Re: Replacement of Cardinality (ubiquitous ordinals)

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:

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Then what *is* restricted comprehension?

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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.

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I am more familiar with unrestricted comprehension
being the antonym of restricted comprehension.
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Unrestricted comprehension grants that
{x:P(x)} exists because
description P(x) of its elements exists.
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Restricted comprehension grants that
{x∈A:P(x)} exists because
description P(x) and set A exist.
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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.

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

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