On Tuesday, May 14, 2019 at 7:29:07 PM UTC-5, Brent wrote:
>
>
>
> On 5/14/2019 9:49 AM, Jason Resch wrote:
>
> What is truth? (Pontus Pilate). Arithmetical statements are true if they
>> are theorems derived from the axioms.
>>
>
> This is false. In every consistent system of axioms there are statements
> that are true but cannot be derived from the axioms.
>
>
> That's not true. There are axiomatic systems that are complete.
>
> In other words truth =/= proof, truth is always greater that what can be
> proved.
>
>
> That's because you have recourse to an idea of "true" that is outside of
> logical inference...such as "empirically true".
>
> Brent
>
An axiom system *A* being complete just means that for every (syntactically
well-formed) sentence *s* in the language of *A*, either *s* or ~*s* can be
proved via the rules of deduction of *A*.
BTW, while Church-Turing is not a useful "thesis", Curry-Howard is.
proofs = programs
(The informal word "model" does not come up in programming language theory
- PLT - that I can surmise, except in the context of a "model of
computation/programming" - lambda calculus, pi calculus, functional,
process-oriented. Its use in physics is a bit confusing. For example,
regarding the equation of the Standard Model as written out by mathematical
physicist Matilde Marcolli:
https://www.sciencealert.com/images/Screen_Shot_2016-08-03_at_3.20.12_pm.png
is the equation itself of the Standard Model here a "model", or is the
interpretation of this equation a "model"?)
@philipthrift
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