Timeline for The true reason for the incompleteness of formal systems
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Nov 18, 2016 at 21:03 | comment | added | Andreas Blass | @Nullachtfünfzehn I had always assumed that Gödel intended "occur" to mean "are mentioned explicitly", but This was just an assumption and he might well have meant something more subtle. | |
| Nov 17, 2016 at 17:15 | comment | added | user99916 | I agree, and find it strange that Gödel asserted that this in fact causes incompleteness. Another question: Gödel said: "whereas in every formal system at most denumerably many types occur". How do you interpret this? Did he mean that every formal system has a countable model (in the sense proven by Löwenheim and Skolem)? | |
| Nov 16, 2016 at 16:28 | comment | added | Andreas Blass | @Nullachtfünfzehn I don't think our ability to strengthen $T$ to $T^+$ (or, more accurately, the existence of $T^+$, never mind our abilities) causes incompleteness of $T$. I'd rather interpret Gödel's remark epistemologically: The existence of $T^+$ helps us to understand what's going on with undecidable sentences (specifically with Gödel's "I am unprovable" example). These sentences express, in the language of $T$, some of the additional information that becomes available when we adopt stronger theories like $T^+$. | |
| Nov 16, 2016 at 15:20 | comment | added | user99916 | And in what sense does the fact that we can strengthen a theory $T$ to a higher type system $T^{+}$ (in which we for example can prove that $T$ is consistent) cause the incompleteness of $T$? I am just curious, since Gödel speaks about this as the "reason of the incompleteness of formal systems". | |
| Jul 9, 2015 at 17:08 | history | answered | Andreas Blass | CC BY-SA 3.0 |