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Author: georgegeorge Date: May 11, 2008 14:17
On May 11, 3:57 pm, kleptomaniac6...@ hotmail.com wrote:
> I meant full second order arithmetic to be the theory PA+comprehension
> +second order induction axiom.
2nd-order logic *already has* a definition, and it *is not* in terms
of
"comprehension". It is just in terms of 2nd-order quantifiers
quantifying
over "all possible" 1st-order subclasses of the obect-level domain.
> I was wondering what you got with PA +comprehension+first order
> arithmetic consequences of second order arithmetic.
Second order arithmetic IS CATEGORICAL, so as soon as you say
"first order arithmetic consequences of second order arithmetic",
you have said ALL that there is to say. That ALREADY IS ALL the
1st-order truths of the 1st-order language of arithmetic.
> In other words, what sort of theorems are provable in
> second order arithmetic but not the latter system.
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Author: kleptomaniac666_kleptomaniac666_ Date: May 11, 2008 15:58
On May 11, 10:17 pm, george cs.unc.edu> wrote:> On May 11, 3:57 pm, kleptomaniac6...@ hotmail.com wrote:
> >> I meant full second order arithmetic to be the theory PA+comprehension
>> +second order induction axiom.
>
> 2nd-order logic *already has* a definition, and it *is not* in terms
> of
> "comprehension". It is just in terms of 2nd-order quantifiers
> quantifying
> over "all possible" 1st-order subclasses of the obect-level domain.
>
>> I was wondering what you got with PA +comprehension+first order
>> arithmetic consequences of second order arithmetic.
>
> Second order arithmetic IS CATEGORICAL, so as soon as you say
> "first order arithmetic consequences of second order arithmetic",
> you have said ALL that there is to say. That ALREADY IS ALL the
> 1st-order truths of the 1st-order language of arithmetic.
> ...
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Author: georgegeorge Date: May 11, 2008 19:59
On May 11, 6:58 pm, kleptomaniac6...@ hotmail.com wrote: > I mean second order arithmetic with comprehension axioms. It is a
> system using first order logic.
Oh.
I apologize.
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Author: kleptomaniac666_kleptomaniac666_ Date: May 12, 2008 05:38
On May 12, 3:59 am, george cs.unc.edu> wrote:> On May 11, 6:58 pm, kleptomaniac6...@ hotmail.com wrote:
> >> I mean second order arithmetic with comprehension axioms. It is a
>> system using first order logic.
>
> Oh.
> I apologize.
Don't sweat it. Look, my basic question, although a kind of weird one,
was what kind of system T would you get if you wrote down the axioms
for full second order arithmetic Z2 (the system from reverse
mathematics), and then scribbled out the induction axiom, and added
"the first order arithmetic consequences of full second order
arithmetic"? If someone thought that Z2 was the correct foundation for
mathematics, and you proposed T instead, what complaints of the form
"but now you can't prove x" would they have?
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Author: georgegeorge Date: May 12, 2008 12:53
>> On May 11, 6:58 pm, kleptomaniac6...@ hotmail.com wrote: >>> I mean second order arithmetic with comprehension axioms.
>>> It is a system using first order logic.
Well, perhaps I finally see which system you are talking about.
Quoting from a review of Simpson:
> Simpson treats second-order arithmetic as a first-order axiomatic
> theory Z2, formulated in a (two-sorted) first-order language L2
> (an extension of the usual first-order language of arithmetic,
> obtained by adding atomic formulas of the form neX,
> where n is a number variable and X is a set variable).
> The axioms of Z2 are the usual first-order axioms
> of Peano Arithmetic plus,
> (i) Induction Axiom:
> (ii) Full 2nd Order Comprehension Scheme:
EXAn[neX<-->phi(n)]
Now that we know that that is the system we are talking
about, I hope I can parse your question.
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Author: Aatu KoskensiltaAatu Koskensilta Date: May 13, 2008 03:04
On 2008-05-11, in sci.logic, george wrote:
> I do not at all approve of the line of answers you have been getting
> to this so far.
Now there's a surprise.
> You can't just say "the induction schema form Peano Arithmetic".
Sure you can. It's perfectly clear what is meant.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Author: Aatu KoskensiltaAatu Koskensilta Date: May 13, 2008 03:12
On 2008-05-11, in sci.logic, kleptomaniac666_@ hotmail.com wrote: > Err, I meant T to be full second order arithmetic, with the second
> order induction axiom deleted, and the first order arithmetic
> consequences of full second order arithmetic added. Wouldn't con(PA)
> then be provable in the resulting system, being a consequence of full
> second order arithmetic?
Quite so. An actual obvious example would be "Pi-1-1 comprehension is
arithmetically sound".
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Author: Aatu KoskensiltaAatu Koskensilta Date: May 13, 2008 03:17
On 2008-05-11, in sci.logic, george wrote:
> On May 11, 3:57 pm, kleptomaniac6...@ hotmail.com wrote:
> >> I meant full second order arithmetic to be the theory PA+comprehension
>> +second order induction axiom.
>
> 2nd-order logic *already has* a definition, and it *is not* in terms
> of "comprehension".
It is apparent you simply don't know what you're talking about. Why,
then, go on about it, shouting random words in capitals with gay
abandon?
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Author: Aatu KoskensiltaAatu Koskensilta Date: May 13, 2008 03:20
On 2008-05-12, in sci.logic, george wrote:
> Unfortunately, I at this point remain confused about what anyone
> might mean by "full second order arithmetic".
Why? You just quoted a description. The qualifier "full" is included
here because in reverse mathematics various fragments, or subsystems,
of second-order arithmetic are considered.
--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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Author: Chris MenzelChris Menzel Date: May 13, 2008 11:17
On Tue, 13 May 2008 10:17:38 GMT, Aatu Koskensilta
said:> On 2008-05-11, in sci.logic, george wrote:
>> On May 11, 3:57 pm, kleptomaniac6...@ hotmail.com wrote:
>> >>> I meant full second order arithmetic to be the theory PA+comprehension
>>> +second order induction axiom.
>>
>> 2nd-order logic *already has* a definition, and it *is not* in terms
>> of "comprehension".
>
> It is apparent you simply don't know what you're talking about. Why,
> then, go on about it, shouting random words in capitals with gay
> abandon?
Curiously, he only used asterisks there. Perhaps his fingers are hoarse.
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