[FOM] First Order Logic

[Charlie]

	  I reproduce below a (SNIPPED version of a) discussion that took place on FOM on Sept. 11th, 2000, between Matt Insall and Harvey Friedman.   I believe this old exchange sums up the matter succinctly and accurately:

Harvey:
So in a sound and complete finitary deductive calculus, semantic
compactness is equivalent to strong completeness.
<SNIP>
Yes, but the hypothesis of completeness is far too strong in the context of
SOL. The set of SOL validities is nowhere near even recursively enumerable.
<SNIP>
SOL is well known not to be semantically compact, and also any complete
deductive system for SOL must be grossly unreasonable.

Charlie Silver


On Sep 2, 2013, at 10:20 AM, Carl Hewitt <hewitt at concurrency.biz> wrote:

> Dear Martin,
>  
> The second-order axiomatization of the Peano Natural Numbers has been a great success since it has become the standard way for mathematicians to prove properties of the Natural Numbers.
>  
> In fact, the axiomatization is basically the only way known to prove that some property is true of the Natural Numbers.
>  
> However, it is computationally undecidable whether or not a sentence is a theorem of the second-order axiomatization of the Peano Natural Numbers.
>  
> Consequently, there is some sentence that is neither provable nor disprovable using the second-order axiomatization of the Peano Natural Numbers.
>  
> Should it be considered a defect of the second-order axiomatization of the Peano Natural Numbers that there is such a sentence?
>  
> Regards,
> Carl
>  
> From: MartDowd at aol.com
> Sent: August 31, 2013
> To: Foundations of Mathematics
> 
> [All] true statements [about the Natural Numbers] are not [provable using the second-order axiomatization of the Peano Natural Numbers].
>  
> - Martin Dowd
>  
>  
> In a message to Foundations of Mathematics on August 30, 2014, Carl Hewitt wrote:
> I am having trouble understanding why the proponents of first-order logic think that second-order systems are unusable.
> [Dedekind 1888] and [Peano 1889] thought they had achieved success because they had presented axioms for natural numbers and real numbers such that models of these axioms are unique up to isomorphism with a unique isomorphism. And later generations of mathematicians were happy to use these axioms.
>  
> The above axiomatizations bar many mathematical monsters that are created by the “first-order thesis” [Barwise 1985].  The articlehttp://arxiv.org/abs/0812.4852 presents a mathematical foundation for Computer Science that is an alternative to the first-order thesis.

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