Re: Replacement of Cardinality (infinite middle)

On 08/19/2024 05:33 PM, Jim Burns wrote:

On 8/19/2024 8:08 PM, Ross Finlayson wrote:

On 08/19/2024 04:18 PM, Jim Burns wrote:

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[...]

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Then, about that
the class of ordinal is an ordinal

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True because of what we mean by 'ordinal'
⎛ which leaves open the other question about
⎜ whether that class or finite ordinals or
⎝ inaccessible cardinal or ... _exist_
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and needn't be given by axiom or relation to an axiom,
yet instead as a matter of comprehension over the class,

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...if the class exists.
How we know that a class exists is by axiom.
It's an abstract object.
What other way could we know?
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This need for some axiom to start off the existing
is harder to paper over in a formal language.
But, with either a formal or natural language,
it's inherent in exploring Plato's realm of Forms.
How else do we enter that realm?
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You mean it's a void or a universe
and one can't know which and its
very contemplation thus inverts it
thus it's some dually-self-infraconsistent
Ding-an-Sich this primary object an ur-element?
I just call it that.
This way both "how do you get something from
nothing" and "how do you get nothing from
something" result the same answer so that
Kant's Sublime is Supreme and Hegel's
Nothing is Being.
Then, Leibnitz doesn't really refer to Plato
is his monadology, nor Wittgenstein in his
Tractatus, yet Gadamer wraps up for them "Amicus Plato".
Here it's simply that axiomless natural deduction is
this thing then axiomless geometry arrives at
it fully suffices for all the Euclidean
then the rest "must" be, "Es muss sein",
and it requires of course that one has arrived
at a theory of a Comenius-like language and then
that that there are no mathematical nor logical
paradoxes, at all, those all being resolved by
dually-self-infraconsistency.
Perhaps you've never left that realm.