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From: Ben Bacarisse <ben.usenet@bsb.me.uk>
Newsgroups: comp.theory
Subject: Re: Simply defining =?iso-8859-1?Q?G=F6del?= Incompleteness and Tarski Undefinability away V33 (Mendelson Satisfiability)
Date: Sun, 02 Aug 2020 00:11:52 +0100
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olcott <NoOne@NoWhere.com> writes:

> On 8/1/2020 2:28 PM, Ben Bacarisse wrote:
>> olcott <NoOne@NoWhere.com> writes:
>>
>>> On 8/1/2020 6:55 AM, Ben Bacarisse wrote:
>>>> olcott <NoOne@NoWhere.com> writes:
>>>>
>>>>> On 7/31/2020 6:47 PM, Ben Bacarisse wrote:
>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>
>>>>>>> On 7/29/2020 8:38 PM, Ben Bacarisse wrote:
>>>>>>>> olcott <NoOne@NoWhere.com> writes:
>>>>>>>>
>>>>>>>>> On 7/29/2020 4:47 PM, Ben Bacarisse wrote:
>>>>>>>> <cut>
>>>>>>>>>> Better would be to stick to the usual definition of a binary relation on
>>>>>>>>>> H which is simply a subset of HxH.
>>>>>>>>>
>>>>>>>>> Where D is the set of humans:
>>>>>>>>> <a26, a87> ∈ Father_of(x, y) is Boolean.
>>>>>>>>
>>>>>>>> More and different junk.
>>>>>>>
>>>>>>> So in other words there is no conventional formal way to assert that
>>>>>>> the relation "father of" exists between two specific individual
>>>>>>> humans.
>>>>>>
>>>>>> You have abandoned any pretence of wanting to know how these things are
>>>>>> done.  Do you plan to do any more exercises from Mendelson?
>>>>>
>>>>> Yes as soon as we have this one issue resolved.
>>>>
>>>> Who's we, and what issue?  Do you mean you want to keep writing your
>>>> math poems using deliberately confusing names and ill-defined
>>>> constructs?  If so, what is there to be resolved?
>>>>
>>>
>>> I want to know the most conventional way to assert that the ordered
>>> pair of two particular individuals are members of the "father or
>>> relation.
>>
>> Explicitly: (x, y) ∈ R.  This is often contracted by using infix
>> notation xRy, particularly if the relation has a symbol for a name
>> (think =, <, ⊆ and so on).
>>
>> Some authors use the contraction R(x, y) especially when the relation is
>> not binary (R(p, q, r, s) is less messy than (p, q, r, s) ∈ R, but I
>> would suggest you avoid this as it look too much like a predicate.
>>
> (1) You are using variables I need to see it with constants.

You really don't know how to make x<y specific by writing 1<2?  If that
is a barrier to you, you need much more basic help than I've been
offering.

But let me try to help.  You tell me what keys I have to type to refer
to the two set members, and what keys to type to refer to the relation.
I will then put first two in ()s with a comma between them, type an ∈
and then the keys you say refer to the relation.  Would that help?

> (2) Isn't the ONLY different between a predicate and a relation that
> the predicate has not been associated with a domain?

Provided you know what the two are, I don't care how you number the
differences.


>>> The way that Mendelson says it is certainly not the most standard way.
>>>
>>> For each individual constant ai of L, an assignment of some fixed
>>> element(a_i)^M of D.
>>
>> Oh dear.  You ask a reasonable question (how does one...) but then it
>> turns out you don't know what's going on.  The line you quote does not
>> "assert that the ordered pair of two particular individuals are members
>> of [some relation]" so you don't, in fact, know how Mendelson does it.
>
> Of course it doesn't. It only refers to the mapping of the constants
> of a formal language to elements of the domain, and it seems to do
> this in a very clumsy way that is probably not standard.

So the problem was just that "it" changed meaning between sentences.

>> Mendelson does it as above except that he uses <...> rather than (...)
>> for an n-tuple: <x, y> ∈ R or xRy.  See page 6.  Have you skipped the
>> important first few pages?
>
> I have skipped everything before page 50 unless I need to refer back
> to it. I already know all this stuff (before page 50) it is Mendelson
> notation of this stuff that I may not know.

You don't know how to say that two thing are in a relation.  I think you
need to review any basic maths book and the first 50 pages of Mendelson
can't help.

> I understood the idea and the notation for ordered pairs of variables
> in a relation of a domain at least a week ago.

You shouldn't.  There are no pairs of variables in any of these
relations.  A relation on some set S is a subset of SxS.

[Aside: Of course, at some meta-level, one can talk about the set of
variables in a language or the set of variables used in a formula and
one might then define relations on these sets, but nothing like that is
happening here.]

> How does this work for constants?
> <a26, a87> ∈ R  ???

Constants are symbols in the formal language.  Pairs of them are never
in any relation in an interpretation.  I don't know how to help you get
past this blockage.  I suspect you just don't mean what you are writing
and rather insist on calling everything in the domain of discourse "a
constant".

-- 
Ben.