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Topic: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Replies: 39   Last Post: Oct 14, 2012 11:56 PM

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Graham Cooper

Posts: 4,348
Registered: 5/20/10
Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Posted: Oct 12, 2012 3:02 AM
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On Oct 12, 2:06 pm, George Greene <gree...@email.unc.edu> wrote:
> On Oct 11, 4:03 pm, Graham Cooper <grahamcoop...@gmail.com> wrote:
>

> > Now you're saying  Predicate Calculus  (with only infererence rules
> > over predicates which you wrongly call FOL)

>
> I'm certainly not THE ONLY one calling it that way, DUMBASS.
>
> http://en.wikipedia.org/wiki/First-order_logic


Hilbert-style systems and natural deduction

A deduction in a Hilbert-style deductive system is a list of formulas,
each of which is a logical axiom, a hypothesis that has been assumed
for the derivation at hand, or follows from previous formulas via a
rule of inference. The logical axioms consist of several axiom schemes
of logically valid formulas; these encompass a significant amount of
propositional logic. The rules of inference enable the manipulation of
quantifiers. Typical Hilbert-style systems have a small number of
rules of inference, along with several infinite schemes of logical
axioms. It is common to have only modus ponens and universal
generalization as rules of inference.

Natural deduction systems resemble Hilbert-style systems in that a
deduction is a finite list of formulas. However, natural deduction
systems have no logical axioms; they compensate by adding additional
rules of inference that can be used to manipulate the logical
connectives in formulas in the proof.

first-order logic is undecidable (although semidecidable), provided
that the language has at least one predicate of arity at least 2
(other than equality). This means that there is no decision procedure
that determines whether arbitrary formulas are logically valid. This
result was established independently by Alonzo Church and Alan Turing
in 1936 and 1937


******************

Also, this is what I was calling "AXIOMS" of predicate calculus the
other day.

http://en.wikipedia.org/wiki/First-order_logic#Provable_identities

Herc


Date Subject Author
10/5/12
Read If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/5/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Frederick Williams
10/7/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Charlie-Boo
10/5/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/5/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Frederick Williams
10/5/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/7/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/8/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/9/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/11/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/12/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/12/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/12/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
camgirls@hush.com
10/12/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Richard Tobin
10/12/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
camgirls@hush.com
10/13/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
george
10/13/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/14/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
george
10/13/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/13/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
george
10/13/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
george
10/13/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/14/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/14/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/14/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper
10/5/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Scott Berg
10/5/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Curt Welch
10/6/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Mike Terry
10/6/12
Read Re: If ZFC is a FORMAL THEORY ... then what is THEOREM 1 ?
Graham Cooper

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