Timeline for Platonic Truth and 1st Order Logic - Take 2
Current License: CC BY-SA 3.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 4, 2017 at 23:41 | comment | added | Joel David Hamkins | This last comment of mine is not quite right, if one is referring to $\Pi^0_1$ sentences in the language of set theory, in the Levy hierarchy. The reason is that every arithmetic assertion is $\Delta^0_1$ in set theory, since the arithmetic quantifiers are bounded by $\omega$. Thus, $T_1$ is already inconsistent, since we will add both the Rosser sentence and its negation. | |
| Dec 29, 2016 at 0:30 | vote | accept | Pace Nielsen | ||
| Dec 28, 2016 at 19:40 | comment | added | Joel David Hamkins | Ah, then my remark about $\omega$-model should be corrected. One thing that we can say is that if in the background theory we have ZFC relative to a truth predicate for first-order truth (and this is provable in Kelley-Morse set theory), so that we can refer to first-order truth, then your theory will again be a stratification of first-order truth, by essentially the same argument as in the PA case. | |
| Dec 28, 2016 at 19:34 | comment | added | Pace Nielsen | The Levy hierarchy. | |
| Dec 28, 2016 at 19:30 | comment | added | Joel David Hamkins | Oh, I may have misunderstood your proposal in the case of ZFC, since I had thought you were only adding arithmetic assertions, but $V=L$ is not arithmetic. Do you intend to add $\Pi^0_n$ statements now in the Levy hierarchy, rather than the arithmetic hiearchy? | |
| Dec 28, 2016 at 19:26 | comment | added | Pace Nielsen | Supposing that there is an $\omega$-model of ZFC, are there any statements known to belong to $T_2$, such as $V=L$ (which, as I understand it, is equivalent to a $\Pi_2^0$ statement in KP)? If any case, what would be a good resource for me to learn more? | |
| Dec 28, 2016 at 19:21 | comment | added | Joel David Hamkins | An essentially similar argument works with ZFC, if one has a strong enough meta theory. For example, if there is an $\omega$-model of ZFC, then you are again adding the true $\Pi^0_n$ sentences of arithmetic at stage $n$. But if ZFC is not arithmetically sound, for example, in a model of ZFC+$\neg$Con(ZFC), then things will go awry already at $T_2$, giving an inconsistent theory. | |
| Dec 28, 2016 at 18:59 | comment | added | Pace Nielsen | Excellent. What happens for ZFC if we try to do the same thing? | |
| Dec 28, 2016 at 18:07 | history | answered | Joel David Hamkins | CC BY-SA 3.0 |