Re: the need for relevance

Han de Bruijn <Han.deBruijn@xxxxxxxxxxxxxx> writes:

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.

None of this has anything to do with self-reference and *no*, we are
not trying to define what infinite means. Your response is nonsense.

And, indeed, if we have a notion of small consistent with the claim
that whenever n is small, so is n+1, then every number is small. This
inference again has not a damned thing to do with self-reference.

But never mind, let's change the property.

Let P be the property "There is an x in X such that x + 1 is not in
X."

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}.

Thus, P is true of N. Hence, there is a natural number (i.e., an n in
N) such that its successor (n+1) is not a natural number (i.e., not in
N).

Happy with that conclusion? This is your method, yes?

--
Jesse F. Hughes

"Besides 'talking trash' is in some ways part of extreme mathematics."
-- James S. Harris
.

Relevant Pages

  • Re: A Possible "solution" to the Halting Problem
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    (sci.logic)
  • Re: A Possible "solution" to the Halting Problem
    ... Pi is an infinite abstract object. ... NAFL and it is very restricted as compared to classical logic. ... *justifying* the self-reference in the proof that you are objecting to. ... Infinite objects are proper classes and so cannot belong to any class. ...
    (sci.logic)
  • Re: Cantor Confusion
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    (sci.math)
  • Re: Cantor Confusion
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  • Re: Cantor Confusion
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