice with this sort of thing, spend some time looking over the math StackExchange question A game with $\delta$, $\epsilon$ and uniform continuity. $\endgroup$ – Dave L. Renfro Mar 14 '14 at 15:00

Yes. The second statement reads "there exists a natural number $m$ such that for all other natural numbers $n$, $m >n$". Clearly this is false.


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