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NNTP-Posting-Date: Thu, 26 Dec 2024 19:18:37 +0000
Subject: Re: Ordinals
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Thu, 26 Dec 2024 11:18:39 -0800
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On 02/19/2024 03:04 PM, Ross Finlayson wrote:
> On 02/19/2024 02:03 PM, Ross Finlayson wrote:
>> On 02/19/2024 12:14 PM, Mild Shock wrote:
>>>
>>> Whats the strategy for writing such nonsense as below?
>>>
>>
>>
>> (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.)
>>
>> Try some sincerety sometime.
>>
>>> What are products of omega? How are paradoxes sets?
>>>
>>> LoL
>>>
>>> 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?
>>>>>
>>>> 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.
>>>>
>>>> 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.
>>>>
>>>> foundational / anti-foundational
>>>> regular / irregular
>>>> well-founded / non-well-founded
>>>> ordinary / extra-ordinary
>>>>
>>>>
>>>> These are about the same.
>>>>
>>>> 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.
>>>>
>>>> ZF defines omega as a constant thus that omega and its products are
>>>> well-founded.
>>
>> You mean "Russell lied to you and you bought it",
>> "Russell's retro-thesis", "Russell's fools"?
>>
>> ORD, is the order type of ordinals, it's among
>> maximal elements and fixed points and universals.
>>
>> It's not non-sense indeed the opposite.
>>
>> 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.)
>>
>> (There's a theory of "ubiquitous ordinals" among
>> all the primordial objects of mathematics a theory
>> of them.)
>>
>> 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.
>>
>> In case you missed it, ....
>>
>>
>> It's pure theory, all theory.
>>
>> It's called foundations, maybe you want to know it.
>>
>> "Conservation of truth", all there is to it.
>>
>>
>>
>
> (Maybe that's just me.)
>