On 5/21/2025 12:56 AM, Richard Heathfield wrote:<snip>On 21/05/2025 06:23, olcott wrote:
Agreed.A self-contradictory input and a proof by contradictionDo you mean like how ZFC resolved Russell's>
Paradox thus converting "set theory" into "naive set theory"?
No, because there is no paradox in the Halting Problem. A proof by contradiction is not a paradox.
>
are not the same thing.
A proof by contradiction wouldA proof by contradiction would conclude that 'by assuming A was possible we have derived a contradiction. We conclude that A is not possible'.
conclude that "this sentence is not true" is true because
it cannot be proved false.
There is no self-contradictory input because such an input is impossible.
ZFC shows how a whole way of examining a problem can beThe Halting Problem shows how there are some problems that cannot be computed by a finite algorithm.
tossed out as incorrect and replaced with a whole new way.
The HP proofs are based on defining a D that canThat an algorithm for ascertaining whether an arbitrary program with arbitrary input halts cannot actually exist is precisely what the Halting Problem proves.
actually do the opposite of whatever value that H returns.
No such D can actually exist.
<snip>
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
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