Re: Replacement of Cardinality (infinite middle)

On 8/19/2024 6:21 PM, Ross Finlayson wrote:

On 08/19/2024 02:43 PM, Jim Burns wrote:

[...]

>
[...] eg where your limit ordinal comes from
as a matter of definition or axiomn if necessary,

Traced back, the limit ordinal comes from an axiom,
at least, it has if we haven't screwed up.
"Where things come from" is the job of axioms.
"What things mean" is the job of definitions.
It's an important distinction.
You are the best authority possible on what you mean.
That gives you a certain freedom of action,
when it comes to _definitions_ which
you don't have when it comes to _axioms_
Axioms have a tendency to being boring.
"Of course THAT exists. Duh!"
That is entirely the point.
You CAN disagree with an axiom,
where you CAN'T with a definition, not sensibly.
I don't WANT you to disagree.
Whence "Duh!"
Given Boolos's axioms for ST, (AKA 'Duh!')
⎛ ∃{}
⎜ ∀x∀y∃z=x∪{y}
⎝ and extensionality
we get that
the natural numbers exist,
finite sequences granting addition and multiplication
exist.
But we can't prove 'Duh!', except from other axioms,
or, if we can, we've proved it's worthless.
(I don't _believe_ it's worthless, but...)

[...] now that you've definitely declared that
your hedgerow has an inside and an outside,

I have declared that my hedgerow has an inside.
I am agnostic with regard to its outside.
That suffices because
I am only talking about its inside.
Note: All is not lost.
There are other hedgerows,
possibly outside of this one.
However, when I talk about another hedgerow,
I'll talk about only its inside.
That's all part of The Plan.