First, Godel’s results are much stronger than the claim that one has undecidable statements within the axiomatic system. But one also has that there’s no algorithm which can in general say whether a given statement is provable in PA (assuming that PA is in fact consistent). This is a much stronger and weirder claim.
This is also different from your group example because the natural numbers are a very concrete object. So our intuition strongly suggests that there should really be only one thing that actually looks like N in some sense. Some (although not all) senses of that are shown to be wrong by Godel’s theorems. Although what is doing more of the work here in some sense are the non-standard models of arithmetic which strictly speaking are distinct from Godel’s results.
Godel’s result about incompleteness COULD mean just that PA is inconsistent.
It would be an interesting way for the incompleteness theorem to be fulfilled. And become quite obsolete at the same time also. Except maybe as an early warning or sign that something is deeply wrong with PA.
Sure, that could be the case but that still is a much weirder situation than what Khoth is talking about. (Also, note that Godel’s theorems apply not just to PA but to the much weaker system of Robinson arithmetic so if there is an inconsistency it is probably happening at an even more fundamental level.)
I am skeptic about any infinity. I am not sure, if it is (always) paradoxical. But a theory which relates to an infinity related axiom is most likely too rich.
The concept of infinity and the concept an (infinite in any sense) god are both too ambitious for this finite world. But very persistent as we see.
There are two major differences:
First, Godel’s results are much stronger than the claim that one has undecidable statements within the axiomatic system. But one also has that there’s no algorithm which can in general say whether a given statement is provable in PA (assuming that PA is in fact consistent). This is a much stronger and weirder claim.
This is also different from your group example because the natural numbers are a very concrete object. So our intuition strongly suggests that there should really be only one thing that actually looks like N in some sense. Some (although not all) senses of that are shown to be wrong by Godel’s theorems. Although what is doing more of the work here in some sense are the non-standard models of arithmetic which strictly speaking are distinct from Godel’s results.
Couldn’t have said it better myself (vote up).
Godel’s result about incompleteness COULD mean just that PA is inconsistent.
It would be an interesting way for the incompleteness theorem to be fulfilled. And become quite obsolete at the same time also. Except maybe as an early warning or sign that something is deeply wrong with PA.
Sure, that could be the case but that still is a much weirder situation than what Khoth is talking about. (Also, note that Godel’s theorems apply not just to PA but to the much weaker system of Robinson arithmetic so if there is an inconsistency it is probably happening at an even more fundamental level.)
Robinson’s arithmetic is still an infinite one. That could be the root of all evils.
How do you feel about the infinite cyclic group?
I am skeptic about any infinity. I am not sure, if it is (always) paradoxical. But a theory which relates to an infinity related axiom is most likely too rich.
The concept of infinity and the concept an (infinite in any sense) god are both too ambitious for this finite world. But very persistent as we see.