Q about Suppes Axiomatic Set Theory
Frederick Williams
reads:
A is complete if and only if every member of A is a subset of A.
He does not express it in his formal language, explaining why
For every x if x \in A then x \subseteq A
won't do. So how would one express the definition of 'complete'
formally?
[1] Patrick Suppes, Axiomatic Set Theory, Dover, 1972, page 130.
--
Which of the seven heavens / Was responsible her smile /
Wouldn't be sure but attested / That, whoever it was, a god /
Worth kneeling-to for a while / Had tabernacled and rested.
David C. Ullrich
<frederick...@tesco.net> wrote:
>This question refers to Definition 3, Section 5.1[1]. The definition
>reads:
>
> A is complete if and only if every member of A is a subset of A.
>
>He does not express it in his formal language, explaining why
>
> For every x if x \in A then x \subseteq A
>
>won't do. So how would one express the definition of 'complete'
>formally?
Why does he call this "complete" instead of "transitive"?
And what's wrong with that formalization?
>[1] Patrick Suppes, Axiomatic Set Theory, Dover, 1972, page 130.
David C. Ullrich
"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
Frederick Williams
>
> On Sun, 22 Nov 2009 12:46:59 +0000, Frederick Williams
> <frederick...@tesco.net> wrote:
>
> >This question refers to Definition 3, Section 5.1[1]. The definition
> >reads:
> >
> > A is complete if and only if every member of A is a subset of A.
> >
> >He does not express it in his formal language, explaining why
> >
> > For every x if x \in A then x \subseteq A
> >
> >won't do. So how would one express the definition of 'complete'
> >formally?
>
> Why does he call this "complete" instead of "transitive"?
Perhaps he wants to keep that for relations. It doesn't matter, does it?
> And what's wrong with that formalization?
An unhelpful, but possibly superior answer, would be 'you'd better read
the book'. Nevertheless, here goes.
Suppes writes (p 130)
Notice that we could not define completeness by the condition:
For every x if x \in A then x \subseteq A,
because the definition of \subseteq was conditional and no decision
can be made about the relation of inclusion for individuals.
You will know (or need to) that x, y, z, etc are "individuals" which
intuitively include sets and urelemente, and A, B, C, etc are sets
only. So by 'conditional' Suppes means
if x is a set then <defintion involving x>.
> >[1] Patrick Suppes, Axiomatic Set Theory, Dover, 1972, page 130.
--
David C. Ullrich
<frederick...@tesco.net> wrote:
>"David C. Ullrich" wrote:
>>
>> On Sun, 22 Nov 2009 12:46:59 +0000, Frederick Williams
>> <frederick...@tesco.net> wrote:
>>
>> >This question refers to Definition 3, Section 5.1[1]. The definition
>> >reads:
>> >
>> > A is complete if and only if every member of A is a subset of A.
>> >
>> >He does not express it in his formal language, explaining why
>> >
>> > For every x if x \in A then x \subseteq A
>> >
>> >won't do. So how would one express the definition of 'complete'
>> >formally?
>>
>> Why does he call this "complete" instead of "transitive"?
>
>Perhaps he wants to keep that for relations. It doesn't matter, does it?
>
>> And what's wrong with that formalization?
>
>An unhelpful, but possibly superior answer, would be 'you'd better read
>the book'. Nevertheless, here goes.
I'd "better" read the book? No, I don't think that would be a very
good answer - it would lead to the question of what consequences
would ensue if I didn't read the book.
The answer to your original question seems quite simple. Or not,
depending on a certain detail of the formal language. Alas I
don't have the book so I can't tell.
>Suppes writes (p 130)
>
> Notice that we could not define completeness by the condition:
>
> For every x if x \in A then x \subseteq A,
>
> because the definition of \subseteq was conditional and no decision
> can be made about the relation of inclusion for individuals.
>
>You will know (or need to) that x, y, z, etc are "individuals" which
>intuitively include sets and urelemente, and A, B, C, etc are sets
>only. So by 'conditional' Suppes means
>
> if x is a set then <defintion involving x>.
>
>> >[1] Patrick Suppes, Axiomatic Set Theory, Dover, 1972, page 130.
David C. Ullrich
Frederick Williams
>
> On Sun, 22 Nov 2009 15:28:33 +0000, Frederick Williams
> <frederick...@tesco.net> wrote:
>
> >"David C. Ullrich" wrote:
> >>
> >> On Sun, 22 Nov 2009 12:46:59 +0000, Frederick Williams
> >> <frederick...@tesco.net> wrote:
> >>
> >> >This question refers to Definition 3, Section 5.1[1]. The definition
> >> >reads:
> >> >
> >> > A is complete if and only if every member of A is a subset of A.
> >> >
> >> >He does not express it in his formal language, explaining why
> >> >
> >> > For every x if x \in A then x \subseteq A
> >> >
> >> >won't do. So how would one express the definition of 'complete'
> >> >formally?
> >>
> >> Why does he call this "complete" instead of "transitive"?
> >
> >Perhaps he wants to keep that for relations. It doesn't matter, does it?
> >
> >> And what's wrong with that formalization?
> >
> >An unhelpful, but possibly superior answer, would be 'you'd better read
> >the book'. Nevertheless, here goes.
>
> I'd "better" read the book? No, I don't think that would be a very
> good answer - it would lead to the question of what consequences
> would ensue if I didn't read the book.
>
> The answer to your original question seems quite simple. Or not,
> depending on a certain detail of the formal language. Alas I
> don't have the book so I can't tell.
Given my ignorance concerning logic (and I do think it's a question of
logic not set theory), my best bet is that someone who has read Suppes
sees my question.
>
> >Suppes writes (p 130)
> >
> > Notice that we could not define completeness by the condition:
> >
> > For every x if x \in A then x \subseteq A,
> >
> > because the definition of \subseteq was conditional and no decision
> > can be made about the relation of inclusion for individuals.
> >
> >You will know (or need to) that x, y, z, etc are "individuals" which
> >intuitively include sets and urelemente, and A, B, C, etc are sets
> >only. So by 'conditional' Suppes means
> >
> > if x is a set then <defintion involving x>.
> >
> >> >[1] Patrick Suppes, Axiomatic Set Theory, Dover, 1972, page 130.
>
> David C. Ullrich
--