Re: Incompleteness of Cantor's enumeration of the rational numbers (standard infinitesimals)

On 11/27/2024 08:10 PM, Ross Finlayson wrote:

On 11/27/2024 12:41 PM, Chris M. Thomasson wrote:

On 11/27/2024 10:50 AM, Ross Finlayson wrote:

On 11/27/2024 10:19 AM, FromTheRafters wrote:

WM explained :

On 27.11.2024 13:32, Richard Damon wrote:

On 11/27/24 5:12 AM, WM wrote:

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Of course. |{1, 2, 3, 4, ...}| = |ℕ| and |{2, 3, 4, ...}| = |ℕ| - 1
is consistent.

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So you think, but that is because you brain has been exploded by the
contradiction.
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We can get to your second set two ways, and the set itself can't know
which.
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We could have built the set by the operation of removing 1 like your
math implies, or we can get to it by the operation of increasing each
element by its successor, which must have the same number of
elements,

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Yes, the same number of elements, but not the same number of natural
numbers.
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Hint: Decreasing every element in the real interval (0, 1] by one
point yields the real interval [0, 1). The set of points remains the
same, the set of positive points decreases by 1.

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If you have a successor function for the real numbers, why don't you
share it with the rest of the world?

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You mean like line-reals and iota-values?
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It's one of Aristotle's continuums, been around forever.
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Oh, you mean stack it up again modern mathematics
and show that a sort of only-diagonal a non-Cartesian
not-a-real-function with surprising and special
real analytical character fits within the theory
otherwise our great axiomatic set theory a descriptive
set theory with a bit of stipulating LUB and measure 1.0?
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I wouldn't say that usenet's "closed" as it were,
though, traffic is usually more directed to the
great maw of mammon's soup-hole, there is though
that each usenet article has a usual unique identifier
according to MLS and Chicago and other usual matters
of agreement in bibliographic reference.
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The infinitesimal analysis of course has been around
for a long, long time, and these days it's called
"non-standard", which it exists at all,
now that you mention it.
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Here it's "line-reals" with "iota-values", at fulfill
being a model of a continuous domain (though, only a
bounded segment, of course), that do have a least positive
iota-value, not to be confused with infinitely-divisible
members of the complete ordered field, that also fulfills
being a "continuous domain" (extent, density, completeness,
measure) after axiomatizing LUB and measure 1.0 above set theory.
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I can say 1.1 is a successor for 1... ;^) That is finite thinking in the
realm of the reals. There are infinite successors for 1, forget about 2
for a moment... ;^)
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1.1
1.01
1.3
1.000001
1.8
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We can say that a successor is greater than its predecessor for the
positive real line...
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(0)->(1)->(2)->(+real_line)
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Well, 1 is greater than 0, 2 is greater than 1 and 0.
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(0)->(.01)->(.010042)->(+real_line)
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.01 is greater than 0, .010042 is greater than .01 and 0.

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When we have "finite thinking" and "infinite things",
it's usually just called "unbounded", yet "infinitary reasoning",
is a thing, and there are several examples of "infinitary reasoning",
like Zeno's arguments usually first, and "the calculus: real analysis
a.k.a. infinitesimal analysis", and "Fourier analysis: analyticity
as it were in bounded regions in expression in infinite series",
these being the usual big three example of "non-standard analysis"
what's also called "super-classical".
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Then there's for example "particle/wave duality", when it's not
just "particles, or waves", it's, "particles, and waves".
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So, "finite thinking" is usually called regular, and,
"infinitary reasoning" has often been called impossible,
because there's an inductive impasse it takes deductive inference
to surmount, yet, anything that arrives "unbounded" is
still an exercise in "infinitary reasoning" in the later account,
while it's called "wishful thinking and axiomatizing the result"
when of course there's an inductive account that it fails.
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So, "infinitary reasoning", includes a) the geometric series,
b) the FTC's, c) Fourier-style analysis, then for example
d) the Dirac delta, though often that's employed itself in
Fourier-style analysis. So there's the geometric series,
methods of exhaustion of course, the FTC's, Dirac delta,
Fourier-style analysis, any of which anybody could call
"non-standard", with regards to the "standard Archimedean:
nothing's infinite" and the "standard non-standard Archimedean:
something's infinite".
>
So, infinitary reasoning is just a usual thing and part of
a fuller dialectic and higher reasoning. The perfect results
of the calculus (the real analysis) are due it.
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The Dirac delta is most people's first, and only,
not-a-real-function with real-analytical-character
given in class, while the geometric series and
iota-values (atoms, say) are most people's first
mental models of infinitary reasoning.
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It's not just "do-ing it", it's, "do-ing do-ing it".
I know that readily makes a doo-doo joke,
it's an obvious mental model of reasoning
after the fact.
This is where there's the prior and posterior, and,
the inductive inference, and, lesser deductive inference
and greater deductive inference, and abductive inference,
yet here just called deductive inference, as to why
linear-only thinkers are kind of like graphical- or visual-
only learners, I forget which of those is a speed-reader
and which is "animal brain only, please". I.e. the
whole reason the primary cerebellum arrives with capacity
for higher mental reason and faculties is to get out of
the mud.
Then, it's often called "non-linear", not just systems
that are non-linear the specific technicalities in
mathematics system with linear solutions, "non-linear thinking",
also "the dialectic" sometimes, where the idea is that all the
classical expositions of mathematical paradoxes are supposed
to arrive at thinking them out to arrive at higher reasoning,
for example, making proofs, besides following them.
In the prior and posterior then it reflects "the virtuous circle"
that in mathematics there's usually inner for outer and
abstraction for reduction, inverses and reciprocals, where
in fact many or most of the methods in mathematics are
where those cross making branches and in singularity theory
working up what's multiplicity theory, analyticity.
Or, in base infinity, 1.1 is like 1.000...001,
sort of like fixed-point in extended-precision
with a least increment, that in the unbounded
in word width, the scalar and what's called super-scalar,
word.