On 08/11/2024 02:38 PM, Jim Burns wrote:
On 8/11/2024 2:10 PM, Ross Finlayson wrote:
On 08/11/2024 09:10 AM, Moebius wrote:
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How do you see omega
as the second constant after empty set
an inductive set in ZF?
It's definitely not "all" infinity.
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ω is defined to be (the set of) all finite ordinals.
In that sense, ω is all of finiteness.
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ω is followed by all the transfinite ordinals.
In that sense, ω is not all of infinity.
Nearly none of it, really.
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----
U and V are inductive sets.
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⋂{ind:U} is the intersection of inductive U.subsets.
⋂{ind:U} is inductive.
for each inductive A ⊆ U: ⋂{ind:U} ⊆ A ⊆ U
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⋂{ind:V} is the intersection of inductive V.subsets.
⋂{ind:V} is inductive.
for each inductive B ⊆ V: ⋂{ind:V} ⊆ B ⊆ V
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In particular, U∩V is an inductive U.subset and V.subset.
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As an inductive V.subset,
⋂{ind:V} ⊆ U∩V ⊆ V
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As an inductive U.subset,
⋂{ind:U} ⊆ ⋂{ind:V} ⊆ U∩V ⊆ U
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⋂{ind:U} ⊆ ⋂{ind:V}
Similarly,
⋂{ind:V} ⊆ ⋂{ind:U}
⋂{ind:U} = ⋂{ind:V}
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⋂{ind:U} = ⋂{ind:V} := ⋂{ind}
the unique intersection of inductive subsets.
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----
{fin} is the set of finite ordinals.
⋂{ind} is the intersection of inductive subsets.
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There is no first finite.ordinal ∉ ⋂{ind}
There is no finite ordinal ∉ ⋂{ind}
{fin} ⊆ ⋂{ind}
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Each inductive set ⊇ ⋂{ind}
{fin} is inductive.
{fin} ⊇ ⋂{ind}
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{fin} ⊆ ⋂{ind}
{fin} ⊇ ⋂{ind}
{fin} = ⋂{ind} := ω
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How do you see omega
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ω is the set of all finite ordinals and is,
for each inductive set,
the intersection of inductive subsets.
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https://www.youtube.com/watch?v=6liqzvagSb8Here's some good listening, a philosophy professor
explains why metaphysics is logically necessary
and may be coherent, and that the rejection
thereof is irrational, or as Spock might put it, "illogical".
He makes good points about the "indivisibility of truth"
and that a conscientious theory has a metaphysical
association with truth "de re" and "de res" not just "de dicto".
Starting with a theory _without_ the constant introduced named omega,
i.e., finite sets, given that there's axiomatized well-foundedness
when otherwise simple comprehension would make the "omega" into
an extra-ordinary or non-well-founded or inconsistent-multiplicity
of a set, starting _without_ omega, the finite sets like ordinals,
are, exactly those sets that don't contain themselves.
Then, omega, as you've defined it, contains itself, again just
quantifying over the specification of what omega purports to
be, including your hereditarily finite build-up which is the
same thing.
So, even with your Russellian retro-thesis, even with "mere induction",
even with axioms otherwise, you're stuck again with both Russell
and Frege, regardless which you pick.
So, omega the constant of ZF or ZFC is just a fragment or extension,
it's a fixed-point, and yes this is correct usage, in a deconstructive
account of your received theory there, why induction doesn't get a say.
So, "omega is not all of infinity", whether it's +-1, besides
the entire cumulative hierarchy, of transfinite cardinals.
There are models of integers where there aren't standard
models of integers.
I'm curious, now that you have a beginning and an end of
the finite, or 0 and omega in ZF, first of all isn't that
a fixed-point and compactification also, as of about the
same thing as a point-at-infinity for either perspective
in geometry or fixed-point in number theory, one of the
regular singular points of the hypergeometric the 0, 1, infinity?
Second, do you just say "see Russell's rule"?
Third, didn't you already have one? Isn't there already one?
If not, where did your otherwise sound naive induction end?
Replacement of Cardinality
Re: Replacement of Cardinality