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The prompt states:

Let us accept as true that a person can always walk an extra mile. Does the Principle of Induction then prove that a person can walk forever? Where is the fallacy?

No. The Principle of Induction says that a person can walk a finite number of miles $n$ for all $n \in \mathbf{N}$, but does not say that a person can walk infinitely many miles (walking forever).

The fallacy is that Induction may be used to prove that a statement of the form $P(n)$ is true for all $n \in \mathbf{N}$. It cannot be used to show that a statement of the form $P(\infty)$ is true.

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There is no fallacy, it is possible to prove that a person can walk forever, under the assumption that he or she can always walk another mile. –  Git Gud Nov 17 '13 at 20:04
    
Please read about accepting answers. –  Git Gud Nov 17 '13 at 20:05
    
"Forever" does not have a clear meaning. If it means can walk more than $n$ for any $n$, then the implication is correct. But why should one accept that a person can always walk an extra mile? –  André Nicolas Nov 17 '13 at 20:06
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@hardmath: you are playing with semantics here. "Walk forever" does not mean "walking infinitely many miles"; it means walking an arbitrarily large number of miles, given enough time. –  Martin Argerami Nov 17 '13 at 20:09
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There's no fallacy. The assumption is false (that a person can always walk another mile). If a person could always walk another mile, a person could walk forever. –  mjqxxxx Nov 17 '13 at 21:01

1 Answer 1

There is no fallacy. The use of induction here shows that, under the assumption, if you give the person enough time, any number of miles will be achieved.

Maybe part of the confusion comes from the fact that you talk about induction, but you don't state your proposition $P(n)$ explicitly. To me, here the natural $P(n)$ is $$ P(n)=\mbox{"a person can walk $n$ miles"}. $$ So the induction principle shows that a person can walk any number of miles. To me, this is what it means "to walk forever". Note that if a person "walks forever" at no time the person would have done infinitely many miles.

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@hardmath A parallel can be drawn with the concept of a function being infinitely differentiable. Of course there does not exist such a thing as $f^{(\infty)}$, but it can happen that a function is such that $f^{(n)}$ exists for all $n\in \Bbb N$ and this is what we call an infinitely differentiable function. –  Git Gud Nov 17 '13 at 20:58

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