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NNTP-Posting-Date: Sun, 04 Aug 2024 14:35:31 +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: Sun, 4 Aug 2024 07:36:16 -0700
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On 08/03/2024 10:25 PM, Jim Burns wrote:
> 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"?
>
>

No, "what can you think", it means.

Usually it just means "construction".