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"?
>
>