From Newsgroup: comp.theory

On 02/05/2026 20:47, Scott Hoge wrote:

In Cantor's theorem, we do not actually construct a diagonal.
Rather, we presuppose that we can enumerate a set, and then,
/purely on the grounds of possibility/, conceive a diagonalized
non-element.

Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.

Cantor's diagonal argument, the one with the binary sequences,
is indeed constructive: a definition of anti-diagonal of *any*
(infinite) list is provided, and the proof that the anti-diagonal
cannot be in the list is quite constructive.

(Namely, we don't need to say "assume ab abdsurdo that
an enumeration is given", we can just say "for *any* list,
we *construct* an element not in the list".)

Just look it up. Here is my own rendition in Rocq: <https://gist.github.com/jp-diegidio/eb05f6265c38b35c85853514ed46ab47>

This links diagonalization to criterion (4) of consciousness.

Rather, and to sum up, it links diagonalisation to the limits
of physicalism...

HTH,

Julio

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From Newsgroup: comp.theory

Julio Di Egidio wrote:

On 02/05/2026 20:47, Scott Hoge wrote:

In Cantor's theorem, we do not actually construct a diagonal.
Rather, we presuppose that we can enumerate a set, and then,
/purely on the grounds of possibility/, conceive a diagonalized
non-element.

Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.

Cantor's diagonal argument, the one with the binary sequences,
is indeed constructive: a definition of anti-diagonal of *any*
(infinite) list is provided, and the proof that the anti-diagonal
cannot be in the list is quite constructive.

(Namely, we don't need to say "assume ab abdsurdo that
an enumeration is given", we can just say "for *any* list,
we *construct* an element not in the list".)

Just look it up.  Here is my own rendition in Rocq: <https://gist.github.com/jp-diegidio/eb05f6265c38b35c85853514ed46ab47>

This links diagonalization to criterion (4) of consciousness.

Rather, and to sum up, it links diagonalisation to the limits
of physicalism...

HTH,

Julio

I guess my question is this: If the diagonal sequence is inadequate,
just what exactly is Cantor attempting to represent with the diagonal
sequence at all?
--
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War in the west
War up north
War down south
War War
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From Newsgroup: comp.theory

On 05/06/2026 12:48 PM, phoenix wrote:

Julio Di Egidio wrote:

On 02/05/2026 20:47, Scott Hoge wrote:

In Cantor's theorem, we do not actually construct a diagonal.
Rather, we presuppose that we can enumerate a set, and then,
/purely on the grounds of possibility/, conceive a diagonalized
non-element.

Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.

Cantor's diagonal argument, the one with the binary sequences,
is indeed constructive: a definition of anti-diagonal of *any*
(infinite) list is provided, and the proof that the anti-diagonal
cannot be in the list is quite constructive.

(Namely, we don't need to say "assume ab abdsurdo that
an enumeration is given", we can just say "for *any* list,
we *construct* an element not in the list".)

Just look it up. Here is my own rendition in Rocq:
<https://gist.github.com/jp-diegidio/eb05f6265c38b35c85853514ed46ab47>

This links diagonalization to criterion (4) of consciousness.

Rather, and to sum up, it links diagonalisation to the limits
of physicalism...

HTH,

Julio

I guess my question is this: If the diagonal sequence is inadequate,
just what exactly is Cantor attempting to represent with the diagonal sequence at all?

Maybe you should ask duBois-Reymond who Cantor cribbed it from,
though accounts of the anti-diagonal are as old as making tables,
then the nested-intervals idea is since the Pythagoreans who
though made an account that the same would go for the rationals,
then Cantor is acribed an "m-w" proof, though that would probably
enough be Dirichlet's, for pigeonhole-principle.

Saying that usually the account of anti-diagonalization is a proof
by contradiction so it's non-constructive, is pretty usual.

Then, getting into accounts otherwise of "quantifying over the
universe" or the usual notions of "equivalence classes" themselves
gets into class/set distinction, is about the domain of discourse
of the universe of mathematical objects, this was a bit too much
for Cantor to bear, though, his paradox or Cantor's paradox is
the usual idea that theories with universes (or rather, "a" universe,
which by definition makes sense only that way) make counter-examples.

Then the powerset-theorem ends up just looking like grounds for
increment itself and why Peano's integers follow from that instead
of being "axiomatized", in what's a "constructive" account.


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From Newsgroup: comp.theory

On 2026-05-06, Julio Di Egidio <[email protected]> wrote:

On 02/05/2026 20:47, Scott Hoge wrote:

In Cantor's theorem, we do not actually construct a diagonal.
Rather, we presuppose that we can enumerate a set, and then,
/purely on the grounds of possibility/, conceive a
diagonalized non-element.

Nope, as explained and re-explained ad nauseam around here:
just the resident trolls won't get it.

Cantor's diagonal argument, the one with the binary sequences,
is indeed constructive: a definition of anti-diagonal of *any*
(infinite) list is provided, and the proof that the
anti-diagonal cannot be in the list is quite constructive.

An AI query seems to agree with you that constructiv-/ist/
mathematics permits Cantor's diagonal argument. I hadn't thought
about that in detail.

What I rather suggested was that Penrose's Orch-OR theory relates
to our ability to cognize through the mode of /possibility/
(where a computer might be confined to /existence/ or
/actuality/). That's why I said we do not /actually/ construct a
diagonal -- we merely think it possible as we enumerate.

Now compare a neuron to a computer chip. The computer chip is
deterministic. The same, macroscopic input invariably yields the
same, macroscopic output. Actuality gives way to actuality. In
contrast, a neuron generates an impulse through an influx of ions
across a voltage-gated ion channel. This behaves like a
"butterfly effect" or "avalanche": the flow of ions provides a /self-reinforcing feedback loop/ on the basis of which infinitely
small differences in the microscopic world may affect the
neuron's "decision." A neuron thus conceives /possibility/ where
a computer is restricted to what is actual.

Then we just relate the infinitely small for the neuron to the
infinitely distant for Gödel's theorem and diagonalization. That
provides us with at least some basis for understanding the
Orch-OR theory.

-- Scott Hoge
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