Re: [OT] Right Books (was re: Excuses, Excuses)


Aatu Koskensilta <aatu.koskensilta@xxxxxxxxx> wrote:
>Acme Diagnostics wrote:
>
>The real problem with your explanation is not, however, its vagueness,
>but the fact that you get the condition of being deductively closed
>wrong. Being deductively closed does not mean that all sentences in T
>can be logically deduced from other sentences in T. Rather, it means
>that whatever is logically deducible from sentences in T is also in T.
>Consider the set T={'0=0'}. All sentences in T can be logically deduced
>from sentences in T, but T is clearly not deductively closed.

Ok, thanks. I see that that is important.

>>>The usual way to present a theory T is
>>>by exhibiting in some form or other an axiomatization, that is a set of
>>>sentences A, such that every sentence in T follows from the sentences in
>>>A.
>>
>> Typically, not every sentence is necessary to make
>> those deductions. The necessary sentences are called an
>> "axiomization."
>
>But for an arbitrary theory T there isn't in general any specific
>collection of 'necessary sentences'. Rather, any set of sentences Ax
>from which all sentences in T follow is an axiomatization. Actually,
>this usage is a bit idiosyncratic - since it implies that T is an
>axiomatization of itself! - since usually axiomatizations are required
>to be recursive or recursively enumrable sets of sentences, i.e. it is
>required that there is a mechanical procedure for verifying when a
>sentence belongs to Ax or at least a mechanical procedure for generating
>the sentences in Ax. When explaining what is meant by 'true axiom' and
>'false axiom' this ramification plays no role, however.

Yeah, I figured that was pretty goofed up. Thanks again.

>> [Probably innaccurate because the coincidence between an
>> axiomization and the associated theory was not explained.
>> I was a confused beginner because I thought the axioms
>> were part of the theory, so that now I would be reading in
>> some state of confusion.]

Well, with your corrections, our beginner would have even
more useful distinctions. One important point of my post was how
the editor and the expert must work together to create
the "right book" and how difficult and tedious that is, i.e.
expensive. E.g. this example from only one of your paragraphs
compared to how many paragraphs are in a textbook.

When I link my example in the future, I intend to link
yours too if correspondents might have interest in the correct
explanations.

I hope that my comments do not reflect in any way on
your confidence in helping newcomers or to explain anything, as
those contributions are so valuable. My post neither intended nor
evidenced anything at all in that regard.

Larry
.

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