Re: How many different unit fractions are lessorequal than all unit fractions? (constructive)

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.