[FOM] certain two papers

[Patrik Eklund]

Just quickly checked your paper, Lukasz, and I should read more, but 
here just a brief observation.

---

You are propositional, so whereas those related logical operators appear 
both as logical and with their 'meta-logical' counterpart, for the 
quantifiers you only have them in the 'meta-logics'.

Whatever the consequences of that, this in itself reveals the problems 
with numbering. You (potentially) number your logical connectives, but 
not the 'meta-logical' ones. Gödel numbers whatever symbol comes his 
way, and then proceeds like logics and meta-logics is one and the same 
thing, in the same bag, as I often say.

---

I still haven't had any FOMer react to my objection to having two 
different negations, namely, one for ~p(x) and one for ~Ex.p(x). The 
first one is typed and is an operator in the underlying signature along 
with p also as an operator. The second one is not the one in the 
signature, simply because Ex.p(x) is not a term. p(x) is a term and has 
a type, and can be used in substitution, but Ex.p(x) is not a term, does 
not have a type, and cannot be used in substitution. Nevertheless, ~ is 
taken to be one and the same (syntactic) symbol. E is not a "formal 
symbol". It's an "informal symbol". Similary, as Church said back in 
1940, lambda is not a "formal symbol". It's an "informal symbol".

So Gödel numbers over formal and informal symbols like painting all over 
the wall including the window frames, even the windows!, with one and 
the same brush and colour. What a house! And the interior is 
inconsistent anyway.

Cheers,

Patrik



On 2017-11-24 00:51, Lukasz T. Stepien wrote:
> Professor Shipman,
> 
>    Thank you very much for your interest in the proof of the 
> consistency
> of the Arithmetic System, mentioned on my website:
> http://www.ltstepien.up.krakow.pl/  . This proof has been included in
> the paper published in _Journal of Mathematics and System Science_: T.
> J. Stępień, Ł. T. Stępień, "On the Consistency of the Arithmetic
> System", _J. Math. Syst. Sci._ 7, No.2, 43-55 (2017)
> http://www.davidpublisher.org/Public/uploads/Contribute/58d876b1d91a2.pdf
> [1]  . From the construction of this proof, it follows that Godel's
> Second Incompleteness Theorem is invalid.
> 
> 
>                                         ŁTS
> 
> ########################################################################################################
> 
> Professor Stepien,
> 
> The link at the bottom of your message leads to a page where you claim
> to have shown that Godel's Second Incompleteness Theorem is invalid and
> that Con(PA) is a theorem of PA. That's much more interesting than what
> you promote in your message; do you stand by that result or have you
> withdrawn it?
> 
> -- JS
> 
> Sent from my iPhone
> 
> --
> 
> Lukasz T. Stepien
> 
> The Pedagogical University of Cracow
> Institute of Computer Science,
> ul. Podchorazych 2
> 30-084 Krakow
> Poland
> 
> tel. +48 12 662-78-54, +48 12 662-78-44
> 
> The URL  http://www.ltstepien.up.krakow.pl
> 
> 
> Links:
> ------
> [1]
> http://www.davidpublisher.org/Public/uploads/Contribute/58d876b1d91a2.pdf
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