Path: ...!local-4.nntp.ord.giganews.com!Xl.tags.giganews.com!local-1.nntp.ord.giganews.com!news.giganews.com.POSTED!not-for-mail
NNTP-Posting-Date: Tue, 13 Aug 2024 03:07:06 +0000
Subject: Re: Replacement of Cardinality (real-valued)
Newsgroups: sci.logic,sci.math
References: <hsRF8g6ZiIZRPFaWbZaL2jR1IiU@jntp>
 <P04FAosjolyjDfgV0JPXWx1mF6o@jntp>
 <40165884-df8f-4614-8644-9161d72fd1cb@att.net>
 <-qUZ96ARwcjh9QPfyWRnijjNwoY@jntp>
 <6b837540-3d9a-4b8e-9a70-88d52e81a1a4@att.net>
 <xQQT0K_Q_k2FbMcCUXF8j3CEg84@jntp>
 <9822f5da-d61e-44ba-9d70-2850da971b42@att.net>
 <p36L63dXamDAkHDhkZhDKqx-h-o@jntp>
 <d8bbe664-a601-4590-9a7f-d5312b4dae54@att.net> <v8j55c$2u09m$2@dont-email.me>
 <w2KNEc6WpgYmvVtoH_VZTRkLnUg@jntp> <v8m8p8$3l66h$1@dont-email.me>
 <v8m98h$3lbdn$1@dont-email.me>
 <pKqcnTZhHeVzSDP7nZ2dnZfqnPudnZ2d@giganews.com> <v8nlsp$nh2$1@dont-email.me>
 <UN-dnYpeY6DnCDL7nZ2dnZfqnPqdnZ2d@giganews.com>
 <UN-dnYVeY6CVCjL7nZ2dnZfqnPoAAAAA@giganews.com>
 <prWcnbZXxt7cPzL7nZ2dnZfqn_ednZ2d@giganews.com>
 <a4af3e21-fa0b-40de-9d1f-e77b7fea976b@att.net>
 <ohKdnWTeloD8hCn7nZ2dnZfqnPednZ2d@giganews.com>
 <c9f82f99-6640-40f0-b14b-8e5424056433@att.net>
 <5FSdnYCK4KhuvCj7nZ2dnZfqn_SdnZ2d@giganews.com>
From: Ross Finlayson <ross.a.finlayson@gmail.com>
Date: Mon, 12 Aug 2024 20:07:13 -0700
User-Agent: Mozilla/5.0 (X11; Linux x86_64; rv:38.0) Gecko/20100101
 Thunderbird/38.6.0
MIME-Version: 1.0
In-Reply-To: <5FSdnYCK4KhuvCj7nZ2dnZfqn_SdnZ2d@giganews.com>
Content-Type: text/plain; charset=utf-8; format=flowed
Content-Transfer-Encoding: 8bit
Message-ID: <GwSdnbf0nMlHUCf7nZ2dnZfqn_cAAAAA@giganews.com>
Lines: 70
X-Usenet-Provider: http://www.giganews.com
X-Trace: sv3-PkzGzlJf3FOp+mzWEmOn+iirY6Egvg0Lg0ujjSr4bwExUEvE9ejy5S1xM2s1gwjnQ+3mFLfur2JG2fh!M/qcN93JM+0Hhy5iN//2NQQXK+60kJnc9xEPYG0ZrQdkJdbXpVO27Ki/75y7CHTQGMrGRxZ7L2wI!Dg==
X-Complaints-To: abuse@giganews.com
X-DMCA-Notifications: http://www.giganews.com/info/dmca.html
X-Abuse-and-DMCA-Info: Please be sure to forward a copy of ALL headers
X-Abuse-and-DMCA-Info: Otherwise we will be unable to process your complaint properly
X-Postfilter: 1.3.40
Bytes: 4641

On 08/08/2024 12:52 PM, Ross Finlayson wrote:
> On 08/08/2024 12:00 PM, Jim Burns wrote:
>> On 8/7/2024 9:03 PM, Ross Finlayson wrote:
>>> On 08/07/2024 01:06 PM, Jim Burns wrote:
>>
>>>> A theory can describe more than one model.
>>>>
>>>> Some claims have proofs.
>>>> Those claims are true in each model.
>>>>
>>>> Some claims are true in each model.
>>>> Those claims have proofs.
>>>> (That is a very nice result, maybe not super.obvious.)
>>>>
>>>> True.and.false in different models
>>>> does not make a theory contradictory.
>>>> The theory is silent, not wrong.
>>>
>>> What theory?
>>
>> | The completeness theorem applies to any first-order theory:
>> | If T is such a theory, and
>> | φ is a sentence (in the same language) and
>> | every model of T is a model of φ,
>> | then there is a (first-order) proof of φ
>> | using the statements of T as axioms.
>> | One sometimes says this as
>> | "anything true in all models is provable".
>> | (This does not contradict Gödel's incompleteness theorem,
>> | which is about a formula φᵤ that is unprovable
>> | in a certain theory T
>> | but true in the "standard" model of the natural numbers:
>> | φᵤ is false in some other, "non-standard" models of T.)
>>
>>> Model theory?
>>> That's exactly what model theory not.is.
>>
>> | The completeness theorem makes a close link between
>> | model theory,
>> | which deals with what is true in different models, and
>> | proof theory,
>> | which studies what can be formally proven
>> | in particular formal systems.
>>
>> https://en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem
>> Gödel's completeness theorem
>>
>>
>
> Well, yeah, proof-theory and model-theory are equi-interpretable,
> then a usual idea that foundations includes the entire space of
> the integer continuum and the linear continuum, or the ubiquitous
> ordinals and the hyper-dimensional holo-gram, makes that model
> theory is the usual milieu with the algebraizations, arithmetizations,
> and geometrizations.
>
> Goedel's pretty simple arithmetization of an algebraization
> and for Goedel's "stop hitting yourself" completion theorem,
> doesn't say much.
>
> That includes that it doesn't say much about the extra-ordinary.
>
> Here it's better to call the extra-ordinary the extra-ordinary
> instead of the non-standard, when, for example, it works out
> that the "standard" model of the integers: _isn't_.
>
> "Is it the universe of numbers or the universe of words? Yeah, it is."
>
>