Re: Replacement of Cardinality (ubiquitous ordinals)

On 8/3/2024 11:51 PM, Ross Finlayson wrote:

On 08/03/2024 08:45 PM, Jim Burns wrote:

On 8/3/2024 9:08 PM, Ross Finlayson wrote:

On 08/03/2024 12:08 PM, Jim Burns wrote:

On 8/2/2024 3:55 PM, Ross Finlayson wrote:

On 08/02/2024 03:39 AM, FromTheRafters wrote:

Then what *is* restricted comprehension?

>
Usually it's just the antonym of
expansion of comprehension.
What I ask,
if that you surpass,
the inductive impasse,
of the infinite super-task.

>

I am more familiar with unrestricted comprehension
being the antonym of restricted comprehension.
>
Unrestricted comprehension grants that
{x:P(x)} exists because
description P(x) of its elements exists.
>
Restricted comprehension grants that
{x∈A:P(x)} exists because
description P(x) and set A exist.
>
The existence of set A might have been granted
because of Restricted.Comprehension or Infinity or
Power.Set or Union or Replacement or Pairing,
but A would be logically prior to {x∈A:P(x)}
by some route.

>
Geometry, axiomatic geometry or Euclid's,
is a classical theory, and it's constructive,
there's only expansion of comprehension,

>
I know what comprehension, restricted.comprehension,
and unrestricted.comprehension are by having seen
set axioms which were called Comprehension,
Restricted.Comprehension, and Unrestricted.Comprehension.
>
What does 'comprehension' mean where there are no sets?

>
What can you think it means.

Your rhetoric suggests that
_you_ don't have something in mind for the term
_you_ introduced,
and you'd like someone else to provide something
to have in mind. Please prove me wrong.
What does 'comprehension' mean where there are no sets?
Specifically,
what does 'expansion of comprehension' mean
in the context of
"geometry, axiomatic geometry or Euclid's"?