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NNTP-Posting-Date: Tue, 17 Sep 2024 20:10:36 +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: Tue, 17 Sep 2024 13:10:27 -0700
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On 09/17/2024 12:26 PM, Jim Burns wrote:
> On 9/16/2024 10:53 PM, Ross Finlayson wrote:
>> On 09/12/2024 11:05 AM, Jim Burns wrote:
>
>>> [...]
>>
>> 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.
>
> Russell's paradox concerns a set which,
> by unrestricted comprehension,
> exists, but which,
> because it both is and isn't self.membered,
> not.exists.
>
> The standard method of resolving the paradox
> is to not.use unrestricted comprehension.
>
> A plain.old.inductive.set is NOT
> a set which both is and isn't self.membered.
> It isn't self.membered.
> I would not say Russell's paradox applies.
>
> A plain.old.inductive.set IS
> banned from the company of merely.finite sets,
> because it isn't finite.
>
> Perhaps you (RF) are drawing a parallel between
> two examples of being banned.
>
> A typical resolution of
> Russell's self.membered and non.self.membered set
> is to modify our axioms so that
> we can avoid claiming it exists.
>
> There is also the Revision Theory of Truth,
> which strives to expand our notion of Truth to
> where it can deal with these tangled references.
> https://plato.stanford.edu/entries/truth-revision/
> https://en.wikipedia.org/wiki/Revision_theory
>
>> I.e., in ZF minus Infinity,
>> comprehending the usual set of v.N. ordinals,
>> results this.
>
> In ZF-Infinity,
> the usual set of finite ordinals might not.exist,
> and, if not.existing, not.causes a paradox.
>
> In ZF-Infinity+Anti.infinity,
> there definitely isn't a set of finite ordinals.
>
>

A class of propositions in "Tertium Non Datur"?


The theory of a "real, concrete, true truth" sort of
requires a Comenius language where everything is true
yet The Liar serves as a prototype of un-truth, making
for "ex falso nihilum" instead of "ex falso quodlibet",
with regards to theories of truth like coherent, correpondent,
pragmatic, now as you've mentioned "revision theory", as
with regards to, for example, the di-aletheic, here then
as with regards to relevance logic fulfilling De Morgan's
laws where classical quasi-modal logic does not.

In my podcasts I often address the notion of "a briefer
metaphysics" and "cyclic metaphysics" to help arrive at
why there can be a theory of "true truth" while it is
yet something we may only _attain_ to yet that it is.
(That it, "Is".)

"A Theory", ....

Mathematical constructions are enduring edifices
and for some for example whose strong mathematical
platonism results a strong logical positivism,
while usual nominalists/fictionalists make a weaker
logical positivism that is always striving yet
to its _origins_, there is that a "structuralist's"
and constructivist overall theory of a "the truth"
makes itself then after a deconstructive account of
modern formalism, quite helping resolve these sorts
issues, when, _there can be no paradoxes at all_,
or that there is one, at all.