Re: the need for relevance

Han de Bruijn wrote:

Jesse F. Hughes wrote:

Yes, I know. You think it's self-evident that N is "potentially
infinite" and that to determine if a property holds on a potentially
infinite set, you check to see how it holds on the finite subsets and
"take a limit". But that doesn't always work, I guess. Here's a
property P: "X is finite (i.e., not potentially infinite)."

P is true of {0}.
P is true of {0,1}.
P is true of {0,1,2}.
...
P is true of {0,1,2,3,...,n}.

so by your reasoning, it follows that P is true of N and hence that N
is not potentially infinite. Geez, where did I go wrong?

A self-referential property P, perhaps? Somehow like the liar paradox?
Anyway, I'm not impressed.

Seems that I've missed a follow up to this. But anyway, your question is
self referential because we yet have to define what infinity MEANS. Thus
you cannot invoke a limit process with that term or its negation, as the
thing to be accomplished. That's a vicious circle. An analogous example:

Prove that all numbers are small

Proof: 1 is small. If n is small then (n+1) is small ==> by mathematical
induction: all numbers are small.

Han de Bruijn

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