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NNTP-Posting-Date: Sun, 04 Aug 2024 03:50:38 +0000
Subject: Re: Replacement of Cardinality (ubiquitous ordinals)
Newsgroups: sci.logic,sci.math
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From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Sat, 3 Aug 2024 20:51:06 -0700
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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.