Re: Analysis of Flibble’s Latest: Detecting vs. Simulating Infinite Recursion ZFC

Richard Heathfield <rjh@cpax.org.uk> writes:

On 21/05/2025 18:47, olcott wrote:

...

*PAY ATTENTION*
I am saying that a key element of the halting problem
proof cannot exist, thus the proof itself cannot exist.

>
Yes, it can, and it does.

You'd think that at some point in decades he's wasted on this he might
have considered looking at the proofs that are not by contradiction.
I've repeatedly suggested looking at Radó's Busy Bee problem and proof.

Watch.
>
Definition: a prime number is an integer >= 2 with no divisors >= 2 except
itself.
>
Hypothesis: there is no largest prime number.
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Proof:
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Assume that a largest prime number Pn exists.
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Itemise all the prime numbers from P1(=2) up to Pn :
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P1 P2 P3 ... Pn
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Insert x symbols all the way along.
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P1 x P2 x P3 ... x Pn
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Add 1.
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The number thus calculated is not divisible by any prime in our list (there
being a remainder of 1 in each case), so the number calculated is (a)
prime, and (b) larger than Pn. Thus Pn is not the largest prime. This
contradicts the assumption made at the beginning, which must therefore be
false. Proof by contradiction.
>
The proof that no largest prime exists despite its assumption that such a
prime /does/ exist - an assumption that turns out to be false.

Interestingly (and not, I think, coincidentally) Euclid's proof is not a
proof by contradiction.  It shows (by case analysis) that any finite
list of primes is incomplete.  There is (in the way Euclid does it) no
need to assume anything (other than some basic axioms).

The same strategy can be used for the halting theorem.  The more direct
proof essentially shows that the infinite list of Turing machines (there
is only one, once we agree a numbering) does not include a halt decider
(just like it does not include uncountably many other deciders that the
cranks are never interested in).

PO seems to like talking to you so you might consider avoiding any
arguments about contradictions by providing an outline of the direct
proof instead.

I've lost count of the times when the proof by contradiction leads
students astray.  Even if they don't think it's invalid, every year one
has to counter the idea that all the decider has to do is "spot the
tricky input" and the decider will work on everything else.

I'm finding it hard to believe that you can really be this stupid. Perhaps
you just get off yanking people's chains.

Hmm...  Don't forget Hanlon's razor.  As for data points, PO has
published a website whose purpose is to "bring new scripture to the
world" and he has claimed, in a court of law, to be God.

Incidentally, there is even a modern proof by construction of the
infinitude of the primes.  The idea being that since n and (n+1) have no
prime factors in common, n(n+1) has more distinct prime factors that n.
This gives a chain of ever larger sets of primes: the unique prime
factors of 2x3, 6x7, 42x43 and so on.

--
Ben.