> On 22 May 2019, at 10:09, 'Cosmin Visan' via Everything List > <[email protected]> wrote: > > Oh, now you say ? So what's the difference ?
Arithmetic meant here the model of the arithmetical theories. The model is, to be informal, everything true about the natural numbers and their definable and non definable relations, be them computable or not. The theories are tools to explore that reality, but after Gödel 1931, we know that any effective theory (effective = those theories where the proof are checkable in finite time) can only scratch the model. Arithmetic is “essentially undecidable”. You can build a theory and add as many axioms as you want, you still only scratch the surface of the truth. Not just because it is infinite (first order real analysis is much more infinite than the natural numbers) but is decidable!, but because it is irreducibly complex. Bruno > > On Tuesday, 21 May 2019 17:30:53 UTC+3, Bruno Marchal wrote: > Arithmetic (not to be confused with human theories about arithmetic) > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/550bcd11-9ad5-4f09-a537-15fb9b0f3c1f%40googlegroups.com > > <https://groups.google.com/d/msgid/everything-list/550bcd11-9ad5-4f09-a537-15fb9b0f3c1f%40googlegroups.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/AFD30D9E-FC89-48C1-9931-EDB3F423D855%40ulb.ac.be.
