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NNTP-Posting-Date: Tue, 17 Sep 2024 02:53:43 +0000
Subject: Re: How many different unit fractions are lessorequal than all unit
 fractions? (constructive)
Newsgroups: sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 16 Sep 2024 19:53:39 -0700
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On 09/12/2024 11:05 AM, Jim Burns wrote:
> On 9/10/2024 8:24 PM, Ross Finlayson wrote:
>> On 09/10/2024 02:21 AM, Jim Burns wrote:
>>> On 9/9/2024 11:11 PM, Ross Finlayson wrote:
>>>> On 09/09/2024 11:23 AM, Jim Burns wrote:
>>>>> On 9/9/2024 12:59 PM, Ross Finlayson wrote:
>>>>>> On 09/08/2024 09:59 PM, Jim Burns wrote:
>>>>>>> On 9/8/2024 6:34 PM, Ross Finlayson wrote:
>
>>>>>>>> It's already been thoroughly elaborated and
>>>>>>>> as attached to formalistic symbolry,
>>>>>>>> that "Russell's thesis, of an antinomy" is that
>>>>>>>> the set of
>>>>>>>> the finite sets that don't contain themselves,
>>>>>>>> exactly like the ordinals are mostly simply modeled to be,
>>>>>>>> does and doesn't contain itself,
>>>
>>>>>>> I wonder what your own conscientious response
>>>>>>> will be to the infiniteness of the set of
>>>>>>> all finite non.self.membered sets.
>>>
>>>> To your question, "are the finites infinite",
>>>> well yeah.
>>>
>>> It seems to me (JB) that
>>> ⎛
>>> ⎜ Russell's thesis is that
>>> ⎜ the set of
>>> ⎜ the finite sets that don't contain themselves
>>> ⎜ does and doesn't contain itself
>>> ⎝
>>> is and is not what you (RF) are claiming.
>>
>> Oh, perhaps, maybe, in a sense, weighing alternatives,
>> it's what I'm claiming is that it isn't and not is
>> what BR Bertrand Russell is claiming.
>
> Bertrand Russel's set is
> the set of all non.self.membered sets.
> not the set of all _finite_ non.self.membered sets.
>
> If you intend to debate that,
> it would be good to get it over and done with
> instead of rushing past it
> to what might be more interesting questions.
>
>> Of course, overgeneralization is generally unsound,
>
> Surely,
> not all generalizations are over.generalizations.
>
> Consider
> ⎛ Each nonempty set of ordinals
> ⎝ holds a first element.
>
> Very general. Very true. We know it is because of
> _what we mean_ by 'ordinal'.
>
>  From such (not.over).generalizations
> further (not.over).generalizations follow
> -- in a not.first.false finite order --
> (not.over).generalizations which we didn't know,
> but we do now, after having seen them in that order.
> We call them 'theorems' and the order 'proofs'.
>
> For example,
> ⎛ Each nonempty set of ordinals
> ⎜ holds a first element.
> ⎜
> ⎜ {α,β: α≠β}
> ⎜ holds a first element, α or β
> ⎜ α≠β  ⇒  α<β ∨ β<α
> ⎜
> ⎜ {α,β,γ: α<β, β<γ}
> ⎜ holds a first element, not β and not γ
> ⎜ α is its first element
> ⎝ α<β ∧ β<γ  ⇒  α<γ
>
> By definition, α≤α and ¬(α<α)
>
> (Not.over).generalizing,
> the ordinals are linearly ordered.
>
> Consider an ordinal as
> the set of all earlier ordinals.
> β = {α:α<β}
>
> Because the ordinals are linearly ordered,
> no ordinal is earlier than itself,
> no ordinal is self.membered.
>
>> while there are absolutes in logic,
>> and for example
>> the unbounded and completions and closures in logic,
>> which you make BR a dupe and dupe you to unfasten.
>>
>> Now, maybe this message will reach you or
>> maybe it won't, ...,
>> maybe it won't.
>
>

Excuse this delay, where as with regards to why
Russell's paradox applies to just a plain old
inductive set when the merely-finite sets are
all the sets in the theory after reverse mathematics
before infinity's axiomatized, that the same telling
blow that Russell used to submarine Frege is just
declared gone away, is what it is.

I.e., in ZF minus Infinity, comprehending the
usual set of v.N. ordinals, results this.

And Russell's like "I was never heeeeerre...".