Re: Ordinals

On 02/19/2024 02:03 PM, Ross Finlayson wrote:

On 02/19/2024 12:14 PM, Mild Shock wrote:

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Whats the strategy for writing such nonsense as below?
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(That sort of mercurial doffed-and-donned presumed jocularity and
familiarity is about the shallowest, vainest, fakest poser's.
That sort of inconstancy isn't "making friends and influencing people",
it's "give 'em nothing to depend on and keep 'em guessing".
It's the most obvious sort of example of a "manipulator",
which is considered a particular variety of pathological.)
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Try some sincerety sometime.
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What are products of omega? How are paradoxes sets?
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LoL
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Ross A. Finlayson schrieb am Samstag, 19. April 2014 um 21:18:08 UTC+2:

On 4/19/2014 12:50 AM, William Elliot wrote:

Does the set of all ordinals exist within ZF?
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This is "Ord", a collection of all ordinals (from among their
representations). The paradox of Cesare Burali-Forti is that
structurally, where membership is used to model order, the
collection itself of the ordinals would be an ordinal, thus
including itself. A "paradox" is not a set in ZF.
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Then there are set theories where it is a set, but those set
theories have anti-foundational infinities as a natural consequence
of definition. Russell has these kinds of sets as "extra-ordinary"
for ordinary.
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foundational / anti-foundational
regular / irregular
well-founded / non-well-founded
ordinary / extra-ordinary
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These are about the same.
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There are roundabout arguments that, for example, the finite ordinals,
as a set, consequently contain themselves, as an element. This is a
direct compactness result.
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ZF defines omega as a constant thus that omega and its products are
well-founded.

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You mean "Russell lied to you and you bought it",
"Russell's retro-thesis", "Russell's fools"?
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ORD, is the order type of ordinals, it's among
maximal elements and fixed points and universals.
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It's not non-sense indeed the opposite.
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My slates for uncountability and paradox,
help itemize how ordinals and sets are together.
(In a theory sets for ordinal relation, uncountability,
then a theory of sets with universes, paradox.)
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(There's a theory of "ubiquitous ordinals" among
all the primordial objects of mathematics a theory
of them.)
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If you study Cohen's "Independence of the Continuum Hypothesis",
right about at the end he introduces a deft consequence of ordinals,
and leaves set theory open about the Continuum Hypothesis.
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In case you missed it, ....
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It's pure theory, all theory.
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It's called foundations, maybe you want to know it.
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"Conservation of truth", all there is to it.
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(Maybe that's just me.)
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