On 03/24/2026 07:32 PM, Ross Finlayson wrote::)On 03/24/2026 07:31 PM, Ross Finlayson wrote:>On 03/24/2026 02:44 PM, Tristan Wibberley wrote:>On 24/03/2026 19:29, WM wrote:>
>Here is the complete proof of a contradiction in set theory:>
>
(1) Cantor's diagonal argument finds for every countable set of
reals a
real number not in that set.
For the definition of "real" in Cantor's argument. I think you need a
lot more for any definition I would accept. As far as I can see,
when we
talk about reals (non-constructive foundationally) we really talk about
an(the?) extension of the rationals to (the smallest subsuming?)
continuum (if it exists See note A) rather than any constructive
definition, so I think you need to provide an argument--that doesn't
depend on Cantor's diagonal argument--that his construction of the
reals
is such.
>
I expect sci.math readers can provide more insight into meaning and
terminology.
>
Note A: we seem to have decided it does exist because we autoexplicate
it with a constructive definition and mathematicians no longer believe
any older concept of the reals describes them except by luck (ie the
reals are what is constructed by their construction and are not what
they were before the construction was conventionally accepted).
>
>(2) According to Cantor's definition of countable set the set of nodes>
of the Binary Tree is countable.
(3) If we map every node onto a path, then the mapped set of paths is
countable.
(4) For every n ∈ ℕ: I map the nth node on a path containing this
node.
I'm not sure I understand your meaning as I'm expecting "to" instead of
"on".
>
>(5) Therefore every npode is covered by this set of paths. There does>
not exist a node which is not covered by these paths.
(6) From the root to every level L(k) the Binary Tree is completely
covered by this set of paths, for every k ∈ ℕ.
(7) These paths represent the real numbers between 0 and 1.
What, all of them? Isn't it limited to some of the rationals being
1/2^m?
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>(8) It is impossible to find a further real number between 0 and 1.^^^^
even if you changed that to "rational" I don't think it would be true
and you provide no argument for the truth of (8).
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How about a constructive definition of reals as just dividing [0,1]
into "standard" infinitesimals or "iota-values", since the limit
of f(n) = n/d for naturals n, d with n -> d and d -> infinity
has the extent, density, completeness (LUB), and measure
what suffice to model a continuous domain?
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Then there's building neatly enough a model of (-oo, oo)
with the integer parts from the integers and each with
a non-integer part [0,1].
>
Then the complete ordered field can exist after the
ordered field of the rationals, as a derivation instead
of a definition (axiom, "non-constructive").
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Such a function as the "natural/unit equivalency function"
falls out of the arguments otherwise for uncountability
as un-contradicted, then also it makes simply that there's
its example as a non-Cartesian function so that thusly
it's free of contradictions with the otherwise usual notion
of transitive cardinality.
>
The rationals by themselves, for example, without presuming
their completion, suffer the arguments for uncountability.
So, having that there's already the LUB and measure 1.0 properties
keeps that afloat.
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Then that "there exists a surjection from the rationals to
the irrationals" has its own sort of account, making for
that there are at least three continuous domains, and such
a function would also be non-Cartesian.
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The standard definition of members of the complete ordered
field is "equivalence classes of series with the property
of being Cauchy".
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This way the line-reals are first, then the field-reals,
then the signal-reals are there own account, three
definitions of continuous domains or definitions of
continuity.
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The usual definition of continuity since Leibnitz is "gaplessness".
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(Excuse, that's "their" not "there", there, pointing to my dotage.)
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If you think reals are Dedekind cuts, iota-values are natural cuts.
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I'm happy to kick Dedekind right out of mathematics, or rather,
I've got no need for Dedekind cuts in an account of real numbers,
and the standard definition of real numbers of the complete ordered
fields as usually sets modeling them descriptively after the
axiomatic set theory is "equivalence classes of sequences, series
that are Cauchy".
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Why lose?
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